PART II
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CONTENTS
PAGE
OPTICAL ILLUSIONS II-1
FREAKS OF FIGURES II-20
CHESS CAMEOS II-26
SCIENCE AT PLAY II-58
CURIOUS CALCULATIONS II-114
WORD AND LETTER PUZZLES II-147
SOLUTIONS II-167
OPTICAL ILLUSIONS
No. I.--SWALLOWED!
Take a small card and place it on its longer edge upon the dotted line. Now set the picture in a good light on the table, and let your head drop gradually towards the card until you almost touch it with your nose. You will see the bird fly into the jaws of the snake!
[Illustration]
AMUSING PROBLEMS
THE CARPENTER’S PUZZLE
1. A carpenter was called in to mend a hole in a wooden floor. The gap was two feet wide, and twelve feet long, while the only board at hand was three feet wide, and eight feet long.
+----------------+ | Fig. 1. | | | | | +----------------+
+------------------------+ | Fig. 2. | | | +------------------------+
This had been put aside as useless, but, on catching sight of it, the carpenter ran his rule over it and said that he could make a perfect fit, and cover all the hole by cutting the board into two pieces. How did he do this?
No. II.--AN ILLUSION OF ROTATION
This most interesting optical illusion was devised by Professor Thompson some years ago:--
[Illustration]
[Illustration]
If the illustration is moved by hand in a small circle on the level, with such motion as is given in rinsing out a bowl, the circles of the larger diagram will seem to revolve in the direction in which the paper is moved, while the cogs of the smaller diagram will apparently turn slowly in the opposite direction.
No. III.--WHIRLING WHEELS
Here is another combination of the clever illusion of the whirling wheels.
[Illustration]
If a rapid rotating motion is given to the diagram, each circle will seem to revolve, and the cog wheel in the centre will appear to move slowly round in the opposite direction.
GOLDEN PIPPINS
2. A man leaves an orchard of forty choice apple trees to his ten sons. On the first tree is one apple, on the second there are two, on the third three, and so on to the fortieth, on which there are forty.
Each son is to have four of the trees, and on them an equal number of the apples. How can they thus apportion the trees, and how many apples will each son have? Here is one way:--
1 2 3 4 5 6 7 8 9 10 20 19 18 17 16 15 14 13 12 11 21 22 23 24 25 26 27 28 29 30 40 39 38 37 36 35 34 33 32 31 ----------------------------- 82 82 82 82 82 82 82 82 82 82
Can you find another perfect solution?
No. IV.--AN ILLUSION OF MOTION
We call very particular attention to this fascinating illustration of the fact that the mind and eye may receive and register false impressions under quite simple conditions:--
[Illustration]
Hold this at rather more than reading distance, upright, and move it steadily up and down. The dark line will soon seem to slide up and down upon the perpendicular line. It will be better seen if drawn to pattern on a card.
No. V.--ARE THEY PARALLEL?
As the eye falls upon the principal lines of this interesting diagram, an immediate impression is formed that they are not parallel.
[Illustration]
This, however, is a most curious illusion, created in the mind entirely by the short sloping lines, as is found at once by the simple test of measurement.
AN AWKWARD FIX
3. With no knowledge of the surrounding district, I was making my way to a distant town through country roads, guided by the successive sign-posts that were provided.
Coming presently to four cross-roads I found to my dismay that some one had in mischief uprooted the sign-post and thrown it into the ditch. In this perplexing fix how could I find my way? A bright thought struck me. What was it?
LINKED SWEETNESS LONG DRAWN OUT
4. As I stood on the platform at a quiet country station, an engine, coming along from my left at thirty miles an hour, began to whistle when still a mile away from me. The shrill sound continued until the engine had passed a mile and a half to my right. For how long was I hearing its whistle?
No. VI.--ILLUSION OF LENGTH
In this curious optical illusion the lines are exactly equal in length.
\ / \ / >--------------------< / \ / \
/ \ / \ <--------------------> \ / \ /
The eye is misled by the effect which the lines drawn outward and inward at their ends produce upon the mind and sight.
ELASTIC QUARTERS
5.
In room marked A two men were placed, A third he lodged in B; The fourth to C was next assigned, The fifth was sent to D. In E the sixth was tucked away, F held the seventh man; For eighth and ninth were G and H, Then back to A he ran. Thence taking one, the tenth and last, He lodged him safe in I; Thus in nine rooms ten men found place, Now can you tell me why?
EXCLUDED DAYS
6. Are there any particular days of the week with which no new century can begin?
No. VII.--ILLUSION OF HEIGHT
These straight lines, at right angles to each other, are, though they do not seem to be, exactly equal in length.
[Illustration]
This and similar illusions are probably due to the variation of the vague mental standard which we unconsciously employ, and to the fact that the mind cannot form and adhere to a definite scale of measurement.
DRIVING POWER
7. Why has a spliced cricket bat such good driving power? and why is the “follow through” of the head of a golf club so telling in a driving stroke?
THE BUSY BOOKWORM
8. On my bookshelf in proper order stand two volumes. Each is two inches thick over all, and each cover is an eighth of an inch in thickness. How far would a bookworm have to bore in order to penetrate from the first page of Vol. I. to the last page of Vol. II.?
No. VIII.--ILLUSION OF DIRECTION
Can you decide at a glance which of the two lines below the thick band is a continuation of the line above it?
[Illustration]
Make up your mind quickly, and then test your decision with a straight edge.
A TERRIBLE TUMBLE
9. From what height must a man fall out of an airship--screaming as he goes overboard--so as to reach the earth before the sound of his cry?
_N.B._--Resistance of the air, and the acoustical fact that sound will not travel from a rare to a dense atmosphere, are to be disregarded.
IN A PREDICAMENT
10. Imagine a man on a perfectly smooth table surface of considerable size, in a vacuum, where there is no outside force to move him, and there is no friction. He may raise himself up and down, slide his feet about, double himself up, wave his arms, but his centre of gravity will be always vertically above the same point of the surface.
How could he escape from this predicament, if it was a possible one?
No. IX.--THINGS ARE NOT WHAT THEY SEEM
It is difficult, even after measurement, to believe that these figures are of the same size.
[Illustration]
But they will stand the test of measurement.
A CLIMBING MONKEY
11. A rope passes over a single fixed pulley. A monkey clings to one end of the rope, and on the other end hangs a weight exactly as heavy as the monkey. The monkey presently starts to climb up the rope. Will he succeed?
GAINING GROUND
12. Seeing that the tension on a pair of traces tends as much to pull the horse backward as it does to pull the carriage forward, why do the traces move on at all?
No. X.--THE SHIFTING BRICK
A very curious and interesting form of optical illusion is well illustrated by what may be called “the shifting brick.”
[Illustration]
The central brick, drawn to show all its edges, as though it were made of glass, will assume the form indicated by one or other of the smaller bricks at its right and left, according to the way in which the eyes accommodate themselves for the moment to one pattern or to the other. If you do not see this at first, look steadily for awhile at the pattern you desire.
ASK A CYCLIST
13. Why does a rubber tyre leave a double rut in dust, and a single one in mud?
TODHUNTER’S UNIQUE PUZZLE PROBLEM
14. If two cats, on opposite sides of a sharply sloping roof, are on the point of slipping off, which will hold on the longest?
No. XI.--AN ILLUSION WITH COINS
If you place four coins in the positions shown at the top of this diagram, and attempt, or challenge some one to attempt, without any measuring, to move the single coin down in a straight line until the spaces from C to D on either side exactly equal the distance from A to B--
[Illustration]
It must drop as far as is shown here, which seems to the unaided eye to be too far.
This excellent illusion can be shown as an after-dinner trick with four napkin-rings.
No. XII.--THE FICKLE BARREL
Here is another excellent optical illusion. Look attentively at the diagram below, and notice in which direction you apparently look into it, as though it were an open cask.
[Illustration]
Now shake the paper, or move it slightly, and you will find, more often than not, that you seem to see into it in quite the opposite direction.
HEADS I WIN!
15. I hold a penny level between my finger and thumb, and presently let it fall from the thumb by withdrawing my finger. It makes exactly a half-turn in falling through the first foot. If it starts “heads,” how far must it fall to bring it “heads” to the floor?
HE DID IT!
16. “They call these safety matches,” said Funnyboy at his club one day, “and say that they strike only on the box. Don’t believe it! I can strike them quite easily on my boot.”
No sooner said than done. He took out a match, struck it on his boot, and--phiz!--it was instantly alight. The box was handed round, and match after match was struck by the bystanders on their boots, but not one of them could succeed.
“You don’t give the magic touch,” said Funnyboy, as he gaily struck another. How did he do it?
No. XIII.--A STRANGE OPTICAL ILLUSION
[Illustration]
[Illustration]
How many cubes can you see as you look at the large diagram? The two smaller ones should be looked at first alternately, and they will assist the eye to see at one time six, and at another time seven, very distinct cubes.
No. XIV.--A CIRCULAR ILLUSION
This curious optical illusion is not easily followed by eye to the finish of the several lines.
[Illustration]
Each short line is, in fact, part of the circumference of a circle, and the circles when completed will be found to be accurately concentric. It would seem at first sight that the lines are taking courses which would eventually meet at some point common to them all.
A CYCLE SURPRISE
17. We commend this curious point to the special attention of cyclists:--
A bicycle is stationary, with one pedal at its lowest point. If this bicycle is lightly supported, and the bottom pedal is pulled backward, what will happen?
No. XV.--THE GHOST OF A COIN
[Illustration]
A most remarkable optical illusion is produced by the blending of the dark and light converging rays of this diagram. Stand with your back to the light, hold the page, or better still, the diagram copied on a card, by the lower right-hand corner, give it a continuous revolving movement in either direction, and the visible ghost of a silver coin, sometimes as large as sixpence, sometimes as large as a shilling, will appear! Where can it come from?
ROUGH AND READY
18. A merchant has a large pair of scales, but he has lost his weights, and cannot at the moment replace them. A neighbour sends him six rough stones, assuring him that with them he can weigh any number of pounds, from 1 to 364. What did each stone weigh?
No. XVI.--A TAME GOOSE
Here is a pretty form of our first illusion:--
[Illustration]
Place the edge of a card on the dotted line, look down upon it in a good light, and, as you drop your face till it almost touches the card, you will see the goose _move towards the sugar_ in the little maiden’s hand.
REJECTED ADDRESSES
19. A wheel is running along a level road, and a small clot of mud is thrown from the hindermost part of the rim. What happens to it? Does it ever renew its acquaintance with the wheel that has thus rejected it?
No. XVII.--ILLUSION OF LENGTH
Here is another method by which an optical illusion of length is very plainly shown:--
A B +--------------+ / \ / \ / \ / \ +------------------------+
A B +--------------+ \ / \ / \ / \ / +----+
Judged by appearances, the line A B in the larger figure is considerably longer than the line A B below it, but tested by measurement they are exactly equal.
THE CARELESS CARPENTER
20. A village carpenter undertook to make a cupboard door. When he began to put it in its place it was too big, so he took it back to his workshop to alter it. Unfortunately he now cut it too little. What could he do? He determined to cut it again, and it at once became a good fit. How was this done?
No. XVIII.--PERPENDICULAR LINES
Here is another excellent illustration that seeing is not always believing.
[Illustration]
No one could suppose at first sight that these four lines are perfectly straight and parallel, but they will stand the test of a straight edge. The divergent rays distract the vision.
BY THE COMPASS
21. If from the North Pole you start sailing in a south-westerly direction, and keep a straight course for twenty miles, to what point of the compass must you steer to get back as quickly as possible to the Pole?
No. XIX.--ILLUSION OF PERSPECTIVE
The optical illusion in the picture which we reproduce is due to the defective drawing of the two men on the platform. In actual size upon the paper the further man looks much taller than the other.
[Illustration]
Measurement, however, shows the figures to be exactly of a height. This illusion is due to the fact that the head of the further man is quite out of perspective. If he is about as tall as the other, and on level ground, both heads should be about on the same line. As drawn, he is, in fact, a monster more than eight feet high.
DICK IN A SWING
22. If Dick, who is five feet in height, stands bolt-upright in a swing, the ropes of which are twenty feet long, how much further in round numbers do his feet travel than his head in describing a semi-circle?
No. XX.--OUR BLIND SPOT
Here is an excellent and very simple illustration of a well-known optical curiosity:--
[Illustration]
Hold this picture at arm’s length in the right hand, hold the left hand over the left eye, and draw the picture towards you gradually, looking always at the black cross with the right eye. The black disc will presently disappear, and then come into sight again as you continue to advance the paper.
A POSER
23. Can you name nine countries in Europe of which the initial letters are the same as the finals?
FREAKS OF FIGURES
A HANDY SHORT CUT
Here is a delightfully simple way in which market gardeners, or others who buy or sell weighty produce, can check their invoices for potatoes or what not.
Say, for example, that a consignment weighs 6 tons, 10 cwts., 1 qr. Then, since 20 cwts. are to a ton as 20s. are to a pound, and each quarter would answer on these lines to 3d., we can at once write down £6 10s. 3d., as the price at £1 the ton. On this sure basis any further calculation is easily made.
No. XXI.--THE PERSISTENCE OF VISION
[Illustration: HOW TO SEE THE GHOST]
Look steadily, in a good light, for thirty seconds at the cross in the eye of the pictured skull; then look up at the wall or ceiling, or look fixedly at a sheet of paper for another thirty seconds, when a ghost-like image of the skull will be developed.
NOT SO FAST!
A gardener, when he had planted 100 trees on a line at intervals of 10 yards, was able to walk from the first of these to the last in a few seconds, for they were set _on the circumference of a circle_!
No. XXII.--THE PERSISTENCE OF VISION
Here is another example of what is known as the persistence of vision:--
[Illustration]
Look fixedly for some little time at this grotesque figure, then turn your eyes to the wall or ceiling, and you will in a few seconds see it appear in dark form upon a light ground.
A PUZZLE NUMBER
The sum of nine figures a number will make, Of which just a third will remain If fifty away from the whole you should take, Thus turning a loss to a gain.
It needs something more than mere arithmetic to discover that the solution to this puzzle is XLV, the sum of the nine digits, for if the L is removed, XV, the third of XLV, remains.
No. XXIII.--ANOTHER PARALLEL FREAK
Here is another curious illusion:--
[Illustration]
The four straight lines are perfectly parallel, but the contradictory herring-bones disturb the eye.
THE MAGIC OF FIGURES
If our penny had been current coin in the first year of the Christian era, and had been invested at compound interest at five per cent., it would have amounted in 1905 to more than £132,010,000,000,000,000,000,000,000,000,000,000,000.
This gigantic sum would afford an income of £101,890,000,000,000,000,000 every second to every man, woman, and child in the world, if we take its population to be 1,483,000,000 souls!
Absurdly small in contrast to these startling figures is the modest eight shillings which the same penny would have yielded in the same time at simple interest.
No. XXIV.--ILLUSION OF LENGTH
Here in another form is shown the illusion of length.
[Illustration]
At first sight it seems that the two upright lines are distinctly longer than the line that slopes, but it is not so.
WHAT IS YOUR AGE?
Here is a neat method of discovering the age of a person older than yourself:--
Subtract your own age from 99. Ask your friend to add this remainder to his age, and then to remove the first figure and add it to the last, telling you the result. This will always be the difference of your ages. Thus, if you are 22, and he is 35, 99 - 22 = 77. Then 35 + 77 = 112. The next process turns this into 13, which, added to your age, gives his age, 35.
No. XXV.--THE HONEYCOMB ILLUSION
In this diagram one hundred and twenty-one circular spots are grouped in a diamond.
[Illustration]
If we half close our eyes, and look at this through our eye-lashes, we find that it takes on the appearance of a section of honeycomb, with hexagonal cells.
MULTIPLICATION NO VEXATION
Here is a ready method for multiplying together any two numbers between 12 and 20.
Take one of the two numbers and add it to the unit digit of the other. Beneath the sum thus obtained, but one place to the right, put the product of the unit digits of the two original numbers.
The sum of these new numbers is the product of the numbers that were chosen. Thus:--
19 × 13. (19 + 3) = 22 (9 × 3) = 27 --- 247
No. XXVI.--CHESS CAMEO
By Dr GOLD
_A Chess Joke_
BLACK +---+---+---+---+---+---+---+---+ | |.b.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| p |...| k |...| |.K.| | +---+---+---+---+---+---+---+---+ | |.n.| |.P.| |...| |...| +---+---+---+---+---+---+---+---+ |.P.| |...| p |...| |...| | +---+---+---+---+---+---+---+---+ | |...| N |...| N |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| B |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
Black has made an illegal move. He must replace this, and move his king as the penalty. White then mates on the move.
[1]SHOT IN A PYRAMID
Here is a method for determining the total number of balls in a solid pyramid built up on a square base:--
Multiply the number of shot on one side of the base line by 2, add 3, multiply by the number on the base line, add 1, multiply again by the number on the base, and finally divide by 6. Thus, if the base line is 12--
12 × 2 + 3 = 27; 27 × 12 + 1 = 325; 325 × 12 = 3900; and 3900 ÷ 6 = 650, which is the required number.
[1] _N.B._--This title does not imply a tragedy.
No. XXVII.--CHESS CAMEO
By A. CYRIL PEARSON
_A Chess Puzzle_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |.p.| |...| |.p.| p | +---+---+---+---+---+---+---+---+ | |.P.| |.P.| |.k.| |...| +---+---+---+---+---+---+---+---+ |...| |.p.| K |.N.| P |...| | +---+---+---+---+---+---+---+---+ | |...| |...| p |...| R |...| +---+---+---+---+---+---+---+---+ |...| |...| |.N.| |...| | +---+---+---+---+---+---+---+---+ | |.q.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White might have given mate on the last move. White now to retract his move, and mate at once.
_Show by analysis the mating position._
No. XXVIII.--CHESS CAMEO
By J. G. CAMPBELL
_A very Clever Device_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| p |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.P.| |.p.| |...| |...| +---+---+---+---+---+---+---+---+ |...| K |...| p |...| |.B.| | +---+---+---+---+---+---+---+---+ | P |...| |.P.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.p.| p | +---+---+---+---+---+---+---+---+ | |.P.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| K |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play and draw.
COMIC ARITHMETIC
“Now, boys,” said Dr Bulbous Roots to class, “you shall have a half-holiday if you prove in a novel way that 10 is an even number.”
Next morning, when the doctor came into school, he found this on the blackboard:--
SIX = 6 IX = 9 --------- By subtraction S = -3
SEVEN = 7 S = -3 ------------ Therefore EVEN = 10
Q. E. D. (_Quite easily done!_)
The half-holiday was won.
No. XXIX.--CHESS CAMEO
By E. B. COOK
_A Fine Example_
BLACK +---+---+---+---+---+---+---+---+ | |.k.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.R.| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | K |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| p |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and draw.
No. XXX.--CHESS CAMEO
By A. F. MACKENZIE
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |.r.| |...| |.b.| +---+---+---+---+---+---+---+---+ |.p.| |...| |...| |.P.| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |.q.| +---+---+---+---+---+---+---+---+ |.N.| b |...| |...| |...| K | +---+---+---+---+---+---+---+---+ | p |.P.| |.k.| P |...| R |...| +---+---+---+---+---+---+---+---+ |...| |.p.| P |.p.| |.B.| R | +---+---+---+---+---+---+---+---+ | |...| n |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.N.| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
There are twelve variations in this beautiful problem.
MAGIC MULTIPLICATION
It will interest all who study short cuts and contrivances to know that a novice at arithmetic who has mastered simple addition, and can multiply or divide by 2, but by no higher numbers, can, by using all these methods, multiply any two numbers together easily and accurately.
This is how it is done:--
Write down the numbers, say 53 and 21, divide one of them by 2 as often as possible, omitting remainders, and multiply the other by 2 the same number of times; set these down side by side, as in the instance given below, and wherever there is an _even_ number on the division side, strike out the corresponding number on the multiplication side. Add up what remains on that side, and the sum is done. Thus:--
53 21 26 (42) 13 84 6 (168) 3 336 1 672 ------ 1113
which is 53 multiplied by 21.
No. XXXI.--CHESS CAMEO
BY A. W. GALITZKY
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.B.| q |.n.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| p | +---+---+---+---+---+---+---+---+ | |...| N |...| R |...| P |.P.| +---+---+---+---+---+---+---+---+ |...| |...| |...| k |...| | +---+---+---+---+---+---+---+---+ | |...| b |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| n |...| |...| |.K.| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
No. XXXII.--CHESS CAMEO
BY S. LOYD
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.N.| |.p.| |.p.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.p.| |...| | +---+---+---+---+---+---+---+---+ | |...| |.k.| N |.q.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.n.| |...| | +---+---+---+---+---+---+---+---+ | |.K.| |...| |.B.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
No. XXXIII.--CHESS CAMEO
BY B. G. LAWS
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.q.| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| n |.p.| |.p.| p |...| k | +---+---+---+---+---+---+---+---+ | N |...| P |...| k |.b.| |.R.| +---+---+---+---+---+---+---+---+ |...| n |...| |.B.| |...| R | +---+---+---+---+---+---+---+---+ | |...| |.r.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
PERFECT NUMBERS
The following particulars about a very rare property of numbers will be new and interesting to many of our readers:--
The number 6 can only be divided without remainder by 1, 2, and 3, excluding 6 itself. The sum of 1 + 2 + 3 is 6. The only exact divisors of 28 are 1, 2, 4, 7, and 14, and the sum of these is 28; 6 and 28 are therefore known as perfect numbers.
The only other known numbers which fulfil these conditions are 496; 8128; 33,550,336; 8,589,869,056; 137,438,691,328; and
2,305,843,008,139,952,128.
This most remarkable rarity of perfect numbers is a symbol of their perfection.
No. XXXIV.--CHESS CAMEO
BY EMIL HOFFMANN
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| q |...| |.B.| +---+---+---+---+---+---+---+---+ |.R.| |...| |...| |.Q.| n | +---+---+---+---+---+---+---+---+ | |...| |...| k |...| |...| +---+---+---+---+---+---+---+---+ |.b.| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |.R.| |...| |...| +---+---+---+---+---+---+---+---+ |.p.| |...| B |...| n |...| b | +---+---+---+---+---+---+---+---+ | K |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves. There are no less than twelve variations!
No. XXXV.--CHESS CAMEO
BY J. POSPICIL
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |.N.| |...| +---+---+---+---+---+---+---+---+ |...| N |...| |...| |...| p | +---+---+---+---+---+---+---+---+ | |...| n |...| B |.p.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.k.| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.K.| |...| |...| P |...| | +---+---+---+---+---+---+---+---+ | |...| |...| P |.n.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.Q.| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
AMICABLE NUMBERS
Somewhat akin to perfect numbers are what are known as amicable numbers, of which there is a still smaller quantity in the realm of numbers.
The number 220 can be divided without remainder only by 1, 2, 4, 5, 10, 11, 22, 44, 55, and 110, and the sum of these divisors is 284. The only divisors of 284 are 1, 2, 4, 71, and 142, and the sum of these is 220.
The only other pairs of numbers which fulfil this curious mutual condition, that the sum of the divisors of each number exactly equals the other number, are 17,296 with 18,416, and 9,363,584 with 9,437,056. No other numbers, at least below ten millions, are in this way “amicable.”
No. XXXVI.--CHESS CAMEO
BY H. J. C. ANDREWS
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| K |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.Q.| | +---+---+---+---+---+---+---+---+ | |.B.| |...| |...| |.R.| +---+---+---+---+---+---+---+---+ |...| |...| k |.N.| |.b.| R | +---+---+---+---+---+---+---+---+ | P |...| |...| |...| P |...| +---+---+---+---+---+---+---+---+ |...| k |...| p |...| |...| | +---+---+---+---+---+---+---+---+ | B |...| |.P.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in two moves.
No. XXXVII.--CHESS CAMEO
BY ALFRED DE MUSSET
_A Gem of the First Water_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| n | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| N |...| |.k.| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |.N.| |...| |.K.| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.R.| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
A SWARM OF ONES
1 × 9 + 2 = 11 12 × 9 + 3 = 111 123 × 9 + 4 = 1111 1234 × 9 + 5 = 11111 12345 × 9 + 6 = 111111 123456 × 9 + 7 = 1111111 1234567 × 9 + 8 = 11111111 12345678 × 9 + 9 = 111111111
No. XXXVIII.--CHESS CAMEO
BY FRANK HEALEY
_A Masterpiece_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.B.| N |...| |...| |...| +---+---+---+---+---+---+---+---+ |.P.| |...| |.p.| |.p.| | +---+---+---+---+---+---+---+---+ | |...| K |...| k |.b.| P |...| +---+---+---+---+---+---+---+---+ |.Q.| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.n.| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
DIVINATION BY FIGURES
There is a pleasant touch of mystery in the following method of discovering a person’s age:--Ask any such subjects of your curiosity to write down the tens digit of the year of their birth, to multiply this by 5, to add 2 to the product, to multiply this result by 2, and finally to add the units digit of their birth year. Then, taking the paper from them, subtract the sum from 100. This will give you their age in 1896, from which their present age is easily determined.
No. XXXIX.--CHESS CAMEO
BY FRANK HEALEY
_The “Bristol Prize Problem”_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| n |...| |...| N |.p.| | +---+---+---+---+---+---+---+---+ | |.N.| |...| |...| Q |...| +---+---+---+---+---+---+---+---+ |...| b |.k.| P |...| |...| | +---+---+---+---+---+---+---+---+ | p |...| p |...| |.p.| |...| +---+---+---+---+---+---+---+---+ |.P.| |.P.| |...| R |...| | +---+---+---+---+---+---+---+---+ | |...| |.P.| |...| P |.K.| +---+---+---+---+---+---+---+---+ |.B.| |...| R |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
FUR AND FEATHERS
As I came in after a day among the birds and rabbits, the keeper asked me--“Well, sir, what sport?” I replied, “36 heads and 100 feet.” It took him some time to calculate that I had accounted for 22 birds and 14 rabbits.
No. XL.--CHESS CAMEO
BY J. E. CAMPBELL
_Splendid Strategy_
BLACK +---+---+---+---+---+---+---+---+ | |...| |.N.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| p |.p.| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.B.| |.k.| p |.P.| |.n.| Q | +---+---+---+---+---+---+---+---+ | |...| p |.N.| |.P.| |...| +---+---+---+---+---+---+---+---+ |...| |...| B |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| K |.R.| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
JUGGLING WITH THE DIGITS
The nine digits can be arranged to form fractions equivalent to
1 1 1 1 1 1 1 - - - - - - - 3 4 5 6 7 8 9
thus:--
5823 1 7956 1 2973 1 2943 1 5274 1 ----- = - ----- = - ----- = - ----- = - ----- = - 17469 3 31824 4 14865 5 17658 6 36918 7
9321 1 8361 1 ----- = - ----- = - 74568 8 75249 9
No. XLI.--CHESS CAMEO
BY W. GRIMSHAW
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.N.| |...| K |...| |...| | +---+---+---+---+---+---+---+---+ | |.b.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| k |...| |...| | +---+---+---+---+---+---+---+---+ | |.P.| p |.p.| p |.P.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.P.| P | +---+---+---+---+---+---+---+---+ | |.Q.| |...| |...| R |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
No. XLII.--CHESS CAMEO
BY S. LOYD
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| P |...| +---+---+---+---+---+---+---+---+ |...| |.K.| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |.p.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.p.| | +---+---+---+---+---+---+---+---+ | |.Q.| |...| |.N.| k |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
A MOTOR PROBLEM
This motor problem will be new and amusing to many readers:--
Let _m_ be the driver of a motor-car, working with velocity _v_. If a sufficiently high value is given to _v_, it will ultimately reach _pc_. In most cases _v_ will then = _o_. For low values of _v_, _pc_ may be neglected; but if _v_ be large it will generally be necessary to square _pc_, after which _v_ will again assume a positive value.
By a well-known elementary theorem, _pc_ + _lsd_ = (_pc_)², but the squaring may sometimes be effected by substituting _x_³ (or × × ×) for _lsd_. This is preferable, if _lsd_ is small with regard to _m_. If _lsd_ be made sufficiently large, _pc_ will vanish.
Now if _jp_ be substituted for _pc_ (which may happen if the difference between _m_ and _pc_ be large) the solution of the problem is more difficult. No value of _lsd_ can be found to effect the squaring of _jp_, for, as is well-known, (_jp_)² is an impossible quantity.
No. XLIII.--CHESS CAMEO
BY J. G. CAMPBELL
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| p |...| Q | +---+---+---+---+---+---+---+---+ | |...| |.k.| N |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.N.| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| P |...| B |...| +---+---+---+---+---+---+---+---+ |...| |...| |.P.| p |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |.K.| |.B.| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
No. XLIV.--CHESS CAMEO
BY FRANK HEALEY
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |.K.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |.R.| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |.P.| k |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |.Q.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
A NEAT METHOD OF DIVISION
To divide any sum easily by 99, cut off the two right-hand figures of the dividend and add them to all the others. Set down the result of this in line below, and then repeat this process until no figures remain on the left to be thus dealt with.
Now draw a line down between the tens and hundreds columns, and add all up on the left of it, thus:--
8694 | 32 120 | 78 87 | 26 1 | 98 1 | 13 | 99 | 14 ------------ ------------- 121 and 99 over. 8782 and 14 over. In other words, 122.
The last number on the right of the lines shows always the remainder. If this should appear as 99 (as in the second example above), add one to the number on the left.
No. XLV.--CHESS CAMEO
BY BLUMENTHAL AND KUND
BLACK +---+---+---+---+---+---+---+---+ | |...| K |...| |.R.| |...| +---+---+---+---+---+---+---+---+ |...| |...| B |...| |...| | +---+---+---+---+---+---+---+---+ | p |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.r.| n |...| |...| |...| | +---+---+---+---+---+---+---+---+ | k |...| p |...| |...| |.Q.| +---+---+---+---+---+---+---+---+ |...| |...| N |...| |...| | +---+---+---+---+---+---+---+---+ | b |.P.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
No. XLVI.--CHESS CAMEO
BY A. F. MACKENZIE
_A Prize Problem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |.R.| |...| +---+---+---+---+---+---+---+---+ |...| |.p.| |.B.| |...| Q | +---+---+---+---+---+---+---+---+ | n |...| |.R.| |...| |.b.| +---+---+---+---+---+---+---+---+ |.p.| |.k.| P |...| |...| | +---+---+---+---+---+---+---+---+ | K |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.n.| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
A SMART SCHOOLBOY
The question, “How many times can 19 be subtracted from a million?” was set by an examiner, who no doubt expected that the answer would be obtained by dividing a million by 19. One bright youth, however, filled a neatly-written page with repetitions of
1,000,000 1,000,000 1,000,000 19 19 19 --------- --------- --------- 999,981 999,981 999,981
and added at the foot of the page, “_N.B._--I can do this as often as you like.”
There was a touch of unintended humour in this, for, after all, the boy gave a correct answer to a badly worded question.
No. XLVII.--CHESS CAMEO
BY A. CYRIL PEARSON
BLACK +---+---+---+---+---+---+---+---+ | |.N.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| r |.n.| |...| n |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |.R.| |...| +---+---+---+---+---+---+---+---+ |...| |.P.| k |...| |...| | +---+---+---+---+---+---+---+---+ | |...| p |.P.| |...| |...| +---+---+---+---+---+---+---+---+ |.Q.| |.P.| |.K.| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| P |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
No. XLVIII.--CHESS CAMEO
BY FRANK HEALEY
_Quite a Gem_
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |.b.| +---+---+---+---+---+---+---+---+ |...| |...| n |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.K.| |...| |...| |.Q.| | +---+---+---+---+---+---+---+---+ | |.P.| |.k.| b |...| P |...| +---+---+---+---+---+---+---+---+ |.R.| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | B |...| |...| |.P.| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
LEWIS CARROLL’S SHORT CUT
Here is a very smart and very simple method of dividing any multiple of 9 by 9, from the fertile brain of Lewis Carroll:--Place a cypher over the final figure, subtract the final figure from this, place the result above in the tens place, subtract the original tens figure from this, and so on to the end. Then the top line, excluding the intruded cypher, gives the result desired. Thus:--
36459 ÷ 9 = 4051,0 = 4051. 36459
No. XLIX.--CHESS CAMEO
BY A. BAYERSDORFER
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| K |...| | +---+---+---+---+---+---+---+---+ | |...| |.k.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |.p.| B |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| Q |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
ANOTHER FREAK OF FIGURES
1 × 8 + 1 = 9 12 × 8 + 2 = 98 123 × 8 + 3 = 987 1234 × 8 + 4 = 9876 12345 × 8 + 5 = 98765 123456 × 8 + 6 = 987654 1234567 × 8 + 7 = 9876543 12345678 × 8 + 8 = 98765432 123456789 × 8 + 9 = 987654321
No. L.--CHESS CAMEO
BY J. DOBRUSKY
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.n.| |...| |...| B |...| | +---+---+---+---+---+---+---+---+ | |...| |...| q |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| k |...| |...| | +---+---+---+---+---+---+---+---+ | |.R.| N |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| K |.R.| |...| |...| | +---+---+---+---+---+---+---+---+ | Q |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.n.| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
DIVINATION BY NUMBERS
Here is one of the methods by which we can readily discover a number that is thought of. The thought-reader gives these directions to his subject: “Add 1 to three times the number you have thought of; multiply the sum by 3; add to this the number thought of; subtract 3, and tell me the remainder.” This is always ten times the number thought of. Thus, if 6 is thought of--6 × 3 + 1 = 19; 19 × 3 = 57; 57 + 6 - 3 = 60, and 60 ÷ 10 = 6.
No. LI.--CHESS CAMEO
BY KONRAD BAYER
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | p |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.k.| |.K.| |...| |...| | +---+---+---+---+---+---+---+---+ | b |.p.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.P.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.R.| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
COINCIDENCES
Here is a curious rough rule for remembering distances and sizes:--
The diameter of the earth multiplied by 108 gives approximately the sun’s diameter. The diameter of the sun multiplied by 108 gives the mean distance of the earth from the sun. The diameter of the moon multiplied by 108 gives the mean distance of the moon from the earth.
No. LII.--CHESS CAMEO
BY J. BERGER
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| B |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| p |...| | +---+---+---+---+---+---+---+---+ | |...| |...| k |...| |...| +---+---+---+---+---+---+---+---+ |.B.| Q |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.p.| |.K.| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
PERSONAL ARITHMETIC
Says Giles, “My wife and I are two, Yet faith I know not why, sir.” Quoth Jack, “You’re ten, if I speak true, She’s one, and you’re a cypher!”
No. LIII.--CHESS CAMEO
BY H. F. L. MEYER
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| R |...| |...| +---+---+---+---+---+---+---+---+ |...| k |...| |...| |...| | +---+---+---+---+---+---+---+---+ | b |.p.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| p |...| P |...| |...| | +---+---+---+---+---+---+---+---+ | |.K.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |.B.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| Q |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
DIVISION BY SUBTRACTION
Here is a curious and quite uncommon method of dividing any multiple of 11 by 11.
Set down the multiple of 11, place a cypher under its last figure, draw a line, and subtract, placing the first remainder under the tens place. Subtract this from the next number in order, and so on throughout, adding in always any number that is carried. Thus:--
363 56408 375034 0 0 0 --- ----- ------ 33 5128 34094
No. LIV.--CHESS CAMEO
BY FRANK HEALEY
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.B.| |...| |...| |...| p | +---+---+---+---+---+---+---+---+ | |.r.| p |.p.| |...| b |...| +---+---+---+---+---+---+---+---+ |...| |.k.| n |...| |.Q.| | +---+---+---+---+---+---+---+---+ | P |...| |...| p |...| |...| +---+---+---+---+---+---+---+---+ |.K.| |...| |.P.| |...| N | +---+---+---+---+---+---+---+---+ | B |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
LUCK IN ODD NUMBERS
Perhaps the old saying, “there is luck in odd numbers,” may have some connection with the curious fact that the sum of any quantity of consecutive odd numbers, beginning always with 1, is the square of that number. Thus:--
1 + 3 + 5 = 9 = 3 × 3. 1 + 3 + 5, etc., up to 17 = 81 = 9 × 9. 1 + 3 + 5, etc., up to 99 = 2500 = 50 × 50.
No. LV.--CHESS CAMEO
BY FRANK HEALEY
BLACK +---+---+---+---+---+---+---+---+ | |.K.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |.n.| |...| |.p.| |...| | +---+---+---+---+---+---+---+---+ | |.N.| |.k.| |...| |...| +---+---+---+---+---+---+---+---+ |...| |.N.| |...| |...| | +---+---+---+---+---+---+---+---+ | |.P.| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |.n.| Q |.P.| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
THE VERSATILE NUMBER
In the number 142857, if the digits which belong to it are in succession transposed from the first place to the end, the result is in each case a multiple of the original number. Thus:--
285714 = 142857 × 2 428571 = 142857 × 3 571428 = 142857 × 4 714285 = 142857 × 5 857142 = 142857 × 6
No. LVI.--CHESS CAMEO
BY J. E. CAMPBELL
BLACK +---+---+---+---+---+---+---+---+ | |...| |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| |...| |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |.p.| |...| +---+---+---+---+---+---+---+---+ |...| |...| p |...| p |...| B | +---+---+---+---+---+---+---+---+ | |...| |...| |.k.| |...| +---+---+---+---+---+---+---+---+ |...| |...| K |...| N |...| | +---+---+---+---+---+---+---+---+ | |...| |...| |...| |.P.| +---+---+---+---+---+---+---+---+ |...| |...| |...| |.R.| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in three moves.
A PARADOX
By the following simple method, a plausible attempt is made to prove that 1 is equal to 2:--
Suppose that _a_ = _b_, then
_ab_ = _a_²
∴ _ab_ - _b_² = _a_² - _b_²
∴ _b_(_a_ - _b_) = (_a_ + _b_)(_a_ - _b_)
∴ _b_ = _a_ + _b_
∴ _b_ = 2_b_
∴ 1 = 2
This process only proves in reality that 0 × 1 = 0 × 2, which is true.
No. LVII.--CHESS CAMEO
_Double First Prize_
BY A. CYRIL PEARSON
BLACK +---+---+---+---+---+---+---+---+ | |...| b |...| |...| |...| +---+---+---+---+---+---+---+---+ |...| |...| p |.K.| |...| p | +---+---+---+---+---+---+---+---+ | |...| p |...| |.p.| |...| +---+---+---+---+---+---+---+---+ |...| |...| B |.k.| |...| | +---+---+---+---+---+---+---+---+ | |.b.| |...| |...| P |...| +---+---+---+---+---+---+---+---+ |.N.| |.N.| |.P.| |...| | +---+---+---+---+---+---+---+---+ | |.R.| |.P.| |.R.| |...| +---+---+---+---+---+---+---+---+ |.b.| q |...| |...| |...| | +---+---+---+---+---+---+---+---+ WHITE
White to play, and mate in four moves.
QUICK CALCULATION
Few people know a very singular but simple method of calculating rapidly how much any given number of pence a day amounts to in a year. The rule is this:--Set down the given number of pence as pounds; under this place its half, and under that the result of the number of original pence multiplied always by five. Take, for example, 7d a day:--
£7 0 0 3 10 0 2 11 --------- £10 12 11 ---------
The reason for this is evident as soon as we remember that the 365 days of a year may be split up into 240, 120, and 5, and that 240 happens to be the number of pence in a pound.
SCIENCE AT PLAY
No. LVIII.--THE GEARED WHEELS
A small wheel with ten teeth is geared into a large fixed wheel which has forty teeth. This small wheel, with an arrow mark on its highest cog, is revolved completely round the large wheel. How often during its course is the arrow pointing directly upwards? Here is a diagram of the starting position.
[Illustration]
No. LIX.--ADVANCING BACKWARDS
Here is a most curious and interesting question:--When an engine is drawing a train at full speed from York to London, what part of the train at any given moment is moving _towards York_?
At any time, when the engine is drawing a train at full speed from York to London, that part of the flange of each wheel which is for the moment at its lowest is actually _moving backwards towards York_.
[Illustration]
For any point, such as A, on the circumference of the tyre, describes in running along a series of curves, as shown by _full_ lines in the diagram; and any point, B, on the outer edge of the flange, follows a path shown by the _dotted_ curves.
If these lines are followed round with a pencil in the direction of the arrows, it will be found that the point on the flange actually moves _backwards_ as it passes _below the track_, while the point A, as it completes each curve, is _at rest_ for the instant on the track, just before it starts afresh. The speed of the train does not affect these very curious facts.
No. LX.--THE FIFTEEN BRIDGES
In the subjoined diagram A and B represent two islands, round which a river runs as is indicated, with fifteen connecting bridges, that lead from the islands to the river’s banks.
[Illustration]
Can you contrive to pass in turn over all these bridges without ever passing over the same one twice?
ARRANGING THE DIGITS
In a school where two boys were taught to think out the bearings of their work, a sharp pupil remarked that 100 is represented on paper by the smallest digit and two cyphers, which are in themselves symbols of nothing. The master, quick to catch any signs of mental activity, took the opportunity to propound to his class the following ingenious puzzle:--How can the sum of 100 be represented exactly in figures and signs by making use of all the nine digits in their reverse order? This is how it is done:--
9 × 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 100.
Another ingenious method of using the nine digits, so that by simple addition they sum up to exactly 100, and each is used once only, is this:--
15 + 36 + 47 = 98 + 2 = 100
Here is another arrangement by which the nine digits written in their inverse order can be made to represent exactly 100:--
98 - 76 + 54 + 3 + 21 = 100.
Here is yet another way of arriving at 100 by using each of the digits, this time with an 0:--
40¹⁄₂ 59³⁸⁄₇₆ -------- 100
No. LXI.--LOOPING THE LOOP
Here is quite a pretty scientific experiment, which any one of a handy turn can construct and arrange:--
[Illustration]
The spiral track is formed of two wires bent, and connected by curved cross-pieces. The upper twist is turned so that the ball starts on a horizontal course.
During the accelerated descent the ball acquires momentum enough to keep it on the vertical track, held outwardly against the wires by centrifugal force.
Convenient proportions are: height of spiral two feet, diameter six inches, and wire rails three-quarters of an inch apart.
No. LXII.--A MECHANICAL BIRD
A close approach to an ideal flying machine can be made with a little ingenuity. Two Y-shaped standards, secured to the backbone rod, support two wires which carry wings of thin silk, provided with light stays, and connected at their inner corners with the backbone by threads.
[Illustration]
Rubber bands are attached to a loop on the inner end of the crank shaft, and secured to a post at the rear. These are twisted by turning the shaft with the cross wire, and when the tension is released the wings beat the air and carry the bird forward. It is known as Penaud’s mechanical bird, and has been sold as an attractive toy.
No. LXIII.--LINE OF SWIFTEST DESCENT
A simple apparatus constructed on the lines of this illustration will give an interesting proof of the laws which govern falling bodies on an inclined plane or on a curved path.
[Illustration]
In the case of the inclined plane the ball is governed by the usual law which controls falling bodies. In that of the concave circular curve, as it is accelerated rapidly at the start, it makes its longer journey in quicker time. In the case of the cycloidal curve it acquires a high velocity. This curve has therefore been called “the curve of swiftest descent,” as a falling body passes over it in less time than upon any path except the vertical.
No. LXIV.--A CENTRIFUGAL RAILWAY
Here is another very simple and pretty illustration of the natural forces which come into play in “looping the loop.”
[Illustration]
This scientific toy on a small scale may be easily made, if care is taken that the height of the higher end of the rails is to the height of the circular part in a greater ratio than 5 to 4.
A ball started at the higher end follows the track throughout, and at one point is held by centrifugal force against the under side of the rails, against the force of gravity.
No. LXV.--A QUESTION OF GRAVITY
If a ball is fired point blank from a perfectly horizontal gun, and travels half a mile over a level plain before it touches ground, and another similar ball is at the same moment dropped from the same height by some mechanical means, the two balls will touch ground simultaneously. The flight, however long, of one through the air has no influence upon the force of gravity, which draws it earthward at the same resistless rate as it draws the other that is merely dropped.
[Illustration]
A SHORT CUT
A quick method of multiplying any number of figures by 5 is to divide them by 2, annexing a cypher to the result when there is no remainder, and if there is any remainder annexing a 5. Thus:--
464 × 5 = 2320; 464 ÷ 2 = 232, annex 0, = 2320. 753 × 5 = 3765; 753 ÷ 2 = 376, annex 5, = 3765.
No. LXVI.--A DUCK HUNT
A duck begins to swim round the edge of a circular pond, and at the same moment a water spaniel starts from the middle of the pond in pursuit of it.
[Illustration]
If both swim at the same pace, how must the dog steer his course so that he is sure in any case to overtake the duck speedily?
THE MAGIC OF DATES
LOUIS NAPOLEON, EMPEROR 1852
1852 1852 1852 date 1 date 1 date 1 of 8 of 8 of 8 his 0 Empress’s 2 their 5 birth 8 birth 6 marriage 3 ---- ---- ---- 1869 1869 1869
Thus, by a most remarkable series of coincidences, the principal dates of the Emperor and Empress of the French added, as is shown above, to the year of the Emperor’s accession, express in each instance the year before his fall.
No. LXVII.--GEOMETRY WITH DOMINOES
In this domino diagram we have a pretty and practical proof that the squares of the sides containing the right angle in any right-angled triangle are together equal to the square of the side opposite to the right angle.
[Illustration]
Each stone forms two squares, and it is easily seen that the number of squares which make up the whole square on the line opposite the right angle are equal to the number of those which make up the two whole squares on the lines which contain that angle.
A second point to be noticed is that the number of pips on the large square are equal to the number on the other two squares combined, an arrangement of the stones which forms quite a game of patience to reproduce, if this pattern is not at hand.
No. LXVIII.--TO COLOUR MAPS
Four colours at most are needed to distinguish the surfaces of separate districts on any plane map, so that no two with a common boundary are tinted alike.
[Illustration]
On this diagram A, B, and C, are adjoining districts, on a plane surface, and X borders, in one way or another, upon each.
It is clearly impossible to introduce a fifth area which shall so adjoin these four districts as to need another tint.
A FREAK OF FIGURES
Here is another freak of figures:--
9 × 1 - 1 = 8 9 × 21 - 1 = 188 9 × 321 - 1 = 2888 9 × 4321 - 1 = 38888 9 × 54321 - 1 = 488888 9 × 654321 - 1 = 5888888 9 × 7654321 - 1 = 68888888 9 × 87654321 - 1 = 788888888 9 × 987654321 - 1 = 8888888888
No. LXIX.--THE TETHERED BIRD
A bird made fast to a pole six inches in diameter by a cord fifty feet long, in its flight first uncoils the cord, _keeping it always taut_, and then recoils it in the reverse direction, rewinding the coils close together. If it starts with the cord fully coiled, and continues its flight until it brings up against the pole, how far does it fly in its double course?
[Illustration]
STRIKE IT OUT
Ask a person to write down in a line any number of figures, then to add them all together as units, and to subtract the result from the sum set down. Let him then strike out any one figure, and add the others together as units, telling you the result.
If this has been correctly done, the figure struck out can always be determined by deducting the final total from the multiple of 9 next above it. If the total happens to be a multiple of 9, then a 9 was struck out.
No. LXX.--THE MOVING DISC AND THE FLY
A fly, starting from the point A, just outside a revolving disc, and always making straight for its mate at the point B, crosses the disc in four minutes, while the disc is revolving twice. What effect has the revolution of the disc on the path of the fly?
[Illustration]
A MAGIC SQUARE
This Magic Square is so arranged that the product of the continued multiplication of the numbers in each row, column, or diagonal is 4096, which is the cube of the central 16.
+---+---+---+ | 8|256| 2| +---+---+---+ | 4| 16| 64| +---+---+---+ |128| 1| 32| +---+---+---+
No. LXXI.--A SHUNTING PUZZLE
The railway, D E F, has two sidings, D B A and F C A, connected at A. The rails at A, common to both, are long enough to hold a single wagon such as P or Q, but too short to admit the whole of the engine R, which, if it runs up either siding, must return the same way.
[Illustration]
How can the engine R be used to interchange the wagons P and Q without allowing any flying shunts?--From _Ball’s Mathematical Recreations_.
No. LXXII.--A CURIOUS FACT
It is a little known and very interesting fact that an equilateral triangle can easily be drawn by rule of thumb in the following way:--Take a triangle of any shape or size, and on each of its sides erect an equilateral triangle. Find and join the centres of these, and a fourth equilateral triangle is always thus formed, as shown by the dotted lines.
[Illustration]
These centres are _centres of gravity_, and they are symmetrically distributed around the centre of gravity of the original triangle.
The figure formed by joining them must therefore be symmetrical, and, as in this case, it is a triangle, it _must be_ always equilateral.
No. LXXIII.--TRY THIS EXPERIMENT
There can be no better instance of how the eye may be deceived than is so strikingly afforded in these very curious diagrams:--
[Illustration]
The square which obviously contains sixty-four small squares, is to be cut into four parts, as is shown by the thicker lines. When these four pieces are quite simply put together, as shown in the second figure, there seem to be sixty-five squares instead of sixty-four.
This phenomena is due to the fact that the edges of the four pieces, which lie along the diagonal A B, do not exactly coincide in direction. In reality they _include a very narrow diamond_, not easily detected, whose area is just equal to that of one of the sixty-four small squares.
FIGURES IN SWARMS
Very curious are the results when the nine digits in reverse order are multiplied by 9 and its multiples up to 81. Thus:--
987654321 × 9 = 8888888889 × 18 = 17777777778 × 27 = 26666666997 × 36 = 35555555556 × 45 = 44444444445 × 54 = 53333333334 × 63 = 62222222223 × 72 = 71111111112 × 81 = 80000000001
It will be seen that the figures by which the reversed digits are multiplied reappear at the beginning and end of each result except the first, and that the figures repeated between them are to be found by dividing the divisors by 9 and subtracting the result from 9. Thus, 54 ÷ 9 = 6, and 9 - 6 = 3.
No. LXXIV.--A TRIANGLE OF TRIANGLES
[Illustration]
In this nest of triangles there are no less than six hundred and fifty-three distinct triangles of various shapes and sizes.
No. LXXV.--PHARAOH’S SEAL
In a chamber of the Great Pyramid an ancient Egyptian jar was found, marked with the device now known as Pharaoh’s seal.
[Illustration]
Can you count the number of triangles or pyramids, of many sizes, but all of similar shape that are expressed on it? Solvers should draw the figure on a larger scale.
MAKING CUBES
It is interesting to note that the repeated addition of odd numbers to one another can be so arranged as to produce cube numbers in due sequence. Thus:--
1 = 1 × 1 × 1 3 + 5 = 2 × 2 × 2 7 + 9 + 11 = 3 × 3 × 3 13 + 15 + 17 + 19 = 4 × 4 × 4 21 + 23 + 25 + 27 + 29 = 5 × 5 × 5
and so on, to any extent.
No. LXXVI.--ROUND THE GARDEN
In a large old-fashioned garden walks were arranged round a central fountain in the shape of a Maltese cross.
[Illustration]
If four persons started at noon from the fountain, walking round the four paths at two, three, four and five miles an hour respectively, at what time would they meet for the third time at their starting-point, if the distance on each track was one-third of a mile?
A NICE SHORT CUT
When the tens of two numbers are the same, and their units added together make ten, multiply the units together, increase one of the tens by unity, and multiply it by the other ten. The result is the product of the two original numbers, if the first result follows the other. Thus:--
43 × 47 = 2021.
No. LXXVII.--A JOINER’S PUZZLE
Can you cut Fig. A into two parts, and so rearrange these that they form either Fig. B or Fig. C?
+--------------+ | | +--------------+ | +--+ | | | | | | +-----------------+ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +--+ | | | | | | | +--------------+ +-----------------+ +--------------+ A B C
The two parts of A must not be _turned round_ to form B or C, but must retain their original direction.
A CALCULATION
Coal may fail us, but we can never run short of material for “words that burn.” It has been calculated that if a man could read 100,000 words in an hour, and there were 4,650,000 men available, they could not pronounce the possible variations which could be formed from the alphabet in 70,000 years!
A PARADOX
It is possible, in a sense, by the following neat method, to take 45 from 45, and find that 45 remains:--
987654321 = 45. 123456789 = 45. --------- 864197532 = 45.
No. LXXVIII.--THE BROKEN OCTAGON
Cut out in stiff cardboard four pieces shaped as Fig. 1, four as Fig. 2, and four as Fig. 3, taking care that they are all exactly true to pattern in shape and proportion to one another.
[Illustration]
Now see whether you can put the twelve pieces together so as to form a perfect octagon.
PROPERTIES OF SEVEN
Here is a proof that 7, if it cannot rival the mystic 9, has quaint properties of its own:--
15873 × 7 = 111111 31746 × 7 = 222222 47619 × 7 = 333333 63492 × 7 = 444444 79365 × 7 = 555555 95238 × 7 = 666666 111111 × 7 = 777777 126984 × 7 = 888888 142857 × 7 = 999999
A SWARM OF EIGHTS
Here is an arithmetical curiosity:--
9 × 9 + 7 = 88 9 × 98 + 6 = 888 9 × 987 + 5 = 8888 9 × 9876 + 4 = 88888 9 × 98765 + 3 = 888888 9 × 987654 + 2 = 8888888 9 × 9876543 + 1 = 88888888 9 × 98765432 + 0 = 888888888
No. LXXIX.--AT A DUCK POND
A farmer’s wife kept a pure strain of Aylesbury ducks for market on a square pond, with a duck-house at each corner. As trade grew brisk she found that she must enlarge her pond. An ingenious neighbour undertook to arrange this without altering the shape of the pond, and without disturbing the duck-houses. What was his plan?
○-------------○ | | | | | | | | | | | | ○-------------○
STRANGE SUBTRACTION
It would seem impossible to subtract 69 from 55, but it can be arranged thus, with six as a remainder:--
SIX IX XL IX X L ------------- S I X =============
No. LXXX.--ALL ON THE SQUARE
Cut out in cardboard twenty triangular pieces exactly the size and shape of this one, and try to place them together so that they form a perfect square.
[Illustration]
ANOTHER MYSTIC NUMBER
The decimal equivalent of ¹⁄₁₃ is .076923. This (omitting the point), multiplied by 1, 3, 4, 9, 10, or 12, yields results in which the same figures appear in varied order, but similar sequence, and multiplied by 2, 5, 6, 7, 8, or 11, it yields a different series, with similar characteristics. Thus:--
76923 × 1 = 76923 × 3 = 230769 × 4 = 307692 × 9 = 692307 × 10 = 769230 × 12 = 923076
76923 × 2 = 153846 × 5 = 384615 × 6 = 461538 × 7 = 538461 × 8 = 615384 × 11 = 846153
DON’T BUY IT TO TRY IT
A kaleidoscope cylinder contains twenty small pieces of coloured glass. As we turn it round, or shake it, so as to make ten changes of pattern every minute, it will take the inconceivable space of time of 462,880,899,576 years and 360 days to exhaust all the possible symmetrical variations. (The 360 days is good!)
No. LXXXI.--PINS AND DOTS
Here is an amusing little exercise for the ingenuity of our solvers.
*-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------*
Take six sharp pins, and puzzle out how to stick them into six of the black dots, so that no two pins, are on the same line, in any direction, vertical, horizontal, or diagonal.
No. LXXXII.--A TRICKY COURSE
The middle of a large playground was paved with sixty-four square flagstones of equal size, which are numbered on this diagram from one to sixty-four.
+----+----+----+----+----+----+----+----+ | _1_| _9_|_17_|_25_|_33_|_41_|_49_|_57_| +----+----+----+----+----+----+----+----+ | _2_|_10_|_18_|_26_|_34_|_42_|_50_|_58_| +----+----+----+----+----+----+----+----+ | _3_|_11_|_19_|_27_|_35_|_43_|_51_|_59_| +----+----+----+----+----+----+----+----+ | _4_|_12_|_20_|_28_|_36_|_44_|_52_|_60_| +----+----+----+----+----+----+----+----+ | _5_|_13_|_21_|_29_|_37_|_45_|_53_|_61_| +----+----+----+----+----+----+----+----+ | _6_|_14_|_22_|_30_|_38_|_46_|_54_|_62_| +----+----+----+----+----+----+----+----+ | _7_|_15_|_23_|_31_|_39_|_47_|_55_|_63_| +----+----+----+----+----+----+----+----+ | _8_|_16_|_24_|_32_|_40_|_48_|_56_|_64_| +----+----+----+----+----+----+----+----+
One of the schoolmasters, who had a head for puzzles, took his stand upon the square here numbered 19, and offered a prize to any boy who, starting from the square numbered 46, could make his way to him, passing through every square once, and only once. It was after many vain attempts that the course was at last discovered. Can you work it out?
No. LXXXIII.--FOR THE CHILDREN
Place twelve draughtsmen, or buttons, in a square, so that you count four along each side of it, thus:--
● ● ● ● ● ● ● ● ● ● ● ●
Now take the same men or buttons, and arrange them so that they form another square, and you can count five along each side of it.
A GAME OF NINES
Here is a good specimen of the eccentricities and powers of numbers:--
153846 × 13 = 1999998 230769 × 13 = 2999997 307692 × 13 = 3999996 384615 × 13 = 4999995 461538 × 13 = 5999994 538461 × 13 = 6999993 615384 × 13 = 7999992 692307 × 13 = 8999991
No. LXXXIV.--TELL-TALE TABLES
She was quite an old maid, and her age was a most absolute secret. Determined to discover it, her scapegrace nephew, on Christmas Eve, produced these tables, and asked her with well simulated innocence on which of them she could see the number of her age.
+--+--+--+ +--+--+--+ +--+--+--+ | 1|23|45| | 2|23|46| |16|27|54| +--+--+--+ +--+--+--+ +--+--+--+ | 3|25|47| | 3|26|47| |17|28|55| +--+--+--+ +--+--+--+ +--+--+--+ | 5|27|49| | 6|27|50| |18|29|56| +--+--+--+ +--+--+--+ +--+--+--+ | 7|29|51| | 7|30|51| |19|30|57| +--+--+--+ +--+--+--+ +--+--+--+ | 9|31|53| |10|31|54| |20|31|58| +--+--+--+ +--+--+--+ +--+--+--+ |11|33|55| |11|34|55| |21|48|59| +--+--+--+ +--+--+--+ +--+--+--+ |13|35|57| |14|35|58| |22|49|60| +--+--+--+ +--+--+--+ +--+--+--+ |15|37|59| |15|38|59| |23|50|61| +--+--+--+ +--+--+--+ +--+--+--+ |17|39|61| |18|39|62| |24|51|62| +--+--+--+ +--+--+--+ +--+--+--+ |19|41| | |19|42| | |25|52| | +--+--+ A| +--+--+ B| +--+--+ C| |21 43| | |22 43| | |26|53| | +--+--+--+ +--+--+--+ +--+--+--+
+--+--+--+ +--+--+--+ +--+--+--+ | 8|27|46| | 4|23|46| |32|43|54| +--+--+--+ +--+--+--+ +--+--+--+ | 9|28|47| | 5|28|47| |33|44|55| +--+--+--+ +--+--+--+ +--+--+--+ |10|29|56| | 6|29|52| |34|45|56| +--+--+--+ +--+--+--+ +--+--+--+ |11|30|57| | 7|30|53| |35|46|57| +--+--+--+ +--+--+--+ +--+--+--+ |12|31|58| |12|31|54| |36|47|58| +--+--+--+ +--+--+--+ +--+--+--+ |13|40|59| |13|36|55| |37|48|59| +--+--+--+ +--+--+--+ +--+--+--+ |14|41|60| |14|37|60| |38|49|60| +--+--+--+ +--+--+--+ +--+--+--+ |15|42|61| |15|38|61| |39|50|61| +--+--+--+ +--+--+--+ +--+--+--+ |24|43|62| |20|39|62| |40|51|62| +--+--+--+ +--+--+--+ +--+--+--+ |25|44| | |21|44| | |41|52| | +--+--+ D| +--+--+ E| +--+--+ F| |26|45| | |22|45| | |42|53| | +--+--+--+ +--+--+--+ +--+--+--+
From her answer he was able to calculate that the old lady was fifty-five.
The tell-tale tables disclosed her age thus:--As it appeared in tables A, B, C, E, and F, he added together the numbers at the top left-hand corners, and found the total to be fifty-five. This rule applies in all cases.
No. LXXXV.--A PAPER PUZZLE
Of the many paper-cutting tricks which appeal to us none is more simple and attractive than this:--
+---+ | | | | +-----+ +-----+ | | +-----+ +-----+ / \ | | / \ \ / | | \ / +-+ | | +-+ | | | | | | | | | | | | | | | | | | | | +---+-+-+---+ | | +-+ | | | +-+ +---+ | +-+-+ | +---+ | +---+---+-+-+---+---+ | | | | +-------------+-------------+
Take a piece of paper, say 5 inches by 3 inches, but any oblong shape and size will do, and after folding it four times cut it lengthways up the centre. Unfold the pieces, and to your surprise you will find a perfect cross and other pieces in pairs of the shapes shown above. The puzzle is how to fold the paper.
+--------------+----------+ |b d¦ | | ¦ | | ¦ | | e¦ | | ¦ | | ¦ | |a c¦ | +--------------+----------+
The paper must be folded first so that B comes upon C, then so that A comes upon D, then from D to C, and lastly from E to C. If it is now cut lengthways exactly along the centre the figures shown on the original diagram will be formed, which resemble a cross and lighted candles on an altar.
No. LXXXVI.--A HOME-MADE PUZZLE
Take a thin board, about eight inches square, and mark it out into thirty-six equal parts; bore a hole in the centre of each part, and then fit in a small wooden peg, leaving about a quarter inch above the surface, as is shown in Fig. 1, the section below the diagram.
[Illustration]
Prepare thirty-six pieces of white or coloured cardboard of the length A to B, and place them over the pegs in any direction in which they will fit so as to form some such symmetrical pattern as is given on the second diagram, putting two holes only on each peg. Chess-players will see that this is the regular knight’s move.
[Illustration]
Quite a number of beautiful designs can be thus formed, and those who have not the means at hand for making a complete set can enjoy the puzzle by merely marking out thirty-six squares, and drawing lines from centre to centre of the exact length from A to B, with black or coloured pencils.
No. LXXXVII.--LOYD’S MITRE PROBLEM
[Illustration]
Divide this figure into four similar and equal parts.
A PRETTY PROBLEM
The solution of the pretty little problem: place three twos in three different groups, so that twice the first group, or half the third group equals the second group, is this:--
2 1 2 2 + 2 ----- = - 2 - - = 1 ----- = 2 2 + 2 2 2 2
No. LXXXVIII.--CONTINUOUS LINES
The following figure, which represents part of a brick wall, cannot be marked out along all the edges of the bricks in less than six continuous lines without going more than once over the same line:--
[Illustration]
Here, in strong contrast to the simple figure given above, which could not be traced without lifting the pen six times from the paper, is an intricate design, the lines of which, on the upper or on the lower half, can be traced without any break at all.
[Illustration]
The general rule that governs such cases is, that where an uneven number of lines meet a fresh start has to be made. In the diagram now given the only such points are at the extremities of the upper and lower halves of the figure at A and X. At all other points two, or four, or six lines converge, and there is no break of continuity in a tracing of the figure.
No. LXXXIX.--CUT OFF THE CORNERS
Can you suggest quite a simple and practical way to fix the points on the sides of a square which will be at the angles of an octagon formed by cutting off equal corners of the square, as shown below?
A E F B +------+--------+------+ | / \ | | / \ | |/ \| M+ +G | | | | | | L+ +H |\ /| | \ / | | \ / | +------+--------+------+ C K I D
MYSTIC FIGURES
Very interesting and curious are the properties of the figures 142857, used in varied order but always in similar sequence, in connection with 7 and 9:--
142857 × 7 = 999999 ÷ 9 = 111111 285714 × 7 = 1999998 ÷ 9 = 222222 428571 × 7 = 2999997 ÷ 9 = 333333 571428 × 7 = 3999996 ÷ 9 = 444444 714285 × 7 = 4999995 ÷ 9 = 555555 857142 × 7 = 5999994 ÷ 9 = 666666
No. XC.--THE FIVE TRIANGLES
The subjoined diagram shows how a square with sides that measure each 12 yards can be divided into five triangles, no two of which are of equal area, and of which the sides and areas can be expressed in yards by whole numbers:--
[Illustration]
The areas of these triangles are 6, 12, 24, 48, and 54 square yards respectively, and the sum of these, 144 square yards, is the area of the square.
CURIOUS COINCIDENCES
Our readers may remember the remarkable fact that the figures of the sum, £12, 12s. 8d., when written thus, 12,128, exactly represent the number of farthings it contains. Now this, so far as we know, is the only instance of the peculiarity, but there are at least five other cases which come curiously near to it. They are these:--
£ s. d. 9 9 6 = 9096 farthings 6 6 4 = 6064 „ 3 3 2 = 3032 „ 10 10 6¹⁄₂ = 10106 „ 13 13 8¹⁄₂ = 13138 „
No. XCI.--PLACING A LADDER
If a ladder, with rungs 1 foot apart, rests against a wall, and its thirteenth rung is 12 feet above the ground, the foot of the ladder is 25 feet from the wall.
[Illustration]
_Proof._--Drop a perpendicular from A to B. Then, as A B C is a right angle, and the squares on A C, A B, are 169 feet and 144 feet, the square on C B must be 25 feet, and the length of C B is 5 feet. We thus move 5 feet towards the wall in going 13 feet up the ladder, and in mounting 65 feet (five times as far) we must cover 25 feet.
No. XCII.--GRACEFUL CURVES
A prettily ingenious method of dividing the area of a circle into quarters, each of them a perfect curve, with perimeter (or enclosing line) equal to the circumference of the circle, and with which four circles can be formed, is clearly shown by the subjoined diagrams:--
[Illustration]
NIGHTS AT A ROUND TABLE
The host of a large hotel at Cairo noticed that his Visitors’ Book contained the names of an Austrian, a Brazilian, a Chinaman, a Dane, an Englishman, a Frenchman, a German, and a Hungarian. Moved by this curious alphabetical list, he offered them all free quarters and the best of everything if they could arrange themselves at a round dining-table so that not one of them should have the same two neighbours on any two occasions for 21 successive days.
The following is one of many ways in which this arrangement can be made, and it seems to be the simplest of them all.
Number the persons 1 to 8; and for our first day set them down in numerical order _except that the two centre ones (4 and 5) change places_:
(1st day)--1 2 3 5 4 6 7 8
Keep the 1 and the 7 unaltered but double each of the other numbers. When the product is greater than 8, divide by 7, and only set down the remainder. Thus we get:
(8th day)--1 4 6 3 8 5 7 2
(Here the fourth figure 3 is 5 × 2 ÷ 7, giving _remainder_ 3, and so on.)
Repeat this operation once more:
(15th day)--1 8 5 6 2 3 7 4
To fill in the intermediate days we have only to keep 1 unchanged and let the remaining numbers run downwards in _simple numerical order_, following 8 with 2, 2 with 3, and so on. Thus:--
1st day--1 2 3 5 4 6 7 8 2nd day--1 3 4 6 5 7 8 2 3rd day--1 4 5 7 6 8 2 3 4th day--1 5 6 8 7 2 3 4 5th day--1 6 7 2 8 3 4 5 6th day--1 7 8 3 2 4 5 6 7th day--1 8 2 4 3 5 6 7 ------------------------- 8th day--1 4 6 3 8 5 7 2 9th day--1 5 7 4 2 6 8 3 10th day--1 6 8 5 3 7 2 4 11th day--1 7 2 6 4 8 3 5 12th day--1 8 3 7 5 2 4 6 13th day--1 2 4 8 6 3 5 7 14th day--1 3 5 2 7 4 6 8 ------------------------- 15th day--1 8 5 6 2 3 7 4 16th day--1 2 6 7 3 4 8 5 17th day--1 3 7 8 4 5 2 6 18th day--1 4 8 2 5 6 3 7 19th day--1 5 2 3 6 7 4 8 20th day--1 6 3 4 7 8 5 2 21st day--1 7 4 5 8 2 6 3
This completes the schedule. It will be found on examination that every number is between every pair of the other numbers once, and once only.
In order to reduce our first-day ring to exact numerical order we have only to interchange the numbers 4 and 5 throughout. The first three lines for example would then become:
1 2 3 4 5 6 7 8 1 3 5 6 4 7 8 2 1 5 4 7 6 8 2 3, etc.
or, by putting letters for figures,
A B C D E F G H A C E F D G H B A E D G F H B C, etc.
An arrangement of the guests is thus arrived at for twenty-one successive days, so that not one of them has the same two neighbours on any two occasions.
No. XCIII.--MAKING MANY SQUARES
Can you apply the two oblongs drawn below to the two concentric squares, so as to produce thirty-one perfect squares?
+------+ +------+ | | | | | | | | +----------------------------------+ +------+ +------+ | | | | | | | | | | | | | +--------------------+ | +------+ +------+ | | | | | | | | | | | | | | | | | | | | +------+ +------+ | | | | | | | | | | | | | | | | | | | | +------+ +------+ | | | | | | | | | | | | | | | | | +--------------------+ | +------+ +------+ | | | | | | | | | | | | +----------------------------------+ +------+ +------+ | | | | | | | | +------+ +------+
No. XCIV.--CUT ACROSS
Take a piece of cardboard in the form of a Greek cross with arms, as shown here, and divide it by two straight cuts, so that the pieces when reunited form a perfect square.
+------+ | | | | +------+ +------+ | | | | +------+ +------+ | | | | +------+
No. XCV.--A PRETTY PUZZLE
The diagrams which we give below show how a hollow square can be formed of the pieces of three-quarters of another square from which a corner has been cut away:--
+-------+ +-------+ +-------+-------+ | | | | | | | | | | +---+ | +---+---+ | | | | | | | | | | | +-------+ +---+ +---+---+ +---+ +---+ | | | | | | | | | | | | | +---+---+ | | +---+---+ | | | | | | | +---------------+ +---------------+ +---------------+
No. XCVI.--A FIVE-FOLD SQUARE
The subjoined diagram shows how a square of paper or cardboard may be cut into nine pieces which, when suitably arranged, form five perfect squares.
[Illustration]
THE SOCIABLE SCHOOLGIRLS
On how many days can fifteen schoolgirls go out for a walk so arranged in rows of three, that no two are together more than once?
Fifteen schoolgirls can go out for a walk on seven days so arranged in rows of three that no two are together more than once.
It is said, on high authority, that there are no less than 15,567,522,000 different solutions to this problem. Here is one of them, given in _Ball’s Mathematical Recreations_, in which _k_ stands for one of the girls, and _a_, _b_, _c_, _d_, _e_, _f_, _g_, in their modifications, for her companions on the seven different days:--
+---------+---------+---------+---------+---------+---------+--------+ | Sunday | Monday | Tuesday |Wednesday| Thursday| Friday |Saturday| +---------+---------+---------+---------+---------+---------+--------+ |_ka₁a₂_ |_kb₁b₂_ |_kc₁c₂_ |_kd₁d₂_ |_ke₁e₂_ |_kf₁f₂_ |_kg₁g₂_ | |_b₁d₁f₁_ |_a₁d₂e₂_ |_a₁d₁e₁_ |_a₂b₂c₂_ |_a₂b₁c₁_ |_a₁b₂c₁_ |_a₁b₁c₂_| |_b₂e₁g₁_ |_a₂f₂g₂_ |_a₂f₁g₁_ |_a₁f₂g₁_ |_a₁f₁g₂_ |_a₂d₂e₁_ |_a₂d₁e₂_| |_c₁d₂g₂_ |_c₁d₁g₁_ |_b₁d₂f₂_ |_b₁e₁g₂_ |_b₂d₁f₂_ |_b₁e₂g₁_ |_b₂d₂f₁_| |_c₂e₂f₂_ |_c₂e₁f₁_ |_b₂e₂g₂_ |_c₁e₂f₁_ |_c₂d₂g₁_ |_c₂d₁g₂_ |_c₁e₁f₂_| +---------+---------+---------+---------+---------+---------+--------+
It is an excellent game of patience, for those who have time and inclination, to place the figures 1 to 15 inclusive in seven such columns, so as to fulfil the conditions.
No. XCVII.--THE THREE CROSSES
It is possible from a Greek cross to cut off four equal pieces which, when put together, will form another Greek cross exactly half the size of the original, and by this process to leave a third Greek cross complete.
[Illustration]
This is how to do it:--
Bisect _C D_ at _N_, _F G_ at _O_, _K L_ at _P_, and _B I_ at _Q_.
Join _N H_, _O M_, _P A_, _Q E_, intersecting at _R_, _S_, _T_, _U_.
Bisect _A R_ at _V_, _E S_ at _W_, _T H_ at _X_, and _M U_ at _Y_.
Join _V Q_, _N Y_, _W P_, _O X_, _N W_, _V O_, _Q X_, _Y P_.
Carefully cut out from the original Greek cross the newly-formed Greek cross, and the odd pieces from around it can be arranged to form another Greek cross.
No. XCVIII.--THE HAMMOCK
The greatest number of plane figures that can be formed by the union of ten straight lines is thirty-six.
[Illustration]
The two equal lines at right angles are first drawn, and each is divided into eight equal parts. The other eight straight lines are then drawn from _a_ to _a_, from _b_ to _b_, and so on, until the hammock-shaped network of thirty-six plane figures is produced.
No. XCIX.--CUTTING A CRUMPET
It will be seen on the diagram below that seven straight vertical cuts with a table-knife will divide a crumpet into twenty-eight parts.
[Illustration]
BALANCE THE SCALES
The nine digits can be so adjusted as to form an equation, or, if taken as weight, to balance the scales. Thus:--
9, 6¹⁄₂ = 3, 5, 7⁴⁄₈
TRUE STRETCHES OF IMAGINATION
How large is the sea? This is a bold big question, and any possible answer involves a considerable stretch of the imagination. Here is a startling illustration of its vast volume:--
If the water of the sea could be gathered into a round column, reaching the 93,000,000 miles which separate us from the sun, the diameter of this column would be nearly two miles and a half!
It is perhaps even more difficult to realise that this mighty mass of waters could be dissipated in a few moments, if the column we have imagined could become ice, and if the entire heat of the sun could be concentrated upon it. All would be melted in _one second of time_, and converted into steam in eight seconds!
No. C.--TO MAKE AN ENVELOPE
Many of our readers may be glad to know an easy way to make an envelope of any shape or size.
[Illustration]
This diagram speaks for itself. When the lines _A B_, _A C_, _D B_, _D C_ have been drawn, the corners of the rectangle, _H E G F_, are folded over, as shown by the dotted lines, after the corners have been rounded, and the margins touched with gum.
No. CI.--SQUARING AN OBLONG
The diagrams below will show how a piece of paper, 15 inches long and 3 inches wide, can be cut into five parts, and rearranged to form a perfect square.
[Illustration]
A PLAGUE OF BLOW-FLIES
The following astounding calculation is answer enough to a question put by one of the authors of “Rejected Addresses:”--“Who filled the butchers’ shops with big blue flies?”
A pair of blow-flies can produce ten thousand eggs, which mature in a fortnight. If every egg hatches out, and there are equal numbers of either sex, which forthwith increase and multiply at the same rapid rate, and if their descendants do the like, so that all survive at the end of six months, it has been calculated that, if thirty-two would fill a cubic inch of space, the whole innumerable swarm would cover the globe, land and sea, half a mile deep everywhere.
No. CII.--BY RULE OF THUMB
Here is quite a neat way to make an equilateral triangle without using compasses:--
[Illustration]
Take a piece of paper exactly square, which we will call _A B C D_, fold it across the middle, so as to form the crease _E F_; unfold it, and fold it again so that the corner _D_ falls upon the crease _E F_ at _G_, and the angle at _G_ is exactly divided. Again unfold the square, and from _G_ draw the straight lines _G C_ and _G D_. Then _G C D_ is the equilateral triangle required.
THE VALUE OF A FRENCHWOMAN
How can we be sure that the value of a Frenchwoman is just 1 franc 8 centimes?
We can be sure that the exact value of a Frenchwoman is 1 franc 8 centimes, for
Two Frenchwomen = Deux Françaises. Deux Françaises = deux francs seize. (2 francs 16). Therefore, One Frenchwoman = 1 franc 8!
No. CIII.--CLEARING THE WALL
If a 52-feet ladder is set up so as just to clear a garden wall 12 feet high and 15 feet from the building, it will touch the house 48 feet from the ground.
[Illustration]
Our diagram shows this, and also, by a dotted line, the only other possible position in which it could fulfil the conditions, if it were then of any practical use.
A BURDEN OF PINS
If one pin could be dropped into a vessel this week, two the next, four the next, and so on, doubling each time for a year, the accumulated quantity would be 4,503,599,627,370,495, and their weight, if we reckon 200 pins to the ounce, would amount to 628,292,358 tons, a full load for 27,924 ships as large as the _Great Eastern_, whose capacity was 22,500 tons.
No. CIV.--MEMORIES OF EUCLID
When at the signpost which said “To _A_ 4 miles, to _B_ 9 miles” on one arm, and on the other “To _C_ 3 miles, to _D_ ---- miles,” and the boy whom I met could only tell me that the farm he worked at was equidistant from _A_, _B_, _C_, and _D_, and nearer to them than to the signpost, and that all the roads ran straight, I found, thanks to memories of Euclid, that I was 12 miles from _D_.
[Illustration]
Since _B A_ and _D C_ intersect outside the circle at the signpost _E_,
therefore _A E_ × _E B_ = _C E_ × _E D_. but _A E_ × _E B_ = 4 × 9 = 36, therefore _C E_ × _E D_ = 36, and _C E_ = 3, therefore _E D_ = 12.
_Q.E.D._
No. CV.--A TRANSFORMATION
This seems to be quite a poor attempt at a Maltese cross, but there is method in the madness of its make.
[Illustration]
It is possible by two straight cuts to divide this uneven cross into four pieces which can be arranged together again so that they form a perfect square. Where must the cuts be made, and how are the four pieces rearranged?
BREVITY IS THE SOUL OF WIT
We all remember that splendidly terse message of success sent home to the authorities by Napier when he had conquered the armies of Scinde--“Peccavi!” (I have sinned).
History had an excellent opportunity for repeating itself when Admiral Dewey defeated the Spaniel fleet, for he might have conveyed the news of his victory by the one burning word--“Cantharides”--“The Spanish fly!”
No. CVI--SHIFTING THE CELLS
In the diagram below a square is subdivided into twenty-five cells.
+-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | |/ \|/ \|/ \|/ \|/ \| | | | | | | | | | | | | +-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | |/ \|/ \|/ \|/ \|/ \| | | | | | | | | | | | | +-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | |/ \|/ \|/ \|/ \|/ \| | | | | | | | | | | | | +-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | |/ \|/ \|/ \|/ \|/ \| | | | | | | | | | | | | +-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | |/ \|/ \|/ \|/ \|/ \| | | | | | | | | | | | | +-----------+-----------+-----------+-----------+-----------+
Can you, keeping always on the straight lines, cut this into four pieces, and arrange these as two perfect squares, in which every semicircle still occupies the upper half of its cell?
A HOME-MADE MICROSCOPE
The following very simple recipe for a home-made microscope has been suggested by a Fellow of the Royal Microscopical Society:--
Take a piece of black card, make a small pinhole in it, put it close to the eye, and look at some small object closely, such as the type of a newspaper. A very decided magnifying power will be shown thus.
No. CVII.--IN A TANGLE
Where on this diagram must we place twenty-one pins or dots so that they fall into symmetrical design and form thirty rows, with three in each row?
[Illustration]
ON ALL FOURS
Those who are fond of figures will find it a most interesting exercise to see how far they are able to represent every number, from one up to a hundred, by the use of four fours. Any of the usual signs and symbols of arithmetic may be brought into use. Here are a few instances of what may thus be done:--
4 + 4 + 4 4 3 = ---------; 9 = 4 + 4 + -; 4 4
4 36 = 4(4 + 4) + 4; 45 = 44 + -; 4
52 = 44 + 4 + 4; 60 = 4 × 4 × 4 - 4.
No. CVIII.--STILL A SQUARE
This figure, which now forms a square, and the quarter of that square, can be so divided by two straight lines that its parts, separated and then reunited, form a perfect square. How is this done?
+------+------+------+ | | | | | | | | +------+------+------+ | | | | | | +------+------+
A MAGIC SQUARE
Here we have arranged five rows of five cards each, so that no two similar cards are in the same lines. Counting the ace as eleven, each row, column, and diagonal adds up to exactly twenty-six.
/-------\ /-------\ /-------\ /-------\ /-------\ | ♣ | | ♥ | | ♠ ♠ | | ♥ ♥ | | ♣ ♣ | | | | | | | | ♥ | | ♣ ♣ | | | | ♥ | | ♠ | | ♥ | | ♣ | | | | | | | | ♥ | | ♣ ♣ | | ♣ | | ♥ | | ♠ ♠ | | ♥ ♥ | | ♣ ♣ | \-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\ | ♦ ♦ | | ♠ ♠ | | ♦ ♦ | | | | ♦ | | | | ♠ | | ♦ ♦ | | | | | | ♦ | | ♠ | | ♦ ♦ | | ♠ | | ♦ | | | | ♠ | | ♦ ♦ | | | | | | ♦ ♦ | | ♠ ♠ | | ♦ ♦ | | | | ♦ | \-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\ | ♥ ♥ | | | | ♠ | | ♥ ♥ | | ♣ ♣ | | ♥ ♥ | | | | | | | | ♣ | | ♥ | | ♣ | | ♠ | | ♥ | | ♣ ♣ | | ♥ ♥ | | | | | | | | ♣ | | ♥ ♥ | | | | ♠ | | ♥ ♥ | | ♣ ♣ | \-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\ | ♣ | | ♥ ♥ | | ♣ ♣ | | ♦ ♦ | | | | | | | | ♣ | | ♦ ♦ | | | | ♣ | | ♥ ♥ | | ♣ | | ♦ | | ♥ | | | | | | ♣ | | ♦ ♦ | | | | ♣ | | ♥ ♥ | | ♣ ♣ | | ♦ ♦ | | | \-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\ | ♦ ♦ | | ♠ ♠ | | | | | | ♣ ♣ | | ♦ | | ♠ ♠ | | | | ♠ ♠ | | | | ♦ | | ♠ | | ♦ | | | | ♣ | | ♦ | | ♠ ♠ | | | | ♠ ♠ | | | | ♦ ♦ | | ♠ ♠ | | | | | | ♣ ♣ | \-------/ \-------/ \-------/ \-------/ \-------/
After you have looked at this Magic Square, and set it out on the table, shuffle the cards, and try to re-arrange them so as to give the same results.
No. CIX.--A TRANSFORMATION
Cut a square of paper or cardboard into seven such pieces as are marked in this diagram.
[Illustration]
Can you rearrange them so that they form the figure 8.
A SECRET REVEALED
There are several ways in which strange juggling with figures and numbers is to be done, but none is more curious than this:--
Ask someone, whose age you do not know, to write down secretly the date and month of his birth in figures, to multiply this by 2, to add 5, to multiply by 50, to add his age last birthday and 365. He then hands you _this total only_ from this you subtract 615. This reveals to you at a glance his age and birthday.
Thus, if he was born April 7 and is 23, 74 (the day and the month) × 2 = 148; 148 + 5 = 153; 153 × 50 = 7650; 7650 + 23 (his age) = 7673; and 7673 + 365 = 8038. If from this you subtract 615, you have 7423, which represents to you the _seventh day of the fourth month, 23 years age_! This rule works out correctly in all cases.
No. CX.--TO MAKE AN OBLONG
Cut out in stiff paper or cardboard two pieces of the shape and size of the small triangle, and four pieces of the shapes and sizes of the other three patterns--fourteen pieces in all.
[Illustration]
The puzzle is to fit these pieces together so that they form a perfect oblong.
SLEEPERS THAT SLIP AND SLEEP
Here is a string of sentences, which may be used as stimulating mental gymnastics when we leave the “Land of Nod.”
A sleeper runs on sleepers, and in this sleeper on sleepers sleepers sleep. As this sleeper carries its sleepers over the sleepers that are under the sleeper, a slack sleeper slips. This jars the sleeper and its sleepers, so that they slip and no longer sleep.
DAYS THAT ARE BARRED
Clever calculation has established a fact which we shall not be able to verify by personal experience. Whatever else may happen, the first day of a century can never fall on Sunday, Wednesday, or Friday.
No. CXI.--SQUARES ON THE CROSS
On this cross there are seventeen distinct and perfect squares marked out at their corners by asterisks.
*-----------* | | | | | | | | | | *-----------* /| \ / |\ / | \ / | \ / | * | \ / | / \ | \ / | / \ | \ *-----------*-----------*-----------*-----------*-----------* | | \ / | \ / | \ / | | | | \ / | \ / | \ / | | | | * | * | * | | | | / \ | / \ | / \ | | | | / \ | / \ | / \ | | *-----------*-----------*-----------*-----------*-----------* \ | \ / | / \ | \ / | / \ | * | / \ | / \ | / \| / \ | / *-----------* | | | | | | | | | | *-----------*
How few, and which, of these can you remove, so that not a single perfect square remains?
STANDING ROOM FOR ALL
On a globe 2 feet in diameter the Dead Sea appears but as a small coloured dot. If it were frozen over there would be standing room on its surface for the whole human race, allowing 6 square feet for each person; and if they were all suddenly engulfed, it would merely raise the level of the lake about 4 inches.
No. CXII.--A CHINESE PUZZLE
Here is quite a good exercise for ingenious brains and fingers. Cut a piece of stiff paper or cardboard into such a right-angled triangle as is shown below.
+ | \ | \ | \ | \ | \ | \ | \ | \ | \ | \ | \ | \ | \ | \ +----------------------------+
+----+ | | +--+ +--+ | | +--+ +--+ | | +--+ +--+ | | +--+ +--+ | | +--+ +--+ | | +--+ +--+ | | +----+
Can you divide this into only three pieces, which, when rearranged, will form the design given as No. 2?
A DEAL IN ROSES
I sent an order for dwarf roses to a famous nursery-garden, asking for a parcel of less than 100 plants, and stipulating that if I planted them 3 in a row there should be 1 over; if 4 in a row 2 over; if 5 in a row 3 over, and if 6 in a row 4 over, as a condition for payment.
The nurseryman was equal to the occasion, charged me for 58 trees, and duly received his cheque.
No. CXIII.--FIRESIDE FUN
This puzzle is not so easy of solution as it may seem at first sight.
Take a counter, or a coin, and place it on one of the points; then push it across to the opposite point, and leave it there. Do this with a second counter or coin, starting on a vacant point, and continue this process until every point is covered, as we place the eighth counter or coin on the last point.
A+ +B | \ / | | \ / | | + | | / \ | | / \ | H+-----------+-----------+-----------+C \ / | | \ / \ / | | \ / + | | + / \ | | / \ / \ | | / \ G+-----------+-----------+-----------+D | \ / | | \ / | | + | | / \ | | / \ | F+ +E
A NOVEL EXERCISE
“Write down,” said a schoolmaster, “the nine digits in such order that the first three shall be one third of the last three, and the central three the result of subtracting the first three from the last.”
The arrangement which satisfies these conditions is, 219, 438, 657.
CURIOUS CALCULATIONS
1. There is a sum of money of such sort that its pounds, shillings, and pence, written down as one continuous number, represent exactly the number of farthings which it contains. What is it?
THE SLIP CARRIAGE
2. If on a level track a train, running all the time at 30 miles an hour, slips a carriage, which is uniformly retarded by the brakes, and which comes to rest in 200 yards, how far has the train itself then travelled?
3. A traveller said to the landlord of an inn, “Give me as much money as I have in my hand, and I will spend sixpence with you.” This was done, and the process was twice repeated, when the traveller had no money left. How much had he at first?
4. How can we obtain eleven by adding one-third of twelve to four-fifths of seven?
FILL IN THE GAPS
5. Can you replace the missing figures in this mutilated long division sum?
215)*7*9*(1** *** ----- *5*9 *5*5 ----- *4* *** =====
6. I buy as many heads of asparagus in the market as can be contained by a string 1 foot long. Next day I take a string 2 feet long, and buy as many as it will gird, offering double the price that I have given before. Was this a reasonable offer?
ALIKE FROM EITHER END
7. As “one good turn deserves another,” first reverse me, then reverse my square, then my cube, then my fourth power. When all this is done no change has been made. What am I?
THE SEALED BAGS
8. How can a thousand pounds be so conveniently stored in ten sealed bags that any sum in pounds from £1 to £1000 can be paid without breaking any of the seals?
9. This is at once a problem and a puzzle:--
Though you twist and turn me over, Yet no change can you discover. Take me thrice, and cut in twain, You will find but one remain.
10. Three gamblers, when they sit down to play, agree that the loser shall always double the sum of money that the other two have before them. After each of them has lost once, it is found that each has eight sovereigns on the table. How much money had each at starting?
FIND THE MULTIPLIER
11. When Tom’s back was turned, the boy sitting next to him rubbed out almost all his sum. Tom could not remember the multiplier, and only this remained on his slate--
345 .. ----- .... .... ----- ..76.
Can you reconstruct the sum?
JUGGLING WITH THE DIGITS
12. The sum of the nine digits is 45. Can you hit upon other arrangements of 1, 2, 3, 4, 5, 6, 7, 8, 9, writing each of them once only, which will produce the same total. Of course fractions may be used.
A BRAIN TWISTER
13. The combined ages of Mary and Ann are forty-four years. Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old, then, is Mary?
14. Mr Oldboy was playing backgammon with his wife on the eve of his golden wedding, and could not make up his mind whether he should leave a blot where it could be taken up by an ace, or one which a tré would hit.
His grandson, at home from Cambridge for the Christmas vacation, solved the question for him easily. What was his decision?
“ASK WHERE’S THE NORTH?”--_Pope._
15. I am aboard a steamer, anchored in a bay where the needle points due north, and exactly 1200 miles from the North Pole. If the course is perfectly clear, and I steam continuously at the rate of 20 miles an hour, always steering north by the compass needle, how long will it take me to reach the North Pole?
16. Three persons, _A_, _B_, and _C_, share twenty-one wine casks of equal capacity, of which seven are full, seven are half full, and seven are empty. How can these be so apportioned that each person shall have an equal number of casks, and an equal quantity of wine, without transferring any of it from cask to cask?
17. A hungry mouse, in search of provender, came upon a box containing ears of corn. He could carry three ears home at a time, and only had opportunity to make fourteen journeys to and fro. How many ears could he add to his store?
18. Take exactly equal quantities of lard and butter; mix a small piece of the butter intimately with all the lard. From this blend take a piece just as large as the fragment removed from the butter, and mix this thoroughly with the butter. Is there now more lard in the butter or more butter in the lard?
WHERE IS THE FALLACY?
19. Here is an apparent proof that 2 = 3:--
4 - 10 = 9 - 15
4 - 10 + ²⁵⁄₄ = 9 - 15 + ²⁵⁄₄
and the square roots of these:--
2 - ⁵⁄₂ = 3 - ⁵⁄₂
therefore
2 = 3.
PARCEL POST LIMITATIONS
20. Our Parcel Post regulations limit the length of a parcel to 3 feet 6 inches, and the length and girth combined to 6 feet. What is the largest parcel of any size that can be sent through the post under these conditions?
21. How can we show, or seem to show, that either four, five, or six nines amount to one hundred?
TEST YOUR SKILL
22. Can you arrange nine numbers in the nine cells of a square, the largest number 100, and the least 1, so that the product by multiplication of each row, column and diagonal is 1000?
CATCHING CRABS
23. Seven London boys were at the seaside for a week’s holiday, and during the six week-days they caught four fine crabs in pools under the rocks, when the tide was out at Beachy Head.
Hearing of this, the twenty-one boys of a school in the neighbourhood determined to explore the pools; but with the same rate of success they only caught one large crab. For how long were they busy searching under the seaweed?
SETTING A WATCH
24. On how many nights could a watch be set of a different trio from a company of fifteen soldiers, and how often on these terms could one of them, John Pipeclay, be included?
“IMPERIAL CÆSAR”
25. If Augustus Cæsar was born on September 23rd, B.C. 63, on what day and in what year did he celebrate his sixty-third birthday, and by what five-letter symbol can we express the date?
26. _A_ was born in 1847, _B_ in 1874. In what years have the same two digits served to express the ages of both, if they are still living?
THIS SHOULD “AMUSE”
27. A hundred and one by fifty divide, To this let a cypher be duly applied; And when the result you can rightly divine, You find that its value is just one in nine.
OVER THE FERRY
28. A man started one Monday morning with money in his purse to buy goods in a neighbouring town. He paid a penny to cross the ferry, spent half of the money he then had, and paid another penny at the ferry on his return home.
He did exactly the same for the next five days, and on Saturday evening reached home with just one penny in his pocket. How much had he in his pocket on Monday before he reached the ferry?
THE MEN, THE MONKEY, AND THE MANGOES
29. Three men gathered mangoes, and agreed that next day they would give one to their monkey and divide the rest equally. The first who arrived gave one to the monkey, and then took his proper share; the second came later and did likewise, and the third later still, neither knowing that any one had preceded him. Finally they met, and, as there were still mangoes, gave one to the monkey, and shared the rest equally. How many mangoes at least must there have been if all the divisions were accurate?
30. I look at my watch between four and five, and again between seven and eight. The hands have, I find, exactly changed places, so that the hour-hand is where the minute-hand was, and the minute-hand takes the place of the hour-hand. At what time did I first look at my watch?
A LARGE ORDER
31. There is a number consisting of twenty-two figures, of which the last is 7. If this is moved to the first place, the number is increased exactly sevenfold. Can you discover this lengthy number?
32. A farmer borrowed from a miller a sack of wheat, 4 feet long and 6 feet in circumference. He sent in repayment two sacks, each 4 feet long and 3 feet in circumference. Was the miller satisfied?
THE FIVE GAMBLERS
33. Five gamblers, whom we will call _A_, _B_, _C_, _D_, and _E_, play together, on the condition that after each hazard he who loses shall give to all the others as much as they then have in hand.
Each loses in turn, beginning with _A_, and when they leave the table each has the same sum in hand, thirty-two pounds. How much had each at first?
34. Knowing that the square of 87 is 7569, how can we rapidly, without multiplication, determine in succession the squares of 88, 89, and 90?
NOT WHOLE NUMBERS
35. Two numbers seek which make eleven, Divide the larger by the less, The quotient is exactly seven, As all who solve it will confess.
AN AMPLE CHOICE
36. If there are twenty sorts of things from which ^99999999999999999999^ different selections can be made, how many of each sort are there?
37. Three women, with no money, went to market. The first had thirty-three apples, the second twenty-nine, the third twenty-seven. Each woman sold the same number of apples for a penny. They all sold out, and yet each received an equal amount of money. How was this?
SIMPLE SUBTRACTION
38. Take five from five, oh, that is mean! Take five from seven, and this is seen.
39. If a bun and a half cost three halfpence, how many do you buy for sixpence?
40. How many times in a day would the hands of a watch meet each other, if the minute-hand moved backward and the hour-hand forward?
41. How can half-a-crown be equally divided between two fathers and two sons so that a penny is the coin of smallest value given to them?
SIZE AND SPEED
42. If the number of the revolutions of the wheel of a bicycle in six seconds is equal to the number of miles an hour at which it is running, what is the circumference of the wheel?
43. Hearing a clock strike, and being uncertain of the hour, I asked a policeman. He had a turn for figures, and replied: “Take half, a third, and a fourth of the hour that has just struck, and the total will be one larger than that hour.” What o’clock was it?
AFTER LONG YEARS
44. If cash is lent at five per cent. To those who choose to borrow, How soon shall I be worth a pound If I lend a crown to-morrow?
ON THE JUMP
45. If I jump off a table with a 20-lb. dumb-bell in my hand, what is the pressure upon me of its weight while I am in the air?
AT A BAZAAR
46. In charitable mood I went recently to a bazaar where there were four tents arranged to tempt a purchaser. At the door of each tent I paid a shilling, and in each tent I spent half of the money remaining in my purse, giving the door-keepers each another shilling as I came out.
It took my last shilling to pay the fourth door-keeper. How much money had I at first, and what did I spend in each tent?
OUT IN THE RAIN
47. Rain is falling vertically, at a speed of 5 miles an hour. I am walking through it at 4 miles an hour. At what angle to the vertical must I hold my umbrella, so that the raindrops strike its top at right angles?
A MONKEY PUZZLER
48. The following interesting problem, translated from the original Sanscrit, is given by Longfellow in his “Kavanagh”:--
“A tree, 100 cubits high, is distant from a well 200 cubits. From the top of this tree one monkey descends, and goes to the well. Another monkey leaps straight upwards from the top, and then descends to the well by the hypotenuse. Both pass over an equal space. How high does the second monkey leap?”
49. A steamboat 105 miles east of Tynemouth Lighthouse springs a leak. She puts back at once, and in the first hour goes at the rate of 10 miles an hour.
More and more water-logged, she decreases her speed each succeeding hour at the rate of one-tenth of what it has been during the previous hour. When will she reach the lighthouse?
50. If a hen and a half lays an egg and a half in a day and a half, how many eggs will twenty-one hens lay in a week?
51. If the population of Bristol exceeds by two hundred and thirty-seven the number of hairs on the head of any one of its inhabitants, how many of them at least, if none are bald, must have the same number of hairs on their heads?
52. A benevolent uncle has in his pocket a sovereign, a half-sovereign, a crown, a half-crown, a florin, a shilling, and a threepenny piece. In how many different ways can he tip his nephew, using only these coins, and how is this most easily determined?
53. Here is a prime problem, in more senses than one, which will tax the ingenuity of our solvers:--I am a prime number of three figures. Increased by one-third, ignoring fractions, I become a square number. Transpose my first two figures and increase me by one-third, and again I am a square number. Put my first figure last, and increase me by one-third, and I am another square number. Reverse my three figures, and increase as before by one-third, and for a fourth time I become a square number. What are my original figures?
54. In how many different ways can six different things be divided between two boys?
55. What is quite the highest number that can be scored at six card cribbage by the dealer, if he has the power to select all the cards, and to determine the order in which every card shall be played?
COVERING THE WALLS
56. A fanciful collector, who bought pictures with more regard to quantity than quality, gave instructions that the area of each frame should exactly equal that of the picture it contained, and that the frames should be of the same width all round.
At an auction he picked up a so-called “old master,” unframed, which measured 18 inches by 12 inches. What width of frame will fulfil his conditions?
A FAMILY REGISTER
57. Our family consists of my mother, a brother, a sister, and myself. Our average age is thirty-nine. My mother was twenty when I was born; my sister is two years my junior, and my brother is four years younger than she is. What are our respective ages?
58. A spider in a dockyard unwittingly attached her web to a mechanical capstan 1 foot in diameter, at the moment when it began to revolve. To hold her ground she paid out 73 feet of thread, when the capstan stopped, and she found herself drained of silk.
To make the best of a bad job she determined to unwind her thread, walking round and round the capstan at the end of the stretched thread. When she had gone a mile in her spiral path she stopped, tired and in despair. How far was she then from the end of her task?
59. A mountebank at a fair had six dice, each marked only on one face 1, 2, 3, 4, 5, or 6, respectively. He offered to return a hundredfold any stake to a player who should turn up all the six marked faces once in twenty throws. How far was this from being a fair offer?
CATS’-MEAT FOR DOGS
60. If ninety groats for twenty cats Will furnish three weeks’ fare, How many hounds for forty pounds, Less one, may winter there?
Just ninety days and one assume The winter’s space to be; And note that what five cats consume Will serve for dogs but three.
(A groat = 4d.)
61. Two wineglasses of equal size are respectively half and one-third full of wine. They are filled up with water, and their contents are then mixed. Half of this mixture is finally poured back into one of the wineglasses. What part of this will be wine and what part will be water?
62. A legend goes that on a stout ship on which St Peter was carried with twenty-nine others, of whom fourteen were Christians and fifteen Jews, he so arranged their places, that when a storm arose, and it was decided to throw half of the passengers overboard, all the Christians were saved. The order was that every ninth man should be cast into the sea. How did he place the Christians and the Jews?
A PROLIFIC COW
63. Farmer Southdown was the proud possessor of a prize cow, which had a fine calf every year for sixteen years. Each of these calves when two years old, and their calves also in their turn, followed this excellent example. How many head did they thus muster in sixteen years?
64. A shepherd was asked how many sheep he had in his flock. He replied that he could not say, but he knew that if he counted them by twos, by threes, by fours, by fives, or by sixes, there was always one over, but if he counted them by sevens, there was no remainder. What is the smallest number that will answer these conditions?
READY RECKONING
65. If a number of round bullets of equal size are arranged in rows one above another evenly graduated till a single bullet crowns the flat pyramid, how can their number be readily reckoned, however long the base line may be?
THE TITHE OF WAR
66. Old General Host A battle lost, And reckoned on a hissing, When he saw plain What men were slain, And prisoners, and missing.
To his dismay He learned next day What havoc war had wrought; He had, at most, But half his host Plus ten times three, six, ought.
One-eighth were lain On beds of pain, With hundreds six beside; One-fifth were dead, Captives, or fled, Lost in grim warfare’s tide.
Now, if you can, Tell me, my man, What troops the general numbered, When on that night Before the fight The deadly cannon slumbered?
67. A farmer sends five pieces of chain, of three links each, to be made into one continuous length. He agrees to pay a penny for each link cut, and a penny for each link joined. What was the blacksmith entitled to charge if he worked in the best interest of the _farmer_?
68. In a parcel of old silver and copper coins each silver piece is worth as many pence as there are copper coins, and each copper coin is worth as many pence as there are silver coins, and the whole is worth eighteen shillings. How many are there of each?
A FEAT WITH FIGURES
69. Take the natural numbers 1 to 11, inclusive, and arrange them in five groups, not using any of them more than once, so that these groups are equal. Any necessary signs or indices may be used.
HOW OLD WAS JOHN?
70. John Bull passed one-sixth of his life in childhood and one-twelfth as a youth. When one-seventh of his life had elapsed he had a son who died at half his father’s age, and John himself lived on four years more. How old was he at the last?
FIGURE IT OUT
71. There are two numbers under two thousand, such that if unity is added to each of them, or to the half of each, the result is in every case a square number. Can you find them?
72. A cheese in one scale of a balance with arms of unequal length seems to weigh 16 lbs. In the other scale it weighs but 9 lbs. What is its true weight?
“DIVISION IS AS BAD!”
73. Can you divide 100 into two such parts that if the larger is divided by the lesser the quotient is also 100?
74. I have marbles in my two side pockets. If I add one to those in the right-hand pocket, and multiply its increased contents by the number it held at first, and then deal in a similar way with those in the other pocket, the difference between the two results is 90. If, however, I multiply the sum of the two original quantities by the square of their difference, the result is 176. How many marbles had I at first in each pocket?
A SURFEIT OF BRIDGE
75. A friendly circle of twenty-one persons agreed to meet each week, five at a time, for an afternoon of bridge, so long as they could do so without forming exactly the same party on any two occasions.
As a central room had to be hired, it was important to have some idea as to the length of time for which they would require it. How long could they keep up their weekly meetings?
76. A herring and a half costs a penny and a half; what is the price of a dozen?
77. What sum of money is in any sense seen to be the double of itself?
COMIC ARITHMETIC
78. At the close of his lecture upon unknown quantities, Dr Bulbous Roots, in playful mood, wrote this puzzle on his blackboard:--
Divide my fifth by my first and you have my fourth; subtract my first from my fifth and you have my second; multiply my first by my fourth followed by my second, and you have my third; place my second after my first and you have my third multiplied by my fourth. What am I?
DROPPED THROUGH THE GLOBE
79. If we can imagine the earth at a standstill for the purpose of our experiment, and if a perfectly straight tunnel could be bored through its centre from side to side, what would be the course of a cannon ball dropped into it from one end, under the action of gravity?
LOVE LETTERS
80. A lady to her lover cried, “How many notes have you of mine?” “Six more I’ve sent,” the youth replied, “Than I have had of thine.”
“But if from one pound ten you take The pennies we on stamps have spent, One eighth their cost you thus will make.” How many had they had and sent?
81. A man, on the day of his marriage, made his will, leaving his money thus:--If a son should be born, two-thirds of the estate to that son and one-third to the widow. If a daughter should be born, two-thirds to the widow and one-third to that daughter. In the course of time twins were born, a boy and a girl. The man fell sick and died without making a fresh will. How ought his estate to be divided in justice to the widow, son, and daughter?
82. My carpet is 22 feet across. My stride, either backwards or forwards, is always 2 feet, and I make a stride every second. If I take three strides forwards and two backwards continuously until I cross the carpet, how long does it take me to reach the end of it?
NO TIME TO BE LOST
83. A merchant at Lisbon has an urgent business call to New York. Taking these places to be, as they appear on a map of the world, on the same parallel of latitude, and at a distance measured along the parallel, of some 3600 miles, if the captain of a vessel chartered to go there sails along this parallel, will he be doing the best that he can for the impatient merchant?
ROUND THE ANGLES
84. Two schoolboys, John and Harry, start from the right angle of a triangular field, and run along its sides. John’s speed is to Harry’s as 13 is to 11.
They meet first in the middle of the opposite side, and again 32 yards from their starting point. How far was it round the field?
AN ECHO FROM THE PAST
85. The following question is given and spelt exactly as it was contributed to a puzzle column by “John Hill, Gent.,” in 1760:--
“A vintner has 2 sorts of wine, viz. A and B, which if mixed in equal parts a flagon of mixed will cost 15 pence; but if they be mixed in a _sesqui-alter_ proportion, as you should take two flagons of A as often as you take three of B, a flagon will cost 14 pence. Required the price of each wine singly.”
PROMISCUOUS CHARITY
86. A man met a beggar and gave him half the money he had in his pocket, and a shilling besides. Meeting another he gave him half of what was left and two shillings, and to a third, he gave half of the remainder and three shillings. This left a shilling in his pocket. How much had he at first?
87. A young clerk wishes to start work at an office in the City on January 1st. He has two promising offers, one from _A_ of £100 a year, with a yearly rise of £20, the other from _B_ of £100 a year with a half-yearly rise of £5. Which should he accept, and why?
HOW CAN I PAY MY BILL?
88. I have an abundance of florins and half-crowns, but no other coins. In how many different ways can I pay my tailor £11, 10s. without receiving change?
89. A monkey climbing up a greased pole ascends 3 feet and slips down 2 feet in alternate seconds till he reaches the top. If the pole is 60 feet high, how long does it take him to arrive there?
“SAFE BIND, SAFE FIND”
90. Old Adze, the village carpenter, who kept his tools in an open chest, found that his neighbours sometimes borrowed and forgot to return them.
To guard against this, he secured the lid of the chest with a letter lock, which carried six revolving rings, each engraved with twelve different letters. What are the chances against any one discovering the secret word formed by a letter on each ring, which will open the lock, and be the only key to the puzzle?
91. Five merry married couples happened to meet at a Swiss hotel, and one of the husbands laughingly proposed that they should dine together at a round table, with the ladies always in the same places so long as the men could seat themselves each between two ladies, but never next to his own wife. How long would their nights at the round table be continued under these conditions?
SOUNDING THE DEPTH
92. In calm water the tip of a stiff rush is 9 inches above the surface of a lake. As a steady wind rises it is gradually blown aslant, until at the distance of a yard it is submerged. Can you decide from these data the depth of the water in which the rush grows?
AMINTA’S AGE
93.
If to Aminta’s age exact Its square you add, and eighteen more, And from her age its third subtract, And to the difference add three score, The latter to the former then Will just the same proportion bear As eighteen does to nine times ten. Can you Aminta’s age declare?
ONE FOR THE PARROT
94. A bag of nuts was to be divided thus among four boys:--Dick took a quarter, and finding that there was one over when he made the division, gave it to the parrot. Tom dealt in exactly the same way with the remainder, as did Jack and Harry in their turns, each finding one nut from the reduced shares to spare for the parrot. The final remainder was equally divided among the boys, and again there was one for the bird. How many nuts, at the lowest estimate, did the bag contain?
95. Here is an easy one:--
If five times four are thirty-three, What will the fourth of twenty be?
96. What fraction of a pound, added to the same fraction of a shilling, and the same fraction of a penny, will make up exactly one pound?
MENTAL ARITHMETIC
97. “Now, boys!” said Dr Tripos, “I think of a number, add 3, divide by 2, add 8, multiply by 2, subtract 2, and thus arrive at twice the number I thought of.” What was it?
98. Two club friends, _A_ and _B_, deposit similar stakes with _C_, and agree that whoever first wins three games at billiards shall take the whole of them. _A_ wins two games and _B_ wins one. Upon this they determine to divide the stakes in proper shares. How must this division be arranged?
99. Not so simple as it sounds is the following compact little problem:--If I run by motor from London to Brighton at 10 miles an hour, and return over the same course at 15 miles an hour, what is my average speed?
100. “I can divide my sheep,” says Farmer Hodge, who from his schooldays had a turn for figures, “into two unequal parts, so that the larger part added to the square of the smaller part shall be equal to the smaller part added to the square of the larger part.”
How many sheep had the farmer?
A HELPFUL BURDEN
101. The following question was proposed in an old book of Mathematical Curiosities published more than a hundred years ago:--
“It often happens that if we take two horses, in every respect alike, yet, if both are put to the draught, that horse which is most loaded shall be capable of performing most work; so that the horse which carries the heavier weight can draw the larger load. How is this?”
102. In the king’s treasury were six chests. Two held sovereigns, two shillings, and two pence, in equal numbers of these coins. “Pay my guard,” said the king, “giving an equal share to each man, and three shares to the captain; give change if necessary.” “It may not be possible,” replied the treasurer, “and the captain may claim four shares.” “Tut, tut,” said the king, “it can be done whatever the amount of the treasure, and whether the captain has three shares or four.”
Was the king right? If so, how many men were there in the guard?
NUTS TO CRACK
103. I bought a parcel of nuts at forty-nine for twopence. I divided it into two equal parts, one of which I sold at twenty-four, the other at twenty-five for a penny. I spent and received an integral number of pence, but bought the least possible number of nuts. How many did I buy? What did they cost? What did I gain?
MONEY MATTERS
104. My purse contained sovereigns and shillings. After I had spent half of its contents there were as many pounds left as I had shillings at first. With what sum did I start?
A DELICATE QUESTION
105. A lady was asked her age in a letter, and she replied by postcard thus:--
If first my age is multiplied by three, And then of that two-sevenths tripled be, The square root of two-ninths of this is four, Now tell my age, or never see me more!
What was her age?
106. If cars run at uniform speed on the twopenny tube, from Shepherd’s Bush to the Bank at intervals of two minutes, how many shall I meet in half an hour if I am travelling from the Bank to Shepherd’s Bush?
PAYMENT BY RESULTS
107. What would it cost me to keep my word if I were to offer my greengrocer a farthing for every different group of ten apples he could select from a basket of a hundred apples?
108. If the minute-hand of a clock moves round _in the opposite direction_ to the hour-hand, what will be the real time between three and four, when the hands are exactly together?
109. Two monkeys have stolen some filberts and some walnuts. As they begin their feast they see the owner of the garden coming with a stick. It will take him two and a half minutes to reach them. There are twice as many filberts as walnuts, and one monkey finishes the walnuts at the rate of fifteen a minute in four-fifths of the time and bolts. The other manages to finish the filberts just in time.
If the walnut monkey had stopped to help him till all was finished, when would they have got away if they ate filberts at equal rates?
110. A cashier, in payment for a cheque, gives by mistake pounds for shillings and shillings for pounds. The receiver spends half-a-crown, and then finds that he has twice as much as the cheque was worth. What was its value?
111. What five uneven figures can be added together so as to make up 14?
112. Three posts which vary in value are vacant in an office. In how many ways can the manager fill these up from seven clerks who apply for the appointments?
113. “It is now ⁵⁄₁₁ of the time to midnight,” said the fasting man, who began his task at noon. What time was it?
A STRIKING TIME
114. If a clock takes six seconds to strike six how long will it take to strike eleven?
WHAT ARE THE ODDS?
115. How would you arrange twenty horses in three stalls so as to have an odd number of horses in each stall?
116. Here is a pretty little problem, which has at any rate an Algebraic form, and is exceedingly ingenious:--
Given _a_, _b_, _c_, to find _q_.
WHEN WAS HE BORN?
117. Tom Evergreen was asked his age by some men at his club on his birthday in 1875. “The number of months,” he said, “that I have lived are exactly half as many as the number which denotes the year in which I was born.” How old was he?
118. Draw three circles of any size, and in any position, so long as they do not intersect, or lie one within another. How many different circles can be drawn touching all the three?
119. We have seen that the nine digits can be so dealt with, using each once, as to add up to 100. How can 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 be arranged so that they form a sum which is equal to 1?
120. How is it possible, by quite a simple method, to find the sum of the first fifty numbers without actually adding them together?
121. Two tram-cars, _A_ and _B_, start at the same time. _A_ runs into a “lie by” in four minutes, and waits there five minutes, when _B_ meets and passes it. Both complete the whole course at the same moment. In what time can _A_ complete it without a rest?
A CURIOUS DEDUCTION
122. Take 10, double it, deduct 10, and tell me what remains.
123. The average weight of the Oxford crew is increased by 2 lbs., when one of them, who weighs 12 stone, is replaced by a fresh man. What is the fresh man’s weight?
ASK ANY MOTORIST
124. A motor car is twice as old as its tyres were when it was as old as its tyres are. When these tyres are as old as the car is now, the united ages of car and tyres will be two years and a quarter. What are their respective ages now?
125. _A_ and _B_ on the edge of a desert can each carry provisions for himself for twelve days. How far into the desert can an advance be made, so that neither of them misses a day’s food?
126. A bottle of medicine and its cork cost half-a-crown, but the bottle and the medicine cost two shillings and a penny more than the cork. What did the cork cost?
IN A FIX
127. A boat’s crew are afloat far from land with no sail or oars. How can they, without making any use of wind or stream, and without any outside help, regain the shore by means of a coil of rope which happens to be at hand.
128. What is the largest sum in silver that I can have in my pockets without being able to give change for a half-sovereign.
129. I have apples in a basket. Without cutting an apple I give half of the number and half an apple to one person; half of what then remains and half an apple to another, and half of what are still left and half an apple to a third. One apple now remains in the basket. How many were there at first?
A QUEER DIVISION
130. A third of twelve divide By just a fifth of seven; And you will soon decide That this must give eleven.
131. A motor goes 9 miles an hour uphill, 18 miles an hour downhill, and 12 miles an hour on the level. How long will it take to run 50 miles and return at once over the same course?
132. In firing at a mark _A_ hits it in two out of three shots, _B_ in three out of four, and _C_ in four out of five. The mark was hit 931 times. If each fired the same number of shots, how many hits did each make, and how many shots were fired?
133. If a cat and a dog, evenly matched in speed, run a race out and back over a course 75 yards in all, and the dog always takes 5 feet at a bound and the cat 3 feet, which will win?
IN A FOG
134. In a fog a man caught up a wagon going in the same direction at 3 miles an hour. If the wagon was just visible to him at a distance of 55 yards, and he could see it for five minutes, at how many miles an hour was he walking?
135. Three horses start in a race. In how many different ways can they be placed by the judge?
NEW ZEALAND FOOTBALL
136. The New Zealanders, winning a match against Oxbridge, scored 34 points, from tries and tries converted into goals.
If every try had been converted they would have made four-fifths of the maximum which a score of 34 points from tries and goals can yield.
What is this maximum, and what was their actual score?
137. What is the smallest number, of which the alternate figures are cyphers, that is divisible by 9 and by 11?
138. Here is an interesting little problem:--_A_, with 8d. in his hand, meets _B_ and _C_, who have five and three loaves respectively. In hungry mood they all agree to share the loaves equally, and to divide the money fairly between _B_ and _C_. How much does each receive?
THE MONEY-BOXES
139. When four money-boxes, containing pennies only, were opened and counted, it was found that the number in the first with half those in all the others, in the second with a third of all the others, in the third with a fourth of all the others, and in the fourth with a fifth of all the others, amounted in each case to 740. How much money did the boxes contain, and how was it divided?
140. Two steamers start together to make a trip to a far-off buoy and back. Steamer _A_ runs all the time at 10 miles an hour. Steamer _B_ does the passage out at 8 miles an hour, and the return at 12 miles. Will they regain port together?
141. A golf player has two clubs mended in London. One has a new head, the other a new leather face. The head costs four times as much as the face. At St Andrews it costs five times as much, and the leather face at St Andrews is half the London price. Including a shilling for a ball he pays twice the St Andrews charges for these repairs. What is the London charge for each?
FROM TWO WRONGS TO MAKE A RIGHT
142. Two children were asked to give the total number of animals in a pasture, where sheep and cattle were grazing. They were told the numbers of sheep and of cattle, but one subtracted, and gave 100 as the answer, and the other arrived at 11,900 by multiplication. What was the correct total?
THE STONE CARRIER
143. Fifty-two stones are placed at intervals along a straight road. The distance between the first and the second is 1 yard, between the second and the third it is 3 yards, between the next two 5 yards, and so on, the intervals increasing each time by 2 yards.
How far would a tramp have to travel to earn five shillings promised to him when he had brought them one by one to a basket placed at the first stone?
144. On a division in the House of Commons, if the Ayes had been increased by fifty from the Noes, the motion would have been carried by five to three. If the Noes had received sixty votes from the Ayes, it would have been lost by four to three. Did the motion succeed? How many voted?
A WATCH PUZZLE
145. How many positions are there on the face of a watch in which the places of the hour and minute-hands can be interchanged so as still to indicate a possible time?
CRICKET SCORES
146. In a cricket match the scores in each successive innings are a quarter less than in the preceding innings. The match was played out, and the side that went in first won by fifty runs. What was the complete score?
HOW HIGH CAN YOU THROW?
147. A boy throws a cricket ball vertically upwards, and catches it as it falls just five seconds later. How high from his hands does the ball go?
MEASURE THE CARPET
148. It may be said of a section of the gigantic carpet at Olympia that had it been 5 feet broader and 4 feet longer it would have contained 116 more feet; but if 4 feet broader and 5 longer the increase would have been but 113 feet. What were its actual breadth and length?
149. In estimating the cost of a hundred similar articles, the mistake was made of reading pounds for shillings and shillings for pence in each case, and under these conditions the estimated cost was £212 18s. 4d. in excess of the real cost. What was the true cost of the articles?
SQUARES IN STREETS
150. The square of the number of my house is equal to the difference of the squares of the numbers of my next door neighbour’s houses on either side.
My brother, who lives in the next street, can say the same of the number of his house, though his number is not the same as mine. How are our houses numbered?
151. Two men of unequal strength are set to move a block of marble which weighs 270 lbs., using for the purpose a light plank 6 feet long. The stronger man can carry 180 lbs. How must the block be placed so as to allow him just that share of the weight?
152. A man has twenty coins, of which some are shillings and the rest half-crowns. If he were to change the half-crowns for sixpences and the shillings for pence he would have 156 coins. How many shillings has he?
COUNT THE COINS
153. Some coins are placed at equal distances apart on a table, so that they form the sides of an equilateral triangle.
From the middle of each side as many are then taken as equal the square root of the number on that side, and placed on the opposite corner coin. The number of coins on each side is then to the original number as 5 is to 4. How many coins are there in all?
PUTTING IN THE POSTS
154. A gardener, wishing to fence round a piece of ground with some light posts, found that if he set them a foot apart there would be 150 too few, but if placed a yard apart there would be 70 to spare. How many posts had he?
155. _A_ gives _B_ £100 to buy 100 animals, which must be cows at £5 each, sheep at £1, and geese at 1s. How many of each sort can he buy?
156. John is twice as old as Mary was when he was as old as Mary is. John is now twenty-one. How old is Mary?
157. In a cricket match _A_ made 35 runs; _C_ and _D_ made respectively half and one-third as many as _B_, and _B_’s score was as much below _A_’s as _C_’s was above _D_’s. What did _B_, _C_, and _D_ each score?
SETTLED BY REMAINDERS
158. What is the least number which, divided by 2, 3, 4, 5, 6, 7, 8, 9, or 10, leaves respectively as remainders 1, 2, 3, 4, 5, 6, 7, 8, and 9?
TABLE TURNING
159. A square table stands on four legs, which are set at the middle point of its sides. What is the greatest weight that this table can uphold upon one of its corners?
BRIBING THE BOYS
160. Well pleased with the inspector’s report, the rector of a country parish came into his school with 99 new pennies in his pocket, and said that he would give them to the five boys in Standard VII. if they could, within an hour, show him how to divide them so that the first share should exceed the second by 3, be less than the third by 10, be greater than the fourth by 9, and less than the fifth by 16. What was the answer which would satisfy these conditions?
SHEEP-STEALING
161. Some Indian raiders carried off a third of a flock of sheep, and a third of a sheep. Another party took a fourth of what remained, and a fourth of a sheep. Others took a fifth of the rest and three-fifths of a sheep. What was the number of the full flock, if there were then 409 left?
162. Three boys begin to fill a cistern. _A_ brings a pint at the end of every three minutes, _B_ a quart every five minutes, and _C_ a gallon every seven minutes. If the cistern holds fifty-three gallons, in what time will it be filled, and who will pour in the last contribution?
ESTIMATE OF AGE
163. A man late in the last century said that his age was the square root of the year in which he was born. In what year did he say this?
A DEAL IN CHINA
164. A dealer in Eastern curios sold a Satzuma vase for £119, and on calculation found that the number which expressed his profit per cent. expressed also the cost price in pounds of the vase. What was this number?
165. What is the chance of throwing at least one ace in a single throw with a pair of dice?
166. A thief starts running from a country house as fast as he can. Four minutes later a policeman starts in pursuit. If both run straight along the road, and the policeman gets over the ground one-third faster than the thief, how soon will he catch him?
167. Twenty-seven articles are exposed for sale on one of the stalls of a bazaar. What choice has a purchaser?
VERY PERSONAL
168. “How old are _you_, dad?” said Nellie on her birthday, as her father gave her as many shillings as she was years old. His answer was quite a puzzle for a time, but with the help of her schoolfellows Nellie worked it out.
This is what he said:--
“I was twice as old as you are The day that you were born. You will be just what I was then When fourteen years are gone.”
How old was Nellie, and how old was her dad?
WORD AND LETTER PUZZLES
A MAGIC COCOON
This Magic Cocoon is so cleverly spun that the word can be traced and read in many ways.
N N O N N O O O N N O O C O O N N O O C O C O O N N O O C O C O C O O N N O O C O C O O N N O O C O O N N O O O N N O N N
How many readings can you discover starting from one or other of the Cs, and passing up and down or sideways, but not diagonally, and never over the same letter twice in a reading? There are 756!
THE CHRONOGRAM
The Chronogram, severely classed by Addison as “a species of false wit” is a sentence in which the salient letters represent in Roman numerals some particular year. A good English specimen is this: “My Day Closed Is In Immortality.” The capital letters in these words give MDCIII., or 1603, the year in which Queen Elizabeth died.
A FRENCH CHRONOGRAM
The battle-cry at Montlhéry in 1465 was:--“à CheVaL, à CheVaL, gendarMes, à CheVaL!” Taking the letters printed in capitals--
M = 1000 CCC = 300 LLL = 150 VVV = 15 ---- we have the date of the battle 1465
A NOTABLE CHRONOGRAM
QVI CHRISTI IAVDES CANTANT SANCTÆ PASSIONIS SVÆ VIRTVTE IN IPSO ET PATRE VNVM SINT.
This curious inscription is placed over the organ at Ober Ammergau. Add together its Roman numerals and they give the date at which the organ was dedicated. The English of it is:--
May those who sing the praises of Christ be by virtue of His Sacred Passion one in The Father and in Himself.
A PERFECT CHRONOGRAM
On a damaged inscription to Bishop Berkeley in Winchester Cathedral are the words--VIXI, LVXI--I have lived, I have shone. Added together in their values as Roman numerals the letters of these two words give his age exactly at his death--eighty-three.
A PRIZE MOTTO
On the return of the C.I.V. from the Boer War a prize was offered by _Truth_ for the best motto appropriate to them. This was to consist of three words of which the first must begin with C the second with I and the third with V.
The prize was taken by the following Latin motto which is singularly happy both in construction and in meaning:--
CIVI IVI VICI _I roused_ _I went_ _I won._
The sequence of events is perfect; no letters but C.I.V. are used and the motto is a palindrome if read by syllables.
THE MUSICAL SCALE
As in olden days some of the Psalms and other writings were constructed in acrostic form, so in the Middle Ages even serious writers would juggle with letters, as though they felt that such tricky methods were an aid to memory.
It was in this spirit that Guido Aretino, a Benedictine monk of Tuscany in 1204, gave names to the notes used in the musical scale from the first syllables of the lines of a Latin hymn. “Ut” is still used in France, though we and the Italians have substituted “do.”
UT queant laxis REsonare fibris, MIra gestorum FAmuli tuorum, SOLve polutis LAbii reatis O Pater alme!
ALL THE ALPHABET!
Many of us know that there is a long verse in the Book of Ezra in which all the letters of the alphabet are used, taking “j” as “i” (_Ezra_ vii., v. 21).
This very curious coincidence also occurs in a comparatively short sentence in “The Beth Book,” by Sarah Grand:--“It was an exquisitely deep blue just then, with filmy white clouds drawn up over it like gauze;” and here “j” is itself in evidence.
APT ALLITERATION
Schopenhauer, the famous German philosopher, who was a confirmed bachelor and misogynist, was compelled while living at Frankfort to support an old lady who had been crippled by his violence. When her death came as a welcome relief to him, he composed the following clever epitaph:--
Obit anus, Abit onus.
which by the interchange of two letters pictured the position. It may be freely rendered:--
Old lady dies, My burden flies.
A DOUBLE SEQUENCE
The following clever composition, which appeared in the pages of _Truth_, contains a _double_ sequence of words, which increase a letter at a time, the same letters appearing in varied order until at last “o” culminates in _thornless_, and “a” in _restrainest_. It is quite a remarkable _tour-de-force_.
O lack-_a_-day! _at_ eve we _sat_, One _star_ had lit its lamps _on_ high. We did _not note_ the circling bat, _Start_ from the _stone_ when flitting nigh. For the _strait_ gate of _honest_ doubt Shut off the _thrones_ of Love and Gain; We dreamed not, as we mourned without, That Time’s swift _transit shortens_ pain. O Thou, Who _trainest_ souls to shine, Though once we craved a _thornless_ lot, This gracious truth we now divine: The bruised reed Thou _strainest_ not; But by _restraints_, that gently tame; _Restrainest_ Passion’s kindling flame.
THE REIGN OF TERROR
During the Reign of Terror, France and her people and position were thus alphabetically described:--
Le peuple Français A B C. (abaissé). La gloire nationale F A C. (effacée). Les places fortes O Q P. (occupées). Quarante trois députés C D. (cédés). L’armée D P C. (dépaysée). Les ministres A J. (agés). La liberté O T. (ôtée). La charte L U D. (éludée).
SEE-SAW
This elaborate method of piling up no less than seven consecutive “thats,” so that they make tolerable sense, was told to his boys during school-time by Dr Moberly, then headmaster of Winchester, and afterwards Bishop of Salisbury, just fifty years ago:--
I saw that C saw.
C saw that that I saw.
I saw that that that C saw was so.
C saw that, that that that I saw was so.
I saw that, that _that_ that that C saw was so.
C saw _that_ that, that _that_ that that I saw was so.
I saw _that_ that, that _that_ that that _that_ C saw was so.
FOR A ROMAN HOLIDAY
If the Roman ladies and children, at their equivalent for Christmas, amused themselves by acting verbal charades, an excellent word was at their disposal, “sustineamus”--“let us endure,” which can be broken up exactly into _sus_, _tinea_, _mus_--_a sow_, _a moth_, _a mouse_.
A WORD SQUARE
1. Can you complete this word square, so that its four words read alike from top to bottom and from left to right?
* E * * E * * * * * * E * * E *
ANOTHER WORD SQUARE
2. Can you fill in this word square?
C * * C * E * N U * E S * U * E S * C * E * S * * E S * * E E * T E * *
DUPLICATE LETTERS
3. In this sentence, when complete,
So****AG****LATI****X****ITH
each group of four missing letters contains two pairs of letters which are alike. Can you on these lines complete the sentence?
Here is a similar sentence by way of illustration:
T****M****TERTAIN****MUND,
which becomes when filled in--
T_wo_ _wo_m_en_ _en_tertain_ed_ _Ed_mund.
ANOTHER WORD SQUARE
4. Can you complete this word square by substituting letters for the dots?
W * * * E * * T * * * T O N * * * N * * E * * * T
WORD BUILDING
5. What word can be made with these?
L S D U D O D U D.
6. A lovelorn youth consulted a married lady on his condition, and was asked by her on a slip of paper:--
“Loruve?”
When he had deciphered this, and had answered in the affirmative, she handed to him another slip, on which this advice was written:--
L “Prove A F and ensure success.” D
What did it all mean?
A DOUBLE ACROSTIC
7. _Saint of Spain, whose daily word Twenty years hath London heard!_ Sweet days, elastic metal, motion sharp. My halls once echoed to an Irish harp. The “son of sorrow,” honourable of yore. Dread goddess, loosing the loud dogs of war. Time’s atom--total of eternities. This name an insect bears, a patriot bore. “So do, yet hear me,” said Themistocles.
ANOTHER WORD SQUARE
8. Can you complete this word square?
* M * N * S M * N * O * * N * B * E N * B * L * * O * L * R S * E * R *
A FRENCH ORACLE
9. A spruce young Frenchman at a _fête_ consulted a modern oracle as to how he could best please the ladies. This was the mystic response:--
MEC DO BIC.
Can you interpret it?
A QUAINT EQUATION
10. In our young days we have often wrestled with vulgar fractions, but apart from Algebra we have had no serious concern with any in which letters take the place of figures. A specimen of this sort, not known to science, is the following curiosity:--
m -- ot -- = mo. y
11. The puzzle in _Truth_ was recently founded upon “ourang-outang,” which had been cleverly buried. We will give a few of the best results. This is one:--
Poor wretch! a moisture filled his eye, “Do not rebuff a lonely boy,” Said he, “If ere I sink and die Your smile--O! pardon will be joy!”
Another is:--
Though I jump, and row, and run, Cap or cup I never won.
What animals are buried in these lines?
LIKE A PEACOCK’S TAIL
12. Fourteen letters here we fix, Vowels only two are spoken; All together these we mix Into what can not be broken.
A WEIRD WORD
13. There is an English word of thirteen letters in which the same vowel occurs four times, the same consonant six times, another consonant twice, and another once. Can you hit upon it?
A CONDENSED PROVERB
14. Though brevity is said to be the soul of wit, we are too often flooded nowadays with a superabundance of words.
Here is an attempt at modest condensation. A familiar English proverb is quite clearly expressed to the solver’s seeing eyes in this brief phrase:--
WE IS DO
What is the proverb?
ANOTHER WORD SQUARE
15. Can you complete this word square?
W * * * * S * R * * R * * * O R * * * * R M * * * R * * N * S * * * * M
CAN YOU DECIPHER IT?
16. The following puzzle lines are attributed to Dr Whewell:--
O O N O O. ----------- U O A O O I O U O N O O O O M E T O O. U O A O I D O S O I O N O O I O U T O O!
A BROKEN DIAMOND
17. Can you fill in the vacancies in this diamond?
P F O * C * R * * F * * C * * * P O R C E L A I N R * * L * * * S * A * * S I * N
Its words must read alike from left to right and from top to bottom.
WHAT IS THIS?
18. Tan HE Edsa VEN in It N Gja SmeTs AsgN aD Az Rett De.
A PHONETIC JOURNEY
19. I can travel first-class on the Great Eastern Railway from 2 2 2 2 2 2 2 2 4 4 4 4 4 5 0 0. What is the cost of my journey, and its length in time?
A CURIOUS OLD INSCRIPTION
20. Seogeh sreve ereh wcisume vahl Lah sehs se otreh nos llebdnas Regni freh nos gnires rohyer Ganoed iryd ale nifae esots sorcy Rub nabot es rohk co caed ir.
Can you decipher it?
IRISH STEW AT SIMPSON’S
21. I wrote the following note recently:--
Dear Jack,--Meet me at Simpson’s to-morrow at 1.30. We will sample their excellent Irish stew. Here are some catchwords that will remind you of the invitation:--
Join me at and i s
Why should they remind him of it?
ENGLISH AS SHE IS SPELT!
22. This was the exact text of a letter sent to the master of an English village school by a labourer as an excuse for his boy’s absence:--
“Cepatomtogoatatrin”
Can you decipher it?
A DOUBLE ACROSTIC
23. This double Acrostic will afford an easy exercise in mental gymnastics for those to whom such pastime appeals:--
Now we are fain To rack your brain.
1. More fit for babes and sucklings than for you.
2. Robbed of externals this is very true.
3. Diminutive in measure and in weight.
4. Pen-name of one a true pen potentate.
5. A palindrome quite plain is here in sight.
6. Sans head and tail it also yields this light.
7. Here is in short what anyone may write.
FIND THE PROVERB
24. c e f h i m n o r s t v y 3 2 2 2 7 1 1 2 6 5 8 3 9 4 4 1 2 1 6 1 3 1 5 8 2 3 5 7 9 6 4
The letters with ones under them are the first letters of words, those with twos under them are second letters of words, and so on.
A PUZZLE WILL
25. Having occasion to make a few slight additions to my will, I called in my lawyer to arrange the matter. How far forward did the instructions contained in the following lines carry him in his work?
Set down a hundred in my will, Add nothing to the text; Five hundred now a space may fill, And one be added next.
Another hundred write as well, And yet another one; Then fifty more, and try to tell The deed that now is done.
ANOTHER WORD SQUARE
26. Can you complete this word square?
* D * * O * D * * I * E * S * A * D T * A * A * * R * A * E R * D * E *
MUSICAL EPITAPHS
27. Over the grave of a French musician, who was choked by a fish bone, the following epitaph was inscribed in notes of music:--A. G. A. E. A.
Over the porch of the house of Gustave Doré these musical notes were placed on a tablet:--C. E. B. A. C. D.
What do these inscriptions signify?
QUITE TOO TOO
28. “Where can we meet to-morrow?” said Jack Spooner to his best girl.
“We will go,” she replied, “at 222222222222 LEY STREET.”
When and where did they meet?
A BROKEN WORD
29. What does this spell?
C ------------------- T T T T T T T T T T
CONTRADICTORY TERMS
30. What English word is it which may be so treated as to affirm or disallow the use of its own initial or final letter?
PRINTERS’ PIE
31. Can you arrange these letters
E I O O O U B C N N R R S S
so that they form the title of a book well-known to boys?
FILL IN THE GAPS
32. Keeping these letters in their present order make a sensible sentence by inserting among them as often as is necessary another letter, which must be in every case the same.
A DEN I I CAN DOCK.
DISTORTED SHAKESPEARE
33. Here is a well-known quotation from Shakespeare, which seems to need some straightening out:--
OXXU8 MAAULGIHCTE NOR
A PHONETIC NIGHTMARE
34. Here, as an awful warning to those who are ready to accept the definition of English spelling given by a former headmaster of Winchester--“Consonants are interchangeable, and vowels do not count”--is a common English word of twelve letters, in “linked sweetness long drawn out.”
Iewkngheaurrhphthewempeighghtips.
Can you decipher it?
ANOTHER WORD SQUARE
35. Can you, by filling in letters, complete this word square so that it shall read alike across and from top to bottom?
* A * * A * * A * E * * * A * E
A QUAINT INSCRIPTION
36. The following curious inscription may be seen on a card hanging up in the bar of an old riverside inn in Norfolk:--
THEM * ILL * ERSLEA * VET * HEMI LLT * HEW * HER * RYMEN * LOW ERTH * EIRS * AILTH * EMA LTS * TER * SLE * AVET * HE * KI LN * FORAD * ROPO * FTH EWHI * TESW * AN * SALE.
Can you decipher it?
A POET’S PI
37. TONDEBNIOTOCHUMFOARYHUR OTDIRECTTHAWHOTERSOFKLSYA; TIKATESTUBALIGHTSTILLETRUFLYR OTBOWLALLNEFESLEAVARFWYAA.
In this printer’s pie the words are in their proper sequence, but the letters are tangled.
BURIED PLACES
38. In the following short sentences five names of places are buried--that is to say, the letters which spell them in proper order form parts of more words than one. Thus, for example “Paris” might be buried in the words “go u_p a ris_e:”
“The men could ride all on donkeys, the skipper, though, came to a bad end.”
When you have discovered these places, try to find out what very unexpected word of more than four letters is buried in the sentence, “On Christmas Eve you rang out angel peals.”
TREASON CONDONED
39. According to an old poet, Sir John Harrington (1561-1612):--
“Treason doth never flourish; what’s the reason? For if it prosper none dare call it treason!”
The classic lines may possibly have been the germ of the flippant modern riddle, “Why is it no offence to conspire in the evening?”
A BIT OF BOTANY
40. Inscribe an _m_ above a line And write an _e_ below, This woodland flower is hung so fine It bends when zephyrs flow.
A PIED PROVERB
41. The following letters, if they are properly rearranged, will fall into the words which form a popular proverb:--
AAEEGGHILLMNNNOOOORRSSSSTT
Can you place them in position?
A DROP LETTER PUZZLE
42. Can you fill in the gaps of this proverb?
E**t* *e*s**s *a*e *h* *o** **i*e.
MULTUM IN PARVO
43. There is an English word of five syllables which has only eight letters, five of them vowels--an a, an e, twice i, and y. What are its consonants?
DOUBLETS
44. Can you turn TORMENT to RAPTURE, using four links, changing only one letter each time, and varying the order of the letters?
A PIED PROVERB
45. Can you arrange these letters so that they form a sentence of five words?
aaceeeffhhiiiiimnnoooprrssttttt.
The result is a well-known English proverb.
WHAT CAN IT BE?
46. Add one letter, and make this into a sensible English sentence:--
GDLDPRTFRRTHDXXFRDDNS
OUT OF PROPORTION
47. One vowel in an English word is found, Which by eight consonants is hedged around.
48. Can you form an English word with these letters?
AAAAABBNNIIRSSTT.
49. What is this? It is found in Shakespeare:--
K I N I.
ALPHA BETA
50. There are two English words which contain each of them ten letters, and six of these are a, b, c, d, e, f, the first six letters of the alphabet. Can you build up either or both of them without looking at the solution?
SHIFTING NUMBERS
51. Of a band of true kinsmen I stand at the head, Who, to keep themselves warm, cluster three in a bed. Put four into gaol and their number has risen, So that six can be counted together in prison. Take the six and recount them, they dwindle to three; Count again, and a change into five you will see. With no number from one to one hundred I mix, Yet with five of my mates I am seen to make six.
AN IMPUDENT PRODIGAL
52. The prodigal son of a wealthy colonial farmer received a letter from his father, to suggest that a considerable part of his inheritance should be safeguarded before he squandered it. His reply ran thus:--“Dear dad, keep 1000050.” As such a sum, even in dollars, was out of the question, the father was completely puzzled.
What did the prodigal mean?
BURIED POETS
53. The names of eight famous British poets are buried in these lines, that is to say, the letters that spell the names form in their proper order parts of different words:--
The sun is darting rays of gold Upon the moor, enchanting spot, Whose purpled heights, by Ronald loved, Up open to his Shepherd cot.
And sundry denizens of air Are flying, aye, each to his nest; And eager make at such an hour All haste to reach the mansions-blest.
Can you dig them up?
A FATEFUL LETTER
54. When _A. B._ gave up the reins of government, and _C. B._ took office in his place, it was found that their political positions could be exactly described by two quite common English verbs, which differ only in this, that the one is longer by one letter than the other, while the rest of the letters are the same, and in the same order. What are these two verbs?
A PRIZE REBUS
55. The following is a prize Rebus:--
done | mutt | a glutt and | i | T. c. d. you make me
A LETTER TANGLE
56. First a _c_ and _a_ _t_, last _a_ _c_ and a _t_, With a couple of letters between, Form a sight that our eyes are delighted to see, Unless in their sight it is seen.
A TRANSPOSITION
57. Cut off my tail and set it at my head, What was an island is a little bear instead.
A REBUS
T S.
58. What English word do these two letters indicate? There are two possible solutions of equal merit.
A PHONETIC PHRASE
59. How can we read this?
I N X I N X I N.
A GOOD END
60.
[Illustration: I F S]
SOLUTIONS
CHESS CAMEOS
No. XXVI
Black has made the false move Kt from Q sq to Kt 3. When this is replaced, and the king is moved as the proper penalty, White mates at once with one or other of the Knights.
No. XXVII
Replace the White Kt at B 7, and a Black Pawn at K 4; then P takes P _en pas_. Mate.
ANALYSIS AND PROOF
It can be proved that Black’s last move _must have been_ P from K 2 to K 4, so that White may take the P _en pas_.
The Black King cannot have moved from any _occupied_ square.
(The White Kt now occupies B 7.)
Nor from Kt 3 or 4, as both are now _doubly_ guarded, so that he cannot have moved _out of a check_.
(The White Kt now helps to guard Kt 5.)
Nor can he have moved from K 2, as the White P on Q 6 cannot have moved _to give check_.
No other P can have moved.
The K P cannot have moved from K 3, became of the position of the White King.
Therefore Black’s last move _must have been_ P from K 2 to K 4, which White can take _en pas_ giving Mate.
No. XXVIII
B--Q 2 B--R 5 P--Kt 4 -------- ---------------- ------- White has no move. P--R 7 P--R 7 becomes Q any
The way in which the B runs to earth and is shut in is most ingenious. Black with the new Q cannot anyhow give White a move.
No. XXIX
R--Kt 7, ch. R--Kt 5 R--B 5, ch. ------------ ----------- ----------------- K moves P becomes Q Q × R, stalemate.
No. XXX
B--Kt 8.
No. XXXI
B--R 4.
No. XXXII
B--R sq.
No. XXXIII
B--Q 4.
No. XXXIV
B--B sq.
No. XXXV
K--R 4.
No. XXXVI
Kt--Kt’s 6.
No. XXXVII
R--Kt 5 Kt--B 6 Kt × Kt mate. ------- ----------- ------------- Kt × R Kt--B 6 ch.
No. XXXVIII
Kt--R 7 Q--KB 8 Q--R 8 mate. ------- --------- ------------ B moves B returns
Any other move of the Kt would impede the movements of the Q.
No. XXXIX
R--R sq. Q--Kt sq. Q to Kt sq. mate. -------- ---------- ----------------- B moves. B returns.
No. XL
This beautiful problem is solved by:--
Q--Kt 6 K--B 2 mates accordingly. ------- ------ ------------------ P × Q any
or
R--B 3 mates accordingly. ------- ------ ------------------ Kt--K 3 any
No. XLI
R--R 2. Q--R sq. Q mates. ------- -------- -------- B × Kt any
if
Kt--QB 8, Q or Kt mates. -------- --------- -------------- B--Q sq. any
if
Kt--QB 6, Q or Kt mates. ----------- --------- -------------- B elsewhere any
No. XLII
Kt--Kt 4, dis. ch. Q--KR 2, ch. Kt--B 2, mate. ------------------ ------------ -------------- K--R 8. P × Q.
There are other variations.
No. XLIII
B--Kt sq. Q--QR 7, Q mates. --------- -------- -------- P × Kt, K × Kt.
If K × Kt, Q × P, and mates next move.
No. XLIV
K--Q 7, R--Q 5, Q mates. ------- ------- -------- K moves K × R
No. XLV
R--Q 8 Q × P, ch. B mates. ------- ---------- -------- K moves K × Q
if
Q--K 7 Q mates. ------- ------ -------- B moves any
No. XLVI
B--B 6 Q--Q 7, ch. R mates. ------ ----------- -------- K × R K × Q
if
R--B 6, ch. Q mates. ----- ----------- -------- B × R K × P
There are other variations.
No. XLVII
Q--R 8 Kt--B 6 mates accordingly. ------ ------- ------------------ Kt × Q any
No. XLVIII
B--Kt 8 Q × B, ch. Q--QR 7, mate. --------- ---------- -------------- B--K Kt 2 Kt--K 4
if
Q--Q 2, ch. P mates. ------ ----------- -------- B--B 3 K moves
No. XLIX
B-Kt 2 Q--K 3 ch. mates. ------ ---------- ------ K--K 4 any
There are other variations.
No. L
Q--KR 2 Q--Q 6, ch. Kt--K 5. double ch. mate. ------------- ----------- ------------------------- K--B 3 or B 4 Q × Q
There are other variations.
No. LI
R--QR sq. R--R 2 P mates. --------- ------ -------- P moves P × R
No. LII
Q--B sq. Q × BP ch. Q mates. -------- ---------- -------- P--K 7 K--Q 3
if
Q × BP ch. Q mates. ------ ---------- -------- K--Q 2 K--B 3
There are other variations.
No. LIII
B--R 8 Q--QR sq. Q--QKt. 7, mate. ------ --------- ---------------- K--R 2 K moves
No. LIV
Q--R 5 Kt--Kt 5 Kt mates. --------------- -------- --------- B × Q or B--B 2 any
No. LV
K--Kt 7 Q--Q 5, ch. Kt. mates. ------- ----------- ---------- Kt--B 3 Kt × Q
if
Q--B 8, ch. Q mates. ------- ------- -------- P--K 3 K moves
No. LVI
R--Kt 6 B--Kt 4 R × P mate. ------- ------- ----------- P moves P × B
No. LVII
Kt from R 3--Kt 5 B--B 4 P--Q 4 ch. R--B 5 mate. ----------------- ------ -------------- ------------ P × Kt P × B P × P _en pas_
if
R--Kt 5 ch. mates accordingly. ----- ----------- ------------------ R × B any
and if
Kt--Q 6 R--B 5 ch. Kt--B 7 mate. ------ ------- ---------- ------------- Q--K 6 Q--Kt 3 Q × R
SCIENCE AT PLAY
No. LVIII.--THE GEARED WHEELS
The arrow head at the top of a small wheel with ten teeth, which is geared into and revolved round a large fixed wheel with forty teeth, will point directly upwards five times in its course round the large wheel. Four of these turnings are due to the rotation of the small wheel on its own axis, and one of them results from its revolution round the large wheel.
No. LX.--THE FIFTEEN BRIDGES
It is possible to pass over all the bridges which connect the islands _A_ and _B_ and the banks of the surrounding river without going over any of them twice.
The course can be shown thus, using capital letters for the different regions of land, and italics for the bridges:--E_a_ F_b_ B_c_ F_d_ A_e_ F_f_ C_g_ A_h_ C_i_ D_k_ A_m_ E_n_ A_p_ B_q_ E_l_D.
This order of the bridges can, of course, be reversed.
No. LXVI.--A DUCK HUNT
In order that a spaniel starting from the middle of a circular pond, and going at the same pace as a duck that is swimming round its edge, shall be sure to catch it speedily, the dog must always keep in the straight line between the duck and the centre of the pond.
The duck can never gain an advantage by turning back, and if it swims on continuously in a circle it will be overtaken when it has passed through a quarter of the circumference, for the dog will in the same time have described a semi-circle whose diameter is the radius of the pond, ending at the point where the duck is caught.
No. LXIX.--THE TETHERED BIRD
When a bird tethered by a cord 50 feet long to a post 6 inches in diameter uncoils the full length of the cord, and recoils it in the opposite direction, keeping it always taut, it flies 10,157 feet, or very nearly 2 miles, in its double course.
To avoid possible misunderstanding, we point out that, in order to pass from the uncoiling to the recoiling position, the bird must fly through a semicircle at the end of the fully extended cord.
No. LXX.--THE MOVING DISC AND THE FLY
[Illustration]
When a fly, starting from the point _A_, just outside the revolving disc, and always making straight for its mate at the point _B_, crosses the disc in four minutes, during which time the disc is turning twice, the revolution of the disc has a most curious and interesting effect on the path of the fly.
The fly is a quarter of a minute in passing from the outside circle to the next, during which the disc has made an eighth of a revolution, and the fly has reached the point marked 1. The succeeding points up to 16 show the position of the fly at each quarter of a minute, until, by a prettily repeated curve, _B_ is reached.
No. LXXI.--A SHUNTING PUZZLE
The following method enables the engine _R_ to interchange the positions of the wagons, _P_ and _Q_, for either of which there is room on the straight rails at _A_, while there is not room there for the engine, which, if it runs up either siding, must return the same way:--
1. _R_ pushes _P_ into _A_. 2. _R_ returns, pushes _Q_ up to _P_ in _A_, couples _Q_ to _P_, draws them both out to _F_, and then pushes them to _E_. 3. _P_ is now uncoupled, _R_ takes _Q_ back to _A_, and leaves it there. 4. _R_ returns to _P_, pulls _P_ back to _C_, and leaves it there. 5. _R_, running successively through _F_, _D_, _B_, comes to _A_, draws _Q_ out, and leaves it at _B._
No. LXXV.--PHARAOH’S SEAL
It is quite puzzling to decide how many similar triangles or pyramids are expressed on the seal of Pharaoh. There are in fact 96.
No. LXXVI.--ROUND THE GARDEN
The four persons who started at noon from the central fountain, and walked round the four paths at the rates of two, three, four, and five miles an hour would meet for the third time at their starting point at one o’clock, if the distance on each track was one-third of a mile.
No. LXXVII.--A JOINER’S PUZZLE
This diagram shows how to divide Fig. _A_ into two parts, and so rearrange these that they form either Fig. _B_ or Fig. _C_, without turning either of the pieces.
+--------------+ | | +--------------+ | +--+ | | | | | +--+ +--------------+--+ | +--+ | | | | | | | | | | +--+ | | +--+ | | +--+ | | | | | | | | | | | +--+ | | +--+ | | +--+ | | | | | | | | | | | +--+ | | +--+ | +--+ | | | | | | | | | +--+ | | +--+ | +--+ | | | | | | | | +--------------+ +--+--------------+ +--------------+ A B C
Cut the five steps, and shift the two pieces as is shown.
No. LXXVIII.--THE BROKEN OCTAGON
The Broken Octagon is repaired and made perfect if its pieces are put together thus:--
[Illustration]
No. LXXIX.--AT A DUCK POND
The pond was doubled in size without disturbing the duck-houses, thus:--
+-----------○-----------+ | / \ | | / \ | | / \ | | / \ | | / \ | ○ ○ | \ / | | \ / | | \ / | | \ / | | \ / | +-----------○-----------+
No. LXXX.--ALL ON THE SQUARE
This is a perfect arrangement:--
[Illustration]
No. LXXXI.--PINS AND DOTS
The pins may be placed thus:--
On the third dot in the top line; on the sixth dot in the second line; on the second dot in the third line; on the fifth dot in the fourth line; on the first dot in the fifth line; on the fourth dot in the sixth line.
*-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------* | \ / | \ / | \ / | \ / | \ / | | \ / | \ / | \ / | \ / | \ / | | * | * | * | * | * | | / \ | / \ | / \ | / \ | / \ | | / \ | / \ | / \ | / \ | / \ | *-----------*-----------*-----------*-----------*-----------*
No. LXXXII.--A TRICKY COURSE
To trace this course draw lines upon the diagram from square 46 to squares 38, 52, 55, 23, 58, 64, 8, 57, 1, 7, 42, 10, 13, 27, and 19. This gives fifteen lines which pass through every square only once.
No. LXXXIII.--FOR THE CHILDREN
Make a square with three on every side, and place the remaining four one on each of the corner men or buttons.
No. LXXXVII.--LOYD’S MITRE PROBLEM
The figure given is thus divided into four equal and similar parts:--
[Illustration]
No. LXXXIX.--CUT OFF THE CORNERS
A E F B +------+--------+------+ | / \ | | / \ | |/ \| M+ +G | | | | | | L+ +H |\ /| | \ / | | \ / | +------+--------+------+ C K I D
A very simple rule of thumb method for striking the points in the sides of a square, which will be at the angles of an octagon formed by cutting off equal corners of the square, is to place another square of equal size upon the original one, so that the centre is common to both, and the diagonal of the new square lies upon a diameter of the other parallel to its side.
No. XCIII.--MAKING MANY SQUARES
The subjoined diagram shows how the two oblongs, applied to the two concentric squares, produce 31 perfect squares, namely, 17 small ones, one equal to 25 of these, 5 equal to 9, and 8 equal to 4.
+------+ | | | | +-------------+------+-------------+ | | | | | | | | | +------+------+------+ | | | | | | | | | | | | | +-----+------+------+------+------+------+------+ | | | | | | | | | | | | | | | | +-----+------+------+------+------+------+------+ | | | | | | | | | | | | | +------+------+------+ | | | | | | | | | +-------------+------+-------------+ | | | | +------+
No. XCIV.--CUT ACROSS
The Greek Cross can be divided by two straight cuts, so that the resulting pieces will form a perfect square when re-set, as is shown in these figures:--
[Illustration]
No. CV.--A TRANSFORMATION
The diagram which is given below shows how the irregular Maltese Cross can be divided by two straight cuts into four pieces, which form when properly rearranged, a perfect square.
[Illustration]
No. CVI.--SHIFTING THE CELLS
The following diagram shows by its dark lines how the whole square can be cut into four pieces, and these arranged as two perfect squares in which every semicircle still occupies the upper half of its cell.
One piece forms a square of nine cells, and it is easy to arrange the other three pieces in a square of sixteen cells by lifting the three cells and dropping the two.
+-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ ∥ / \ | | / \ | / \ | / \ | / \ ∥ / \ | |/ \|/ \|/ \|/ \∥/ \| | | | | ∥ | | | | | ∥ | +-----------+-----------+-----------+-----------+-----------+ | / \ | / \ | / \ | / \ ∥ / \ | | / \ | / \ | / \ | / \ ∥ / \ | |/ \|/ \|/ \|/ \∥/ \| | | | | ∥ | | | | | ∥ | +-----------+-----------+===========+===========+===========+ | / \ | / \ ∥ / \ | / \ | / \ | | / \ | / \ ∥ / \ | / \ | / \ | |/ \|/ \∥/ \|/ \|/ \| | | ∥ | | | | | ∥ | | | +-----------+===========+-----------+-----------+-----------+ | / \ ∥ / \ ∥ / \ | / \ | / \ | | / \ ∥ / \ ∥ / \ | / \ | / \ | |/ \∥/ \∥/ \|/ \|/ \| | ∥ ∥ | | | | ∥ ∥ | | | +===========+-----------+-----------+-----------+-----------+ | / \ | / \ ∥ / \ | / \ | / \ | | / \ | / \ ∥ / \ | / \ | / \ | |/ \|/ \∥/ \|/ \|/ \| | | ∥ | | | | | ∥ | | | +-----------+-----------+-----------+-----------+-----------+
No. CVII.--IN A TANGLE
[Illustration]
It will be seen, on the subjoined diagram, how twenty-one counters or coins can be placed on the figure so that they fall into symmetrical design, and form thirty rows, with three in each row.
No. CVIII.--STILL A SQUARE
A +------+------+------+ | | | | | | | | +------+------+------+ C | | | | | | +------+------+ B
In order that a square and an additional quarter may be divided by two straight lines so that their parts, separated and then reunited, form a perfect square, lines must be drawn from the point _A_ to the corners _B_ and _C_. Draw the figure on paper, cut through these lines, and you will find that the pieces can be so reunited that they form a perfect square.
No. CIX.--A TRANSFORMATION
The diagram below shows how the seven parts of the square can be rearranged so that they form the figure 8.
[Illustration]
No. CX.--TO MAKE AN OBLONG
Here is an oblong formed by piecing together two of the smaller triangles, and four of each of the other patterns--
[Illustration]
Here is another:--
[Illustration]
No. CXI.--SQUARES ON THE CROSS
This diagram shows how every indication of the seventeen squares is broken up by the removal of seven of the asterisks which mark their corners.
[Illustration]
Those surrounded by circles are to be removed.
No. CXII.--A CHINESE PUZZLE
|\ | \ | \ | \ | \ | \ | +--\ | | \ | +--+--+ \ | | | \ | +--+ +--+\ | | | \ +--+ +--+\ | | \ +-----------------+--+
The dotted lines on the triangular figure show how a piece of cardboard cut to the shape of Fig. 1 can be divided into three pieces, and rearranged so that these form a star shaped as in Fig. 2.
No. CXIII.--FIRESIDE FUN
To solve this puzzle slip the first coin or counter from _A_ to _D_, then the others in turn from _F_ to _A_, from _C_ to _F_, from _H_ to _C_, from _E_ to _H_, from _B_ to _E_, from _G_ to _B_, and place the last on _G_. It can only be done by a sequence of this sort, in which each starting point is the finish of the next move.
[Illustration]
AMUSING PROBLEMS
1. THE CARPENTER’S PUZZLE
The carpenter cleverly contrived to mend a hole 2 feet wide and 12 feet long, by cutting the board which was 3 feet wide and 8 feet long, as is shown in Fig. 1, and putting the two pieces together as is shown in Fig. 2.
Fig. 1. +----------------+ | ________| |________| | | | +----------------+
Fig. 2. +------------------------+ | ________¦ | | ¦ | +------------------------+
2. GOLDEN PIPPINS
Here is another solution:--
1| 3| 5| 7| 9|11|13|15|17|19 39|37|35|33|31|29|27|25|23|21 2| 4| 6| 8|10|12|14|16|18|20 40|38|36|34|32|30|28|26|24|22 --+--+--+--+--+--+--+--+--+-- 82|82|82|82|82|82|82|82|82|82
3. AN AWKWARD FIX
I was able to find my way in a strange district, when the sign-post lay uprooted in the ditch, without any difficulty. I simply replaced the post in its hole, so that the proper arm, with its lettering, _pointed the way that I had come_, and then, of necessity, the directions of the other arms were correct.
4. LINKED SWEETNESS LONG DRAWN OUT
The train was whistling for 5 minutes. Sound travels about a mile in 5 seconds, so the first I heard of it was 5 seconds after it began. Its last sound reached me 7¹⁄₂ seconds after it ceased, so I heard the whistle for 5 minutes, 2¹⁄₂ seconds.
5. These quarters were not so elastic as they are made to appear. In good truth, considering that the second man who was placed in _A_ was afterwards removed to _I_, no real _second_ man was provided for at all.
6. The first day of a new century can never be Sunday, Wednesday, or Friday. The cycle of the Gregorian calendar is completed in 400 years, after which all dates repeat themselves.
As in this cycle there are only four first days of a century, it is clear that three of the seven days of the week must be excluded. Any perpetual calendar shows that the four which do occur are Monday, Tuesday, Thursday, and Saturday, so that Sunday, Wednesday, and Friday are shut out.
A neat corollary to this proof is that Monday is the only day which may be the first, or which may be the last, day of a century.
7. A cricket bat with spliced handle has such good driving power, because the elasticity of the handle allows the ball to be in contact with the blade of the bat for a longer time than would otherwise be possible.
With similar effect the “follow through” of the club head at golf maintains contact with the ball, when it is already travelling fast.
8. When two volumes stand in proper order on my bookshelf, each 2 inches thick over all, with covers ¹⁄₈ of an inch in thickness, a bookworm would only have to bore ¹⁄₄ of an inch, to penetrate from the first page of Vol. I, to the last page of Vol. II, for these pages would be in actual contact if there was no binding. This very pretty and puzzling question combines in its solution all the best qualities of a clever catch with solid and simple facts.
9. A man would have to fall from a height of nearly 15 miles to reach earth before the sound of his cry as he started. The velocity of sound is constant, while that of a falling body is continually accelerated. At first the cry far outstrips the falling man, but he _overtakes and passes through his own scream_ in about 14¹⁄₂ miles, for his body falls through the 15 miles in 70 seconds, and sound travels as far in 72 seconds. Air resistance, and the fact that sound cannot pass from a rare to a dense atmosphere, are disregarded in this curious calculation.
10. A man on a perfectly smooth table in a vacuum, and where there was no friction, though no contortions of his body would avail to get away from this position, could escape from the predicament by throwing from him something which he could detach from his person, such as his watch or coat. He would himself instantly slide off in the opposite direction!
11. The monkey clinging to one end of a rope that passes over a single fixed pulley, while an equal weight hangs on the other end, cannot climb up the rope, or rise any higher from the ground.
If he continues to try to climb up, he will gradually pull the balancing weight on the other end of the rope upwards, and the slack of the rope will drop below him, while he remains in the same place.
If, after some efforts, he rests, he will sink lower and lower, until the weight reaches the pulley, because of the extra weight of rope on his side, if friction is disregarded.
12. Though the tension on a pair of traces tends as much to pull the horse backward as it does to pull the carriage forward, it is the initial pull from slack to taut which sets the traces in motion; and this, once started, must continue indefinitely until checked by a counter pull.
13. Some say that a rubber tyre leaves a double rut in dust and a single one in mud, because the air, rushing from each side into the wake of the wheel, piles up the loose dust. Others hold that the central ridge is caused by the continuous contraction of the tyre as it passes its point of contact with the road.
A correspondent, writing some years ago to “Knowledge,” said:--“It is our old friend the sucker. The tyre being round, the weight on centre of track only is great enough to enable the tyre to draw up a ridge of dust after it.”
14. If two cats on a sloping roof are on the point of slipping off, one might think that whichever had the longest paws (pause) would hold on best. Todhunter, in playful mood, saw deeper into it than that, and pronounced for the cat that had the _highest mew_, for to his mathematical mind the Greek letter _mu_ was the coefficient of friction!
15. If a penny held between finger and thumb, and released by withdrawing the finger, starts “heads” and makes half a turn in falling through the first foot, it will be “heads” again on reaching the floor, if it is held four feet above it at first.
16. HE DID IT!
Funnyboy had secretly prepared himself for the occasion by rubbing the chemical coating from the side of the box on to his boot.
17. THE CYCLE SURPRISE
If a bicycle is stationary, with one pedal at its lowest point, and that pedal is pulled backwards, while the bicycle is lightly supported, the bicycle will move backwards, and the pedal relatively to the bicycle, will move forwards. This would be quite unexpected by most people, and it is well worth trying.
18. The rough stones, by which any number of pounds, from 1 to 364, can be weighed, are respectively 1 ℔., 3 ℔s., 9 ℔s., 27 ℔s., 81 ℔s., and 243 ℔s. in weight.
19. If we disregard the resistance of the air, a small clot of mud thrown from the hindermost part of a wheel would describe a parabola, which would, in its descending limb, bring it back into kissing contact with the wheel which had rejected it.
20. THE CARELESS CARPENTER
When the carpenter cut the door _too little_, he did not in fact _cut it enough_, and he had to cut it again, so that it might fit.
21. If from the North Pole you start sailing in a south-westerly direction, and keep a straight course for twenty miles, you must steer due north to get back as quickly as possible to the Pole, if, indeed, it has been possible to start from it in any direction other than due south.
22. DICK IN A SWING
Dick’s feet will travel in round numbers nearly 16 feet further than his head, or to be exact, 15·707,960 feet.
23. A POSER
The initial letters of Turkey, Holland, England, France, Italy, Norway, Austria, Lapland, and Spain spell, and in this sense are the same as, “the finals.”
CURIOUS CALCULATIONS
1. The only sum of money which satisfies the condition that its pounds, shillings, and pence written down as a continuous number, exactly give the number of farthings which it represents, is £12, 12s., 8d., for this sum contains 12,128 farthings.
2. If, when a train, on a level track, and running all the time at 30 miles an hour, slips a carriage which is uniformly retarded by brakes, and this comes to rest in 200 yards, the train itself will then have travelled 400 yards.
The slip carriage, uniformly retarded from 30 miles an hour to no miles an hour, has an average speed of 15 miles an hour, while the train itself, running on at 30 miles an hour all the time, has just double that speed, and so covers just twice the distance.
3. The traveller had fivepence farthing when he said to the landlord, “Give me as much as I have in my hand, and I will spend sixpence with you.” After repeating this process twice he had no money left.
4. This is the way to obtain eleven by adding one-third of twelve to four-fifths of seven--
TW(EL)VE + S(EVEN) = ELEVEN
5. Here is the completed sum:--
215)*7*9*(1** 215)37195(173 *** 215 ---- ---- *5*9 1569 *5*5 1505 ---- ---- *4* 645 *** 645 === ===
The clue is that no figure but 3, when multiplied into 215, produces 4 in the tens place.
6. If I attempt to buy as many heads of asparagus as can be encircled by a string 2 feet long for double the price paid for as many as half that length will encompass, I shall not succeed. A circle double of another in circumference is also double in diameter, and its area is four times that of the other.
7. If, when you reverse me, and my square, and my cube, and my fourth power, you find that no changes have been made, I am 11, my square is 121, my cube 1331, and my fourth power 14641.
8. A thousand pounds can be stored in ten sealed bags, so that any sum in pounds up to £1,000 can be paid without breaking any of the seals, by placing in the bags 1, 2, 4, 8, 16, 32, 64, 128, 256, and 489 sovereigns.
9. It is the fraction ⁶⁄₉ which is unchanged when turned over, and which, when taken thrice, and then divided by two becomes 1.
10. When the three gamblers agreed that the loser should always double the sum of money that the other two had before them, and they each lost once, and fulfilled the conditions, remaining each with eight sovereigns in hand, they had started with £13, £7, and £4 as the following table shows:--
A B C £ £ £ At starts 13 7 4 When A loses 2 14 8 When B loses 4 4 16 When C loses 8 8 8
11. Tom’s sum, which his mischievous neighbour rubbed almost out, is reconstructed thus:--
345 345 ** 37 ----- ----- **** 2415 **** 1035 ----- ----- **76* 12765
12. Here are two other arrangements of the nine digits which produce 45, their sum; each is used once only:--
5 × 8 × 9 × (7 + 2) ------------------- = 45 1 × 3 × 4 × 6
7² - 5 × 8 × 9 -------------- + 1 = 45 3 × 4 × 6
13. If, when the combined ages of Mary and Ann are 44, Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann, Mary is 27¹⁄₂ years old, and Ann is 16¹⁄₂.
For, tracing the question backwards, when Ann was 5¹⁄₂ Mary was 16¹⁄₂. When Ann is three times that age she will be 49¹⁄₂. The half of this is 24³⁄₄, and when Mary was at that age Ann was 13³⁄₄. Mary’s age, by the question, was twice this, or 27¹⁄₂.
14. It is safer at backgammon to leave a blot in the tables which can be taken by an ace than one which a three would hit. In either the case of an actual ace or a three the chance is one in eleven; but there are two chances of throwing deuce-ace, the equivalent of three.
15. If I start from a bay, where the needle points due north, 1200 miles from the North Pole, and the course is perfectly clear, I can never reach it if I steam continuously 20 miles an hour, steering always north by the compass needle. After about 200 miles I come upon the Magnetic Pole, which so affects the needle that it no longer leads me northward, and I may have to steer south by it to reach the geographical Pole.
16. The 21 casks, 7 full, 7 half full, and 7 empty, were shared equally by A, B, and C, as follows:--
Full cask. Half full. Empty. A 2 3 2 B 2 3 2 C 3 1 3
or--
A 3 1 3 B 3 1 3 C 1 5 1
Thus each had 7 casks, and the equivalent of 3¹⁄₂ caskfuls of wine.
17. The foraging mouse, able to carry home three ears at a time from a box full of ears of corn, could not add more than fourteen ears of corn to its store in fourteen journeys, for it had each time to carry along two ears of its own.
18. If, with equal quantities of butter and lard, a small piece of butter is taken and mixed into all the lard, and if then a piece of this blend of similar size is put back into the butter, there will be in the end exactly as much lard in the butter as there is butter in the lard.
19. The fallacy of the equation--
4 - 10 = 9 - 15
4 - 10 + ²⁵⁄₄ = 9 - 15 + ²⁵⁄₄
and the square roots of these--
2 - ⁵⁄₂ = 3 - ⁵⁄₂
therefore 2 = 3
is explained thus:--The fallacy lies in ignoring the fact that the square roots are _plus or minus_. In the working we have taken both roots as _plus_. If we take one root plus, and the other minus, and add ⁵⁄₂, we have either 2 = 2, or 3 = 3.
20. The largest possible parcel which can be sent through the post under the official limits of 3 feet 6 inches in length, and 6 feet in length and girth combined, is a cylinder 2 feet long and 4 feet in circumference, the cubic contents of which are 2⁶⁄₁₁ cubic feet.
21. We can show, or seem to show, that either four, five, or six nines amount to 100, thus:--
IX IX 99⁹⁄₉ = 100 IX 9 × 9 + 9 + 9⁹⁄₉ = 100 IX IX --- 100
22. This is the magic square arrangement, so contrived that the products of the rows, columns, and diagonals are all 1,000.
+---+---+---+ | 50| 1| 20| +---+---+---+ | 4| 10| 25| +---+---+---+ | 5|100| 2| +---+---+---+
23. If seven boys caught four crabs in the rock-pools at Beachy Head in six days, the twenty-one boys who searched under the seaweed and only caught one crab with the same rate of success were only at work for _half a day_.
24. A watch could be set of a different trio from a company of fifteen soldiers for 455 nights, and one of them, John Pipeclay, could be included ninety-one times.
25. If Augustus Cæsar was born September 23, B.C. 63, he celebrated his sixty-third birthday on September 23, B.C. 0; or, writing it otherwise, September 23, A.D. 0; or again, if we wish to include both symbols, B.C. 0 A.D. It is clear that his sixty-second birthday fell on September 23, B.C. 1, and his sixty-fourth on September 23, A.D. 1, so that the intervening year may be written as above.
26. The difference of the ages of _A_ and _B_ who were born in 1847 and 1874, is 27, or 30 - 03. Hence, when _A_ was 30 _B_ was 03. And _A_ was 30 in 1877. Eleven years later _A_ was 41 and _B_ 14, and eleven years after that _A_ was 52 and _B_ 25. Thus the same two digits served to express the ages of both in 1877, 1888, and 1899. This can only happen in the cases of those whose ages differ by some multiple of nine.
27 A hundred and one by fifty divide, To this let a cypher be duly applied; And when the result you can rightly divine, You find that its value is just one in nine--
is solved by CLIO, one of the nine Muses.
28. The man who paid a penny on Monday morning to cross the ferry, spent half of what money he then had left in the town, and paid another penny to recross the ferry, and who repeated this course on each succeeding day, reaching home on Saturday evening with one penny in his pocket, started on Monday with £1 1s. 1d. in hand.
29. When the three men agreed to share their mangoes equally after giving one to the monkey, and when each helped himself to a third after giving one to the monkey, without knowing that anyone had been before him, and they finally met together, gave one to the monkey, and divided what still remained, there must have been at least seventy-nine mangoes for division at the first.
30. If, after having looked at my watch between 4 and 5, I look again between 7 and 8, and find that the hour and minute-hands have then exactly changed places, it was 36 12-13 minutes past 4 when I first looked. At that time the hour-hand would be pointing to 23 1-13 minutes on the dial, and at 23 1-13 minutes past 7 the hour hand would be pointing to 36 12-13 minutes.
31. The number consisting of 22 figures, of which the last is 7, which is increased exactly sevenfold if this 7 is moved to the first place, is 1,014,492,753,623,188,405,797.
32. The two sacks of wheat, each 4 feet long and 3 feet in circumference, which the farmer sent to the miller in repayment for one sack 4 feet long and 6 feet in circumference, far from being a satisfactory equivalent, contained but half the quantity of the larger sack, for the area of a circle the diameter of which is double that of another is equal to four times the area of that other.
33. The five gamblers, who made the condition that each on losing should pay to the others as much as they then had in hand, and who each lost in turn, and had each £32 in hand at the finish, started with £81, £41, £21, £11, and £6 respectively.
34. If we know the square of any number, we can rapidly determine the square of the next number, without multiplication, by adding the two numbers to the known square. Thus if we know that the square of 87 is 7569,
then the square of 88 = 7569 + 87 + 88 = 7744; so too the square of 89 = 7744 + 88 + 89 = 7921; and the square of 90 = 7921 + 89 + 90 = 8100.
35. The two numbers which solve the problem--
Two numbers seek which make eleven, Divide the larger by the less, The quotient is exactly seven, As all who find them will confess--
are 1³⁄₈ and 9⁵⁄₈, for 1³⁄₈ + 9⁵⁄₈ = 11, and ⁷⁷⁄₈ ÷ ¹¹⁄₈ = 7.
36. There must be nine things of each sort, in order that ^99999999999999999999^ different selections may be made from twenty sorts of things.
37. The women who had respectively 33, 29, and 27 apples, and sold the same number for a penny, receiving an equal amount of money, began by selling at the rate of three a penny. The first sold ten pennyworth, the second eight pennyworth, and the third seven pennyworth.
The first had then left three apples, the second five, and the third six. These they sold at one penny each, so that they received on the whole--
The first 10d. + 3d. = 13d. The second 8d. + 5d. = 13d. The third 7d. + 6d. = 13d.
38. The puzzle--
Take five from five, oh, that is mean! Take five from seven, and this is seen--
is solved by _fie_, _seen_.
39. If a bun and a half cost three halfpence, it is plain that each bun costs a penny, but, by general custom, you buy seven for sixpence.
40. The hands of a watch would meet each other twenty-five times in a day, if the minute-hand moved backwards and the hour-hand forwards. They are, of course, together at starting.
41. The only way in which half-a-crown can be equally divided between two fathers and two sons, so that a penny is the smallest coin made use of, is to give tenpence each to a grandfather, his son, and his grandson.
42. If the number of the revolutions of a bicycle wheel in six seconds is equal to the number of miles an hour at which it is running, the circumference of the wheel is 8⁴⁄₅ feet.
43. The hour that struck was twelve o’clock.
44. Sixty years.
45. If I jump off a table with a 20lb dumb-bell in my hand there is no pressure upon me from its weight while I am in the air.
46. If at a bazaar I paid a shilling on entering each of four tents, and another shilling on leaving it, and spent in each tent half of what was in my pocket, and if my fourth payment on leaving took my last shilling, I started with 45s., spending 22s. in tent 1, 10s. in tent 2, 4s. in tent 3, and 1s. in tent 4, having also paid to the doorkeepers 8s.
47. When rain is falling vertically at 5 miles an hour, and I am walking through it at 4 miles an hour, the rain drops will strike the top of my umbrella at right angles if I hold it at an angle of nearly 39 degrees.
As I walk along, meeting the rain, the effect is the same as it would be if I was standing still, and the wind was blowing the rain towards me at the rate of 4 miles an hour.
48. When one monkey descends from the top of a tree 100 cubits high, and makes its way to a well 200 yards distant, while another monkey, leaping upwards from the top, descends by the hypotenuse to the well, both passing over an equal space, the second monkey springs 50 cubits into the air.
49. The steamboat which springs a leak 105 miles east of Tynemouth Lighthouse, and, putting back, goes at the rate of 10 miles an hour the first hour, but loses ground to the extent in each succeeding hour of one-tenth of her speed in the previous hour, never reaches the lighthouse, but goes down 5 miles short of it.
50. Twenty-one hens will lay ninety-eight eggs in a week, if a hen and a-half lays an egg and a-half in a day and a-half. Evidently one egg is laid in a day by a hen and a-half, that is to say three hens lay two eggs in a day. Therefore, twenty-one hens lay fourteen eggs, in a day, or ninety-eight in a week.
Q. E. D. (Quite easily done!)
51. If the population of Bristol exceeds by 237 the number of hairs on the head of anyone of its inhabitants that are not bald, at least 474 of them must have the same number of hairs on their heads.
52. In tipping his nephew from seven different coins, the uncle may give or retain each, thus disposing of it in two ways, or of all in 2 × 2 × 2 × 2 × 2 × 2 × 2 ways. But as one of these ways would be to retain them all, there are not 128, but only 127 possible variations of the tip.
53. The prime number which fulfils the various conditions of the question is 127. Increased by one-third, excluding fractions, it becomes 169, the square of 13. If its first two figures are transposed, and it is increased by one-third, it becomes 289, the square of 17. If its first figure is put last, and it is increased by one-third, it becomes 361, the square of 19. If, finally, its three figures are transposed, and then increased by one-third, it becomes 961, the square of 31.
54. Six things can be divided between two boys in 62 ways. They could be _carried_ by two boys in 64 ways (2 × 2 × 2 × 2 × 2 × 2), but they are not _divided_ between two boys if all are given to one, so that two of the 64 ways must be rejected.
55. The highest possible score that the dealer can make at six cribbage, if he is allowed to select the cards, and to determine the order of play, is 78. The dealer and his opponent must each hold 3, 3, 4, 4, the turn-up must be a 5, and crib must have the knave of the suit turned up, and 5, 5, 5. It will amuse many of our readers to test this with the cards.
56. The picture frame must be 3 inches in width all round, if it is exactly to equal in area the picture it contains, which measures 18 inches by 12 inches.
57. If my mother was 20 when I was born, my sister is two years my junior, and my brother is four years younger still, our ages are 56, 36, 34, and 30.
58. The spider in the dockyard, whose thread was drawn from her by a revolving capstan 1 foot in diameter, until 73 feet of it were paid out, after walking for a mile round and round the capstan at the end of the stretched thread in an effort to unwind it all, had, when she stopped in her spiral course, 49 more feet to walk to complete her task.
59. The mountebank at a fair, who offered to return any stake a hundredfold to anyone who could turn up all the sequence in twenty throws of dice marked each on one face only with 1, 2, 3, 4, 5, or 6, should in fairness have engaged to return 2332 times the money; for of the 46,656 possible combinations of the faces of the dice, only one can give the six marked faces uppermost. Thus the chance of throwing them all at one throw is expressed by ¹⁄₄₆₆₅₆, and in twenty throws by about ¹⁄₂₃₃₂.
60. If 90 groats (each = 4d.) feed twenty cats for three weeks, and five cats consume as much as three dogs, seventy-two hounds can be fed for £39 in a period of ninety-one days.
61. When equal wine-glasses, a half and a third full of wine, are filled up with water, and their contents are mixed, and one wine-glass is filled with the mixture, it contains ⁵⁄₁₂ wine and ⁷⁄₁₂ water.
62. The arrangement by which St Peter is said to have secured safety for the fifteen Christians, when half of the vessel’s passengers were thrown overboard in a storm, is as follows:--
XXXXIIIIIXXIXXXIXIIXXIIIXIIXXI
Each Christian is represented by an X, and if every ninth man is taken until fifteen have been selected, no X becomes a victim.
63. If Farmer Southdown’s cow had a fine calf every year, and each of these, and their calves in their turn, at two years old followed this example, the result would be no less than 2584 head in sixteen years.
64. The number of the flock was 301. This is found by first taking the least common multiple of 2, 3, 4, 5, 6, which is 60, and then finding the lowest multiple of this, which with 1 added is divisible by 7. This 301 is exactly divisible by 7, but by the smaller numbers there is 1 as remainder.
65. The rule for determining easily the number of round bullets in a flat pyramid, with a base line of any length, is this:--
Add a half to half the number on the base line, and multiply the result by the number on that line. Thus, if there are twelve bullets as a foundation--
12 + ¹⁄₂ = ¹³⁄₂; and ¹³⁄₂ × ¹²⁄₁ = 78.
The same result is reached by multiplying the number on the base line by a number larger by one, and then halving the result. Thus--
12 × 13 = 156, 156 ÷ 2 = 78.
66. We can gather from the lines--
Old General Host A battle lost, And reckoned on a hissing, When he saw plain What men were slain, And prisoners, and missing.
To his dismay He learned next day What havoc war had wrought; He had, at most, But half his host Plus ten times three, six, ought.
One-eighth were lain On beds of pain, With hundreds six beside; One-fifth were dead, Captives, or fled, Lost in grim warfare’s tide.
Now, if you can, Tell me, my man, What troops the general numbered, When on that night Before the fight The deadly cannon slumbered?
that old General Host had an army 24,000 strong.
67. When the farmer sent five pieces of chain of 3 links each, to be made into one continuous length, agreeing to pay a penny for each link cut, and a penny for each link joined, the blacksmith, if he worked in the best interest of the farmer, could only charge sixpence: for he could cut asunder one set of 3 links, and use these three single links between the other four sets.
68. If, in a parcel of old silver and copper coins, each silver piece is worth as many pence as there are copper coins, and each copper coin is worth as many pence as there are silver coins, there are eighteen silver and six copper coins, when the whole parcel is worth eighteen shillings.
69. These are five groups that can be arranged with the numbers 1 to 11 inclusive, so that they are all equal:--
(8² - 5² + 1) = (11² - 9²) = (7² - 3²) = (6² + 2²) = 4(10).
70. John Bull, under the conditions given, lived to the age of eighty-four years.
71. The two numbers to each of which, or to the halves of which, unity is added, forming in every case a square number, are 48 and 1680.
72. The true weight of a cheese that seemed to weigh 16 ℔s. in one scale of a balance with arms of unequal length, and only 9℔s. in the other, is 12℔. This is found by multiplying the 16 by the 9, and finding the square root of the result.
73. The two parts into which 100 can be divided, so that if one of them is divided by the other the quotient is again exactly 100 are 99¹⁄₁₀₁ and ¹⁰⁰⁄₁₀₁.
74. If, with marbles in two pockets, I add one to those in that on the right, and then multiply its contents by the number it held at first, and after dealing in a similar way with those on the left, find the difference between the two results to be 90; while if I multiply the sum of the two original quantities by the square of their difference the result is 176, I started with twenty-three in the right-hand pocket and twenty-one in the other.
75. The circle of twenty-one friends who arranged to meet each week five at a time for Bridge so long as exactly the same party did not meet more than once, and who wished to hire a central room for this purpose, would need it for no less than 20,349 weeks, or more than 390 years, to carry out their plan.
76. If a herring and a half costs (not cost) a penny and a half, the price of a dozen such quantities is eighteenpence.
77. The sum of money which in a sense appears to be the double of itself is 1s. 10d., for we may write it _one_ and _ten_ pence or _two_ and _twenty_ pence.
78. The “comic arithmetic” question set by Dr Bulbous Roots--
Divide my fifth by my first, and you have my fourth; subtract my first from my fifth, and you have my second; multiply my first by my fourth followed by my second, and you have my third; place my second after my first, and you have my third multiplied by my fourth--is solved by COMIC.
79. If the earth could stand still, and a straight tunnel could be bored through it, a cannon ball dropped into it, if there is no air or other source of friction, would oscillate continually from end to end.
Taking air into account, the ball would fall short of the opposite end at its first lap, and in succeeding laps its path would become shorter and shorter, until its initial energy was exhausted, when it would come to rest at the centre.
80. He sent 163. She sent 157.
81. When twins were born the estate was properly divided thus:--
Taking the daughter’s share as 1 The widow’s share would be 2 And the son’s share 4 - Total 7 shares.
So the son takes four-sevenths, the widow two-sevenths, and the daughter one-seventh of the estate.
82. If each of my strides forwards or backwards across a 22 feet carpet is 2 feet, and I make a stride every second; and if I take three strides forwards and two backwards until I cross the carpet, I reach the end of it in forty-three seconds. In three steps I advance 6 feet. Then in two steps I retrace 4 feet, thus gaining only 2 feet in five steps, _i.e._, in five seconds. I therefore advance 16 feet in forty seconds, and three more strides cover the remaining 6 feet.
83. If the captain of a vessel chartered to sail from Lisbon to New York, which appear on a map of the world to be on the same parallel of latitude, and which are, along the parallel, about 3600 miles apart, takes his ship along this parallel, he will not be doing his best for the impatient merchant who has had an urgent business call to New York.
The shortest course between the two points is traced by a segment of a “great circle,” having its centre at the centre of the earth, and touching the two points. This segment lies wholly north of the parallel, and is the shortest possible course.
84. When John and Harry, starting from the right angle of a triangular field, run along its sides, and meet first in the middle of the opposite side, and again 32 yards from their starting point, if John’s speed is to Harry’s as 13 to 11, the sides of the field measure 384 yards.
85. If two sorts of wine when mixed in a flagon in equal parts cost 15d., but when mixed so that there are two parts of _A_ to three of _B_ cost 14d., a flagon of _A_ would cost 20d., and a flagon of _B_ 10d.
86. If, when a man met a beggar, he gave him half of his loose cash and a shilling, and meeting another gave him half what was left and two shillings, and to a third half the remainder and three shillings, he had two guineas at first.
87. The clerk who has two offers of work from January 1, one from _A_ of £100 a year, with an annual rise of £20, and the other from _B_ of £100 a year, with a half-yearly rise of £5, should accept _B_’s offer.
The half-yearly payments from _A_ (allowing for the rise), would be 50, 50, 60, 60, 70, 70, etc., etc.; and from _B_ they would be 50, 55, 60, 65, 70, 75, etc., etc., so that _B_’s offer is worth £5 a year more than _A_’s always.
88. If I have a number of florins and half-crowns, but no other coins, I can pay my tailor £11, 10s. in 224 different ways.
This can be found thus by rule of thumb: Start with 0 half-crowns and 115 florins. Then 4 half-crowns and 110 florins. Add 4 half-crowns and deduct 5 florins each time till 92 half-crowns and 0 florins is reached.
89. The monkey climbing a greased pole, 60 feet high, who ascended 3 feet, and slipped back 2 feet in alternate seconds, reached the top in 1 minute, 55 seconds, for he did not slip back from the top.
90. When Adze, the carpenter, secured his tool-chest with a puzzle lock of six revolving rings, each engraved with twelve different letters, the chances against any one discovering the secret word formed by a letter on each ring was 2,985,983 to 1; for the seventy-two letters may be placed in 2,985,984 different arrangements, only one of which is the key.
91. The five married couples who arranged to dine together in Switzerland at a round table, with the ladies always in the same places, so long as the men could seat themselves each between two ladies, but never next to his own wife, were able under these conditions to enjoy thirteen of these nights at the round table.
92. If in a calm the tip of a rush is 9 inches above the surface of a lake, and as the wind rises it is gradually blown aslant, until at the distance of a yard it is submerged, it is growing in water that is 5 feet 7¹⁄₂ inches deep.
93. Aminta was eighteen.
94. When Dick took a quarter of the bag of nuts, and gave the one over to the parrot, and Tom and Jack and Harry dealt in the same way with the remainders in their turns, each finding a nut over from the reduced shares for the bird, and one was again over when they divided the final remainder equally, there were, at the lowest estimate, 1021 nuts in the bag.
95. Eight and a quarter is the answer to the nonsense question--
If five times four are thirty-three, What will the fourth of twenty be?
96. The similar fraction of a pound, a shilling, and a penny which make up exactly a pound are as follows:--
s. d. ²⁴⁰⁄₂₅₃ of £1 = 18 11¹⁶⁹⁄₂₅₃ ²⁴⁰⁄₂₅₃ of 1s. = 11⁹⁷⁄₂₅₃ ²⁴⁰⁄₂₅₃ of 1d. = ²⁴⁰⁄₂₅₃ --------------- £1 0 0 ========
97. When Dr Tripos thought of a number, added 3, divided by 2, added 8, multiplied by 2, subtracted 2, and thus arrived at double the number, he started with 17.
98. When _A_ and _B_ deposited equal stakes with _C_, and agreed that the one who should first win three games of billiards should take all, but consented to a division in proper shares when _A_ had won two games and _B_ one, it was evident that if _A_ won the next game all would go to him, while if he lost he would be entitled to one half. One case was as probable as the other, therefore he was entitled to _half of these sums taken together_; that is, to three quarters of the stakes, and _B_ to a quarter only.
99. The average speed of a motor which runs over any course at 10 miles an hour, and returns over the same course at 15 miles an hour, is 12 miles an hour, and not 12¹⁄₂, as might be imagined. Thus a run of 60 miles out takes, under the conditions, six hours, and the return takes four hours; so that the double journey of 120 miles is done in ten hours, at an average speed of 12 miles an hour.
100. Farmer Hodge, who proposed to divide his sheep into two unequal parts, so that the larger part added to the square of the smaller part should equal the smaller part added to the square of the larger part, had but one sheep.
Faithful to his word, he divided this sheep into two unequal parts, ²⁄₃ and ¹⁄₃, and was able to show that ²⁄₃ + ¹⁄₉ = ⁷⁄₉, and that ¹⁄₃ + ⁴⁄₉ = ⁷⁄₉. He was heard to declare further, and he was absolutely right, that _no number larger than_ 1 can be so divided as to satisfy the conditions which he had laid down.
The fact that _sheep_ is both singular and plural, adds much to the perplexing points of this attractive problem.
Here is a very simple proof that the number _must be_ 1:--
Let a + b = no. of sheep
then a² + b = b² + a
a² - b² = a - b
or (a + b)(a - b) = a - b
therefore a + b = 1.
101. A horse that carries a load can draw a greater weight _up the shaft of a mine_ than a horse that bears no burden. The load holds him more firmly to the ground, and thus gives him greater power over the weight he is raising from below.
102. In the six chests, of which two contained pence, two shillings, and two pounds, there must have been at least the value of 506 pence. This can be divided into 22 (or 19 + 3) shares of 23d. each, or 23 (19 + 4) shares of 22d. each. Evidently then the treasure can be divided so that 19 men have equal shares, while their captain has either 3 shares or 4 shares.
103. If I bought a parcel of nuts at 49 for 2d., and divided it into two equal parts, one of which I sold at 24, the other at 25 a penny; and if I spent and received an integral number of pence, but bought the least possible number of nuts, I bought 58,800 nuts, at a cost of £10, and I gained a penny.
104. When, with a purse containing sovereigns and shillings, after spending half of its contents, I found as many pounds left as I had shillings at first, I started with £13, 6s.
105. When the lady replied to a question as to her age--
If first my age is multiplied by three, And then of that two-sevenths tripled be, The square root of two-ninths of this is four; Now tell my age, or never see me more--
she was 28 years old.
106. If cars run, at uniform speed, from Shepherd’s Bush to the Bank, at intervals of two minutes, and I am travelling at the same rate in the opposite direction, I shall meet 30 in half-an-hour, for there are already 15 on the track approaching me, and 15 are started from the other end during my half hour’s course.
107. If it was possible to carry out my offer of a farthing for every different group of apples which my greengrocer could select from a basket of 100 apples, he would be entitled to the stupendous sum of £18,031,572,350 19s. 2d.
108. If the minute-hand of a clock moves round between 3 and 4 in the opposite direction to the hour-hand, the hands will be exactly together when it is really 41⁷⁄₁₃ minutes past 3.
109. If the walnut monkey had stopped to help the other, and they had eaten filberts at equal rates, they would have escaped in 2¹⁄₄ minutes.
110. The value of the cheque, for which the cashier paid by mistake pounds for shillings, was £5, 11s. 6d. The receiver to whom £11, 5s. 6d. was handed, spent half-a-crown, and then found that he had left £11, 3s., just twice the amount of the original cheque.
111. The number 14 can be made up by adding together five uneven figures thus:--11 + 1 + 1 + 1. It will be seen that although only four _numbers_ are used, 11 is made up of _two figures_.
Here is another, and quite a curious solution, 1 + 1 + 1 + 1 = 4, and with another 1 we can make up 14!
112. A business manager can fill up three vacant posts of varying value from seven applicants in 210 different ways. For the first post there would be a choice among 7, for the second among 6, and for the third among 5, so that the possible variations would amount to 7 × 6 × 5 = 210.
113. If the fasting man, who began his task at noon, said it is now ⁵⁄₁₁ of the time to midnight, he spoke at 3.45 p.m., meaning that ⁵⁄₁₁ of the remaining time till midnight had elapsed since noon.
114. If a clock takes six seconds to strike 6, it will take 12 seconds to strike 11, for there must be ten intervals of 1¹⁄₁₅ seconds each.
115. Twenty horses can be arranged in three stalls, so that there is an odd number in each, by placing one in the first stall, three in the second, and sixteen (an odd number to put into any stall!) in the third.
116. The little problem, “Given _a_, _b_, _c_, to find _q_,” is solved, without recourse to algebra, thus: _a_, _b_, _c_, = _c_, _a_, _b_; take a cab and go over Kew Bridge, and you find a phonetic _Q_!
117. Tom Evergreen was 75 years old when he was asked his age by some men at his club in 1875, and said--“The number of months that I have lived are exactly half as many as the number which denotes the year in which I was born.”
118. Eight different circles can be drawn. A circle can have one of the three inside and two outside in three ways, or one outside and three inside in three ways (each of the three being inside or outside in turn), or all three may be inside, or all three may be outside, the touching circle.
119. The way to arrange 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, so that used once each they form a sum which is equal to 1 is this:--
35 148 -- + --- = 1. 70 296
120. The sum of the first fifty numbers may be found without any addition thus:--The first fifty numbers form twenty-five pairs of fifty-one each (1 + 50, 2 + 49, etc., etc.), and 51 × 25 is practically 51 × 100 ÷ 4 = 1275.
121. The tramcar _A_, which started at the same time as _B_, but ran into a “lie by” in four minutes, and waited there five minutes till _B_ came along, when they completed their courses at the same moment in opposite directions, could have run the whole distance in ten minutes.
122. What remains will be 8 if we take 10 and double it by writing one 10 over another so as to form 18, and then deduct 10.
123. If the average weight of the Oxford crew is increased by 2℔s., when one of them who weighs 12 stone, is replaced by a fresh man, the weight of that substitute is 13 stone 2℔s.
124. If a motor-car is twice as old as its tyres were when it was old as its tyres are, and if, when these tyres are as old as the car itself is now, their united ages will be 2¹⁄₄ years, the car is now 12 months old, and the tyres have had 9 months’ wear.
125. _A_ and _B_, who could each carry provisions for himself for twelve days, started to penetrate as far as possible into a desert, on the understanding that neither of them should miss a day’s food. After an advance of four days, each had provisions still for eight days. One gave four portions of his store to his companion, which did not overload him, and returned with the other four. His comrade was then able to advance another four days’ journey, and still have rations for the eight days’ return. Thus the furthest possible penetration into the desert under the conditions was an eight days’ march.
126. If, when a bottle of medicine and its cork cost half-a-crown, the bottle and the medicine cost two and a penny more than the cork, the cork cost twopence half-penny.
127. A boat’s crew far from land, with no sail or oars, and with no assistance from wind or stream, or outside help of any kind, can regain the shore by means of a coil of rope. Motion is given to the boat by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. This curious illustration of mechanical principles is from “Ball’s Mechanical Recreations.” (_Macmillan._)
128. It will be found that after a crown and as many four-shilling pieces as possible have been crammed into our pockets, there would still be room for one sixpence and one threepenny-piece in some corner or cranny. We can, therefore, have one crown, one sixpence, one threepenny-piece, and as many four-shilling pieces as our pockets will hold, and yet be unable to give change for a half-sovereign.
129. There were fifteen apples in the basket. Half of these and half an apple, _i.e._, eight were first given, then half the remainder and half an apple, _i.e._, four, then on similar lines two, leaving one in the basket.
130. The Queer Division--
A third of twelve divide By just a fifth of seven; And you will soon decide That this must give eleven--
is solved by LV ÷ V, or 55 ÷ 5 = 11.
131. A motor that goes 9 miles an hour uphill, 18 miles an hour downhill, and 12 miles an hour on the level, will take 8 hours and 20 minutes to run 50 miles out and return at once over the same course.
132. The number of shots fired at a mark was 420 each by _A_, _B_, and _C_. _A_ made 280 hits, _B_ 315, and _C_ 336.
133. If a dog and a cat, evenly matched in speed, run a race out and back over a course of 75 yards in all, and the dog always takes 5 feet at a bound, and the cat 3 feet, the cat will win, because at the turning point the dog overleaps the half distance more than the cat does, and so has a longer run in.
134. When a man caught up a wagon going at 3 miles an hour, which was just visible to him in a fog at a distance of 55 yards, and which he saw for five minutes before reaching it, he was walking at the rate of 3³⁄₈ miles an hour.
135. Three horses, _A B C_, can be placed after a race in thirteen different ways, thus:--_A B C_, _A C B_, _B A C_, _B C A_, _C A B_, _C B A_, or _A B C_ as a dead heat; or _A B_, _A C_, or _B C_ equal for the first place; or _A_ first with _B C_ equal seconds; or _B_ first with _A C_ equal seconds; or _C_ first with _A B_ equal seconds.
136. The 34 points scored against Oxbridge by the New Zealanders can be made up in two ways, either by 8 tries and 2 converted tries, or by 3 tries and 5 converted tries.
The highest possible score on these lines is 10 tries converted, equalling 50 points, and as the New Zealanders’ score, if all tries are converted, becomes four-fifths of this, their actual score was 3 tries and 5 converted into goals.
137. The smallest number, of which the alternate figures are cyphers, which is divisible by 9 and by 11 is 909090909090909090909!
138. Our problem in which it is stated that _A_ with 8d. met _B_ and _C_ with five and three loaves, and asked how the cash should be divided between _B_ and _C_, if all agreed to share the loaves. Now each eats two loaves and two-thirds of a loaf, and _B_ gives seven-thirds of a loaf to _A_, while _C_ gives him one-third of a loaf. So _B_ receives 7d. and _C_ 1d.
139. When, on opening four money-boxes containing pennies only, it was found that those in the first with half of all the rest, those in the second with a third of the others, those in the third with a fourth, and those in the fourth with a fifth of all the rest, amounted in each case to 740, the four boxes held £6. 1s. 8d., and the numbers of pennies were 20, 380, 500, and 560.
140. If two steamers, _A_ and _B_, start together for a trip to a distant buoy and back, and _A_ steams all the time at ten knots an hour, while _B_ goes outward at eight knots and returns at twelve knots an hour, _B_ will regain port later than _A_, because its loss on the outward course will not have been recovered on the run home.
141. If in London a new head to a golf club costs four times as much as a new leather face, while at St Andrews it costs five times as much, and if the leather face costs twice as much in London as in St Andrews, and if, including a shilling paid for a ball, the charges in London were twice as much as they would have been at St Andrews, the London cost of a new head is four shillings, and of a leather face a shilling.
142. When two children were asked to give the total number of sheep and cattle in a pasture, from the number of each sort, and one by subtraction answered 10, while the other arrived at 11,900 by multiplication, the true numbers were 170 sheep, 70 cattle, 240 in all.
143. If a man picks up one by one fifty-two stones, placed at such intervals on a straight road that the second is a yard from the first, the third 3 yards from the second, and so on with intervals increasing each time by 2 yards, and bring them all to a basket placed at the first stone, he has to travel about 52 miles, or, to be quite exact, 51 miles, 1292 yards.
144. When, in the House of Commons, if the Ayes had been increased by 50 from the Noes, the motion would have been carried by 5 to 3; and if the Noes had taken 60 votes from the Ayes it would have been lost by 4 to 3, the motion succeeded; 300 voted “Aye,” and 260 “No.”
145. There are 143 positions on the face of a watch in which the places of the hour and minute-hands can be interchanged, and still indicate a possible time. There would be 144 such positions but for the fact that at twelve o’clock the hands occupy the same place.
146. If in a cricket match the scores in each successive innings are a quarter less than in the preceding innings, and the side which goes in first wins by 50 runs, the complete scores of the winners are 128 and 72, and of the losers 96 and 54.
147. When a ball is thrown vertically upwards, and caught five seconds later, it has risen 100 feet. It takes the same time to rise as to fall, and when a body falls from rest, it travels a number of feet represented by sixteen times the square of the time in seconds.
Hence comes the rule that the height in feet of a vertical throw is found by squaring the time in seconds of its flight, and multiplying by four.
148. The carpet which, had it been 5 feet broader and 4 feet longer, would have contained 116 more feet, and if 4 feet broader and 5 longer 113 more, was 12 feet long and 9 feet broad.
149. When, in estimating the cost of a hundred similar articles, shillings were read as pounds, and pence as shillings, and the estimated cost was in consequence £212, 18s. 4d. in excess of the real cost, the true cost of each article was 2s. 5d.
150. If the square of the number of my house is equal to the difference of the squares of the numbers of my next door neighbours’ houses, and if my brother in the next street can say the same of his house, though its number is not the same as that of mine, our houses are numbered 8 and 4. In my street the even numbers are all on one side, in my brother’s street they, run odd and even consecutively, and so 8² = 10² - 6², and 4² = 5² - 3².
151. When two men of unequal strength have to move a block which weighs 270 ℔s., on a light plank 6 feet long, if the stronger man can carry 180 ℔s., the block must be placed 2 feet from him, so that he may have that share of the load.
152. If a man who had twenty coins, some shillings, and the rest half-crowns, were to change the half-crowns for sixpences, and the shillings for pence, and then found that he had 156 coins, he must have had eight shillings at first.
153. If, when coins are placed on a table at equal distances apart, so as to form sides of an equilateral triangle, and when as many are taken from the middle of each side as equal the square root of the number on that side, and placed on the opposite corner, the number on each side is then to the original number as five is to four, there are forty-five coins in all.
154. When the gardener found that he would have 150 too few if he set his posts a foot apart, and seventy to spare if he set them at every yard, he had 180 posts.
155. In order to buy with £100 a hundred animals, cows at £5, sheep at £1, and geese at 1s. each, the purchaser must secure nineteen cows, one sheep, and eighty geese.
156. If John, who is 21, is twice as old as Mary was when he was as old as Mary is, Mary’s age now is 15³⁄₄ years.
157. If in a cricket match _A_ makes 35 runs, and _C_ and _D_ make respectively half and a third of _B_’s score, and if _B_ scores as many less than _A_ as _C_ scores more than _D_, _B_ made 30, _C_ 15, and _D_ 10 runs.
158. The least number which, divided by 2, 3, 4, 5, 6, 7, 8, 9, or 10, leaves remainders 1, 2, 3, 4, 5, 6, 7, 8, 9, is 2519, their least common multiple less 1.
159. A square table standing on four legs, which are set at the middle points of its sides, can at most uphold its own weight upon one of its corners.
160. The division of ninety-nine pennies, so that share 1 exceeds share 2 by 3, is less than share 3 by 10, exceeds share 4 by 9, and is less than share 5 by 16, is 17, 14, 27, 8, and 33.
161. If Indians carried off a third of a flock and a third of a sheep, and others took a fourth of the remainder and a fourth of a sheep, and others a fifth of the rest and three-fifths of a sheep, and there were then 409 left, the full flock was 1025 sheep.
162. When a cistern which held fifty-three gallons was filled by three boys, _A_ bringing a pint every three minutes, _B_ a quart every five minutes, and _C_ a gallon every seven minutes, it took 230 minutes to fill it, and _B_ poured in the final quart, _A_ and _C_ _coming up one minute too late_ to contribute at the last.
163. A man who said, late in the last century, that his age then was the square root of the year in which he was born, was speaking in the year 1892.
164. If when a dealer in curios sold a vase for £119, his profit per cent., and the cost price of the vase, were expressed by the same number, it had cost him £70.
165. The chance of throwing at least one ace in a single throw with a pair of dice is ¹¹⁄₃₆, for there are five ways in which each dice can be thrown so as not to give an ace, so that twenty-five possible throws exclude aces. Hence the chance of _not_ throwing an ace is ²⁵⁄₃₆, which leaves ¹¹⁄₃₆ in favour of it.
166. The policeman who ran after a thief starting four minutes later, and running one-third faster, if they both ran straight along the road, caught him in twelve minutes.
167. At a bazaar stall, where twenty-seven articles are exposed for sale, a purchaser may buy one thing or more, and the number of choices open to him is one less than the continued product of twenty-seven twos, or 134217727.
VERY PERSONAL
168. When Nellie’s father said:--
I was twice as old as you are The day that you were born. You will be just what I was then When fourteen years are gone--
he was 42, and she was 14.
WORD AND LETTER PUZZLES
1. The word square is completed thus:--
MEAD EDGE AGUE DEED
2. The word square filled in is:--
CIRCLE INURES RUDEST CREASE LESSEE ESTEEM
_Notice the curious diagonal of E’s._
3. In the incomplete sentence,
SO****AG****LATI****X****ITH
the duplicate letters are filled in thus:--
SOME MEAGRE RELATIVE VEXED EDITH
The two last letters of each word are repeated as the two first of the word that follows.
4. The word square is completed thus:--
WASTE ACTOR STONE TONIC ERECT
5. By twice building up two Ds into a B we make BULBOUS.
6. The question put on paper to the love-lorn youth, “Loruve?” is, when interpreted, “Are you in love?” and the advice given to him on another slip, “Prove
L A F D
and ensure success,” reads into, “Prove a fond lover, and ensure success” (a f _on_ d l _over_).
7. DOUBLE ACROSTIC ST. JAMES’ GAZETTE
S . . . prin . . . G T . . . ar . . . A J . . . abe . . . Z A . . . t . . . E M . . . omen . . . T E . . . mme . . . T S . . . trik . . . E
8. The completed word square is--
AMENDS MINION ENABLE NIBBLE DOLLAR SNEERS
9. The oracular response to a young Frenchman at a _fête_, who inquired how he could best please the ladies--
MEC DO BIC
conceals this sage advice--
Aimez, cédez, obéissez! Love, yield, obey!
10. The solution to our Letter Fraction Problem is of a verbal character. The original statement 15844
m -- ot = mo -- y
is dealt with thus:--
m _on_ ot _on_ y = mo _not on_ y, and so the word monotony solves the equation.
11. The buried beasts are _chamois_, _buffalo_, _heifer_, and _leopard_; and when the Oxford athlete cries--
“Though I jump and row and run, Cap or cup I never won”
he introduces us to a _porcupine_.
12. The lines--
Fourteen letters here we fix, Vowels only two are spoken; All together these we mix Into what can not be broken--
is solved by _indivisibility_, which has many an _i_, like a peacock’s tail.
13. The English word of thirteen letters in which the same vowel occurs four times, the same consonant six times, another twice, and another once, is _Senselessness_.
14. The condensed proverb “WE IS DO” reads at its full length as “Well begun is half done.”
15. This is the completed word square:--
WASHES ARTERY STORMS HERMIT ERMINE SYSTEM
16. Dr Whewell’s puzzle lines--
O O N O O. ---------- U O A O O I O U, O N O O O O M E T O O. U O A O I D O S O I O N O O I O U T O O!
read thus:--
OH SIGH FOR NO CIPHER
You sigh for a cipher, O, I sigh for you, Sigh for no cipher, O sigh for me too. You sigh for a cipher, I decipher so, I sigh for no cipher, I sigh for you too!
17. This is the completed diamond:--
P POR CORES FORCEPS PORCELAIN REFLECT SPACE SIT N
18. The medley--
Tan HE Edsa VEN in It N Gja SmeTs AsgN aD Az Rett De
is read by taking first the capitals in their order, and then the small type. It comes put out as “The Evening Standard and Saint James’s Gazette.”
19. The statement, I can travel first-class on the G.E.R. from 2222222244444500, reads into--from 22 to 2 to 22 to 4 for 44 4d; or, in plain terms, from 1.38 to 3.38 for 14s. 8d. This works out at about 3d. a mile, the usual allowance for first-class, for two hours, at about 29 miles an hour.
20. A CURIOUS OLD INSCRIPTION
Read the inscription backwards, and it resolves itself into the lines familiar to us in our childhood:--
Ride a cock horse to Banbury Cross To see a fine lady ride on a grey horse. Rings on her fingers and bells on her toes, She shall have music wherever she goes!
21. The reason why the invitation to Jack to sample the Irish stew at Simpson’s was to be kept in mind by the catch words--
Join me at and i s,
is because if you join _me_ and _at_, and note that _and_ is _on i_, which is _on s_, you arrive at the suggestive sentence--_meat and onions_.
22. The labourer’s quaint letter, which ran “Cepatomtogoatatrin,” was, in plainer English, “Kept at home to go a tatering.”
23. Our double acrostic comes out thus:--
PROBLEM--PUZZLES
P ........a....... P (T) R ................ U (E) O ................ Z B ........o....... Z L .......eve...... L E ........v....... E M ................ S
24. The Hidden Proverb is--
“Necessity is the mother of invention.”
25. The “deed done” in our Will puzzle is the making in Roman numerals of “Codicil.” The lawyer was to set down, a hundred, to add nothing, to set down five hundred, then one, then another hundred, and then one more, and, finally, fifty, and accordingly he wrote upon the parchment the one word CODICIL.
26. The word square is--
EDITOR DESIRE ISLAND TIARAS ORNATE REDSEA
27. The notes of music A. G. A. E. A. over the grave of a French musician, who was choked by a fish bone, are in the French notation, “La sol la mi la,” which reads into “La sole l’a mis 1à.”
Similarly the inscription over the porch of Gustave Doré’s house C. E. B. A. C. D. is equivalent to “Do, mi, si, la, do, re,” which may be taken to represent “Domicile à Doré.”
28. When his best girl said to Jack Spooner, “We can go to-morrow at 222222222222 LEY STREET,” he understood her to mean, “We can go to-morrow, at two minutes to two, to two twenty-two, to 222 Tooley Street.”
29
C ---------- spells _contents_ (c on ten ts!). TTTTTTTTTT
30. We can treat the word _disused_ so as to affirm or to disallow the use of its initial or final d, for we can write it _d is used_, or _disuse d_!
31. The title of the book shaken up into
EIOOOU BCNNRRSS
is “Robinson Crusoe.”
32. If the letter M only is inserted in the proper places in the line--
A DEN I I CAN DOCK
it will read: _Madmen mimic and mock_.
33. The quotation from Shakespeare--
OXXU8 MAAULGIHCTE NOR
is by interpretation:--“Nothing extenuate, nor set down aught in malice.”
34. The phonetic nightmare--
Ieukngheaurrhphthewempeighghteaps--
is merely the word _unfortunates_. It can be justified thus by English spelling of similar sound taken letter by letter:--
u--iew in view; n--kn in know; f--gh in tough; o--eau in beau; r--rrh in myrrh; t--phth in phthisis; u--ewe; n--mp in comptroller; a--eigh in neigh; t--ght in light; e--ea in tea; and s--ps in psalm.
35. The word square is completed thus:--
FARM AREA RENT MATE
36. A QUAINT INSCRIPTION
The millers leave the mill, The wherrymen lower their sail; The maltsters leave the kiln, For a drop of the White Swan’s ale.
37. A POET’S PI
TONDEBNIOTOCHUMFOARYHUR OTDIRECTTHAWHOTERSOFKLSYA; TIKATESTUBALIGHTSTILLETRUFLYR OTBOWLALLNFESLEAVARFWYAA
is disentangled thus:--
Don’t be in too much of a hurry To credit what other folks say; It takes but a slight little flurry To blow fallen leaves far away.
38. BURIED PLACES
The five buried places are Deal, London, Esk, Perth, and Baden. The word is Ourangoutang.
39. It is no offence to conspire in the evening, because what is treasonable is _reasonable after t_!
40. The bit of botany--
Inscribe an _m_ above a line, And write an _e_ below, This woodland flower is hung so fine It bends when zephyrs blow--
is solved by,
m -, e
_an em on e_, the delicately hung wind-flower.
41. A PIED PROVERB
The pied proverb, is “A rolling stone gathers no moss.”
42. The Drop Letter Proverb--
E..t. .e.s..s .a.e .h. .o.. ..i.e, is--_Empty vessels make the most noise_.
43. The English word of five syllables, which has eight letters, five of them vowels--namely an a, an e, twice i, and y--is _Ideality_.
44. TORMENT may be turned into RAPTURE, using four links, changing only one letter each time, and varying the order of the letters, thus: TORMENT, _portent_, _protest_, _pratest_, _praters_, RAPTURE.
45. The pied sentence--
a a c e e e f f h h i i i i i m n n o o o p r r s s t t t t t
can be cast into the proverb--
“Procrastination is the thief of time.”
46. The English sentence, when the letter _o_ is added, reads:--
“Good old port for orthodox Oxford dons.”
47
One vowel in an English word is found, Which by eight consonants is hedged around--
is solved by _Strengths_.
48. The letters AAAAABBNNIIRSSTT form the word _Antisabbatarians_.
49. The quotation from Shakespeare,
KINI
stands for “A little more than kin, and less than kind.”
50. The two English words which have the first six letters of the alphabet among their ten letters are _fabricated_ and _bifurcated_.
51. SHIFTING NUMBERS
The letter _A_ stands at the head of the letters of the alphabet. For _bed_ 3 of these are used; for _goal_, 4; for _prison_, 6; for _six_, 3; for _three_, 5. The letter _A_ is not used in the spelling of the name of any number from 1 to 100, but it makes up, with the other vowels, the number 6.
52. The prodigal’s letter to his father, “Dear Dad, keep 1000050,” in reply to a suggestion for safeguarding some of his prospects, was written in playful impudence; and its interpretation is, “Dear Dad, keep cool!” for the figures in Roman numerals are COOL.
53. BURIED POETS
The poets’ names buried in the lines--
The sun is darting rays of gold Upon the moor, enchanting spot; Whose purpled heights, by Ronald loved, Up open to his Shepherd cot.
And sundry denizens of air Are flying, aye, each to his nest; And eager make at such an hour All haste to reach the mansions blest.
are Gray, Moore, Byron, Pope, Dryden, Gay, Keats, and Hemans.
54. When _A. B._ gave up the reins of government, and _C.B._ took office in his place, the two verbs, similar in all respects, except that the one is longer by one letter than the other, which expressed the change, were _resigns_--_reigns_.
55
Underdone mutton and onion make between you and me, A glutton a little seedy after a capital tea.
56
First a _c_ and _a t_, last _a c_ and a _t_, With a couple of letters between, Form a sight that our eyes are delighted to see, Unless in their sight it is seen--
is solved by _Cataract_. The first line reads, First a _c_ and _a t_, last _a c_ and a _t_, that is _cat_ and _act_.
57. Cuba.
58. The Rebus T S is solved by the words _tones_ and _tans_, _t_ before _one s_, or _t_ before _an s_.
59. The phonetic phrase--
INXINXIN--
is, _Ink sinks in_!
A GOOD END
60. Finis (F IN IS).
=