Part 9
Solvers may like to study the position given, which is one of many that are possible, and to discover for themselves the ruling conditions which are its characteristics.
No. XVII
When the boy’s father came up just in time to stop him from breaking out of bounds, and said, “Never throw a leg, lad,”
[Illustration]
the rest of the sentence, spelt with _exactly the same letters_, was “over the garden wall!”
No. XVIII
“Catastrophe,” the title of the tragedy foreshadowed, can be recast into “_A cat! stop her!_” By similar process the words, “New parrot stand in a house,” become “_He turns on a soda-water tap!_”
[Illustration]
The parrot’s ready resource and triumph is depicted here with striking effect.
No. XIX
When the judge at a baby show said to the mother of the small boy whose thumb was in his mouth, “Your lad Tommy likes such tit-bits,” the precocious child replied, as he removed his comforting hand, in a sentence spelt with _exactly the same letters_, “So to-day, sir, I suck my little thumb.”
[Illustration]
No. XX
When the lady sitting at the back of this overloaded wagonette said to her husband, “This big load quite hinders his pull,” in her sympathy with the struggling horse,
[Illustration]
he made this very practical reply, in a sentence spelt with _exactly the same letters_: “Do sit quiet, girl; I shall push behind!”
No. XXI
When a bystander whispered to the marker, “Eh! what a stout player is striking!”
[Illustration]
an appropriate reply, spelt with _exactly the same letters_, would have been: “He plays without taking a rest, sir.”
No. XXII
The two English words appropriate to this picture--
[Illustration]
which have as their anagrams “Or not a man first,” and “O I love nuts!” are _Transformation and Evolutions_.
No. XXIII
This is a fancy portrait of William--
[Illustration]
We decide by anagram whether this is _William or dear Jack_, for these words, when recast, spell “I am Will, a card joker!”
No. XXIV
The word indicated by this picture in combination with the lines below it--
[Illustration]
Begin with the end of my first, And then you will find out the rest; For it all will appeal to your thirst, Or point to a ponderous guest.
is _Stout_.
No. XXV
The words of Jigger’s wife, when she said that he seemed to be in a “sad pet,” were true by anagram.
[Illustration]
His ball hugs the cushion so closely as to be completely _pasted_.
No. XXVI
When, as they held on to the fractious cow, the farmer exclaimed, “See, we hold this cow’s horns and tail,”
[Illustration]
his foreman, using exactly the same letters in his sentence, said--
“She cannot toss, her wild head is low.”
No. XXVII
While the horse shown in this picture might be saying, if it could speak, “I’m a train’d stepper!”--
[Illustration]
the driver, from his point of view, might say, as he held him in check, “Spirit and a temper!” making use in his words of _exactly the same letters_.
No. XXVIII
When one onlooker, seeing the artist working with his feet, said--
“Why, now I see this fine artist has no hand!”
[Illustration]
the other replied, using exactly the same letters,
“He draws in any fashion with his ten toes!”
No. XXIX
When her husband, showing this picture
[Illustration]
said to his wife, “This is a wine bottle, dear, on a lure,” she, knowing that temptation in this form would fail, said, as she glanced at his illustration of their aims, in words spelt with exactly the same letters:--
“And see, he will not rise at our bait!”
No. XXX
The sturdy musician, who had said, “What shall I play?” to which some one replied, “Any strains of Beethoven, he charms all!” as this was not an acceptable suggestion, struck up a piece after his own heart.
[Illustration]
He said, as he struck the strings, in a sentence composed of exactly the same letters--“Nay, for this ’cello heaven sent a Brahms!”
No. XXXI
Here is the picture of a parsnip lying across a swede readjusted and reversed.
[Illustration]
We gave as a clue the anagram--
“Here is our parsnip on swede.”
ANAGRAM
_Wise and superior person he!_
but this is now hardly needed to show who is thus represented in friendly caricature: (With apologies to G. B. S.)
No. XXXII
The letter puzzle is solved thus--
L E V E L E E E E V V V E E E E L E V E L
Within this square the word LEVEL runs in twelve different directions, being itself a palindrome.
No. XXXIII
The sentence formed with the ten letters above the line, which is the key to this sum, is _Do your best_. If these letters are numbered consecutively 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and the corresponding figures are substituted for the letters, the sum works out as is shown in the second diagram--
S B 9 7 R E 6 8 Y D 3 1 O T 4 0 U O 5 2 ------ ----- O E E 2 8 8 ===== =====
No. XXXIV
The twelve names of flowers and foliage that may be gathered within these borders, by moving in any direction one square at a time,
╔═══╤═══╤═══╤═══╤═══╤═══╤═══╗ ║ 1 │ 2 │ 3 │ 4 │ 5 │ 6 │ 7 ║ ║ L│ L│ B│ H│ P│ E│ F║ ╟───┼───┼───┼───┼───┼───┼───╢ ║ 8 │ 9 │10 │11 │12 │13 │14 ║ ║ L│ Y│ E│ L│ O│ R│ N║ ╟───┼───┼───┼───┼───┼───┼───╢ ║15 │16 │17 │18 │19 │20 │21 ║ ║ I│ V│ B│ R│ I│ V│ K║ ╟───┼───┼───┼───┼───┼───┼───╢ ║22 │23 │24 │25 │26 │27 │28 ║ ║ A│ L│ E│ T│ O│ N│ I║ ╟───┼───┼───┼───┼───┼───┼───╢ ║29 │30 │31 │32 │33 │34 │35 ║ ║ C│ N│ A│ S│ U│ L│ P║ ╚═══╧═══╧═══╧═══╧═══╧═══╧═══╝
are 18, 26, 32, 24, _Rose_; 25, 33, 34, 28, 35, _Tulip_; 35, 28, 27, 21, _Pink_; 31, 32, 25, 24, 18, _Aster_; and, in similar ways, _Verbena_; _Salvia_; _Ivy_; _Lily_; _Lilac_; _Heliotrope_; _Fern_; and _Bell_.
No. XXXV
The solution of this little problem, set by Dr Puzzlewitz on his blackboard to test the powers of his young pupils--“What are the values of A and B, when 4 is the result of dividing A by B, or of subtracting B from A?”--
┏━━━━━━━━━━━━━┓ ┃ ┃ ┃ A - B = 4 ┃ ┃ ┃ ┃ A ÷ B = 4 ┃ ┃ ┃ ┗━━━━━━━━━━━━━┛
is that A = 5¹⁄₃ and B = 1¹⁄₃.
No. XXXVI
This is the diamond squared:--
┌───┐ │ s │ ┌───┼───┼───┐ │ h │ i │ s │ ┌───┼───┼───┼───┼───┐ │ h │ i │ n │ t │ s │ ┌───┼───┼───┼───┼───┼───┼───┐ │ s │ i │ n │ u │ o │ u │ s │ └───┼───┼───┼───┼───┼───┼───┘ │ s │ t │ o │ r │ m │ └───┼───┼───┼───┼───┘ │ s │ u │ m │ └───┼───┼───┘ │ s │ └───┘
in which the words read alike from top to bottom, and from left to right.
No. XXXVII
This is the arrangement of the 32 letters in the 64 cells--
╔═══╤═══╤═══╤═══╤═══╤═══╤═══╤═══╗ ║ │ A │ │ E │ I │ │ O │ ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ E │ O │ │ │ │ │ A │ I ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ │ │ A │ I │ E │ O │ │ ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ I │ │ O │ │ │ A │ │ E ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ O │ │ I │ │ │ E │ │ A ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ │ │ E │ O │ A │ I │ │ ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ A │ I │ │ │ │ │ E │ O ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ │ E │ │ A │ O │ │ │ I ║ ╚═══╧═══╧═══╧═══╧═══╧═══╧═══╧═══╝
No A is in the same column, row, or diagonal with another A, no E with another E, no I with another I, and no O with another O.
No. XXXVIII
This is the anagram square, with the letters, which in the former diagram spelt the words _vote_, _wove_, _prow_, _call_, _stew_, _news_, _core_, _nape_, recast into fresh words which now read alike from top to bottom and from left to right of the square.
╔═══╤═══╤═══╤═══╤═══╤═══╤═══╤═══╗ ║###│ C │###│ R │###│ O │###│ W ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ C │###│ L │###│ A │###│ W │###║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║###│ L │###│ O │###│ V │###│ E ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ R │###│ O │###│ P │###│ E │###║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║###│ A │###│ P │###│ E │###│ S ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ O │###│ V │###│ E │###│ N │###║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║###│ W │###│ E │###│ N │###│ T ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ W │###│ E │###│ S │###│ T │###║ ╚═══╧═══╧═══╧═══╧═══╧═══╧═══╧═══╝
The empty squares and diagonal setting are necessary for this particular puzzle, as the words would not form a word square if their letters were placed below one another in the usual way.
No. XXXIX
The sentence formed with the ten letters above the line, which is the key to this sum, is--_Add these up_. If these letters are numbered consecutively 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and the corresponding figures are substituted for the numbers, the sum works out as is shown below.
D U 2 9 E H 6 5 E D 8 3 A P 1 0 S T 7 4 ----- ----- D E A 2 6 1 ===== =====
No. XL
The four words, seek, slab, leek, moan, which were placed on the white squares when recast form the following combination:--
╔═══╤═══╤═══╤═══╤═══╤═══╤═══╤═══╗ ║###│ M │###│ A │###│ S │###│ K ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ A │###│ B │###│ L │###│ E │###║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║###│ S │###│ L │###│ O │###│ E ║ ╟───┼───┼───┼───┼───┼───┼───┼───╢ ║ K │###│ E │###│ E │###│ N │###║ ╚═══╧═══╧═══╧═══╧═══╧═══╧═══╧═══╝
These fresh words read alike from side to side, and zigzag from top to bottom.
No. XLI
These are the four words, recast by anagram from afar, task, seat, leal, and which now form a perfect word square.
╔═══╤═══╤═══╤═══╗ ║ F │ A │ S │ T ║ ╟───┼───┼───┼───╢ ║ A │ R │ E │ A ║ ╟───┼───┼───┼───╢ ║ S │ E │ A │ L ║ ╟───┼───┼───┼───╢ ║ T │ A │ L │ K ║ ╚═══╧═══╧═══╧═══╝
No. XLII
The word square is recast thus--
╔═══╤═══╤═══╤═══╤═══╗ ║ c │ r │ e │ s │ s ║ ╟───┼───┼───┼───┼───╢ ║ r │ e │ a │ c │ h ║ ╟───┼───┼───┼───┼───╢ ║ e │ a │ g │ e │ r ║ ╟───┼───┼───┼───┼───╢ ║ s │ c │ e │ n │ e ║ ╟───┼───┼───┼───┼───╢ ║ s │ h │ r │ e │ d ║ ╚═══╧═══╧═══╧═══╧═══╝
Its words are spelt with the same letters as the words _chess_, _greed_, _canes_, _rears_, _cheer_, which formed the original square, but did not read alike from top to bottom, and from left to right, as these do.
No. XLIII
The five familiar proverbs hidden in this square of 169 letters are: A rolling stone gathers no moss. Too many cooks spoil the broth. A live dog is more to be feared than a dead lion. You cannot eat your cake and have it. Peace hath her victories no less renowned than war.
R E N O W N E D T H A N W S Y O U R C A K E A N D A S T E T O B E F E A R H R E A R K S S P O I L E A F L E O O H E R S N T D V O O T M O T L I N O H T E U N O S C A L A G M E H I R S N I Y G O R S O B A T S E N G N E N O T S R N P A I A O A M O O T S O A E W R C D E V I L A H T D A S O U O Y N O I L D A E C A T C I V R E H H T A H E Z
No. XLIV
[Illustration]
If the shaded circles are cut out and the diagram is placed squarely over the jumbled letters, with the I., II., III., IV. in turn at the top left-hand corner, this sentence is disclosed--
Le premier Supplément du Journal de la Jeunesse a été publié dans le Numéro du Dix-neuf Juin Mil huit cent soixante-quinze.
No. XLV
This is the way to reconstruct Sam Loyd’s black pony--so that, while its legs and tail are strangely misplaced, they form the spirited outline of a white galloping horse.
[Illustration]
No. XLVI
Here is the key to Sam Loyd’s ingenious puzzle--
[Illustration]
which shows the jockeys and horses in full racing trim.
No. XLVII
This is the inevitable result of the boy’s attempt to annex with his mouth the sugar on the chair--
[Illustration]
No. XLVIII
The leap-frog puzzle is solved in nine hops thus:--
[Illustration]
First jump from stool 2, then from 5, 3, 6, 7, 1, 3, and 6 in turn to the vacant stools.
No. LV
This diagram shows that the seven wheels, which spin so merrily when the paper is rotated in the hand, can be divided off into separate enclosures by only three straight lines--
[Illustration]
No. LVI
The diagram below shows how the market-gardener, keeping one-fourth of his square field for himself in the shape of a triangle, was able to divide the remainder so that each of his four sons had an equal portion of similar shape--
[Illustation]
No. LVII
Here is a drawing of the perfect Latin cross--
[Illustation]
The position of the two long pieces does not readily suggest itself to those who try to arrange the five on paper with a pencil.
No. LVIII
This diagram shows the effectual means taken by four rich men, whose houses were further afield, to exclude four poor men from all access to a central lake, that they might reserve the fishing for themselves.
[Illustration]
They built a high wall on the lines that are indicated which, while it left a way for each of them to the water, altogether shut it away from their poor neighbours.
No. LIX
This is the square that can be formed with the ten pattern pieces given--
[Illustration]
No. LX
The dotted lines in this diagram show how the figure can be divided into nine parts by four straight cuts
[Illustration]
which can be reunited to form a perfect cross.
No. LXI
This is a simple way by which the figure given can be divided by four straight cuts into four equal and similar parts--
[Illustration]
No. LXII
This is the way to draw twenty-two straight lines within the circle at right-angles to each other, so that they divide it into four similar parts--
[Illustration]
and each part has three dots within its borders.
No. LXIII
These diagrams show how the upper triangle is divided into five parts, which can be rearranged to form the equilateral triangle below.
[Illustration]
The originator of this ingenious novelty says, “The method of construction is not shown, but its application is general, and the result is easily verified by measurement.”
No. LXVI
This is an arrangement of the twenty-seven counters in nine rows, six in a row, within the borders of an equilateral triangle.
* ╱ ╲ ╱ ╲ * * ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲ * * * ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲ ╱ * * ╲ ╱ ╱ ╲ ╱ ╲ ╲ ╱ ╱ ╲ ╱ ╲ ╲ ╱ ╱ * ╲ ╲ ╱ ╱ ╱ ╲ ╲ ╲ ╱ ╱ ╱ ╲ ╲ ╲ ╱ ╱ ╱ ╲ ╲ ╲ *─────*─────*───────*─────*─────* ╱ ╱ ╱ ╲ ╲ ╲ *─────*─────*───────────*─────*─────* ╱ ╱ ╱ ╲ ╲ ╲ *─────*─────*───────────────*─────*─────*
No. LXVII
All the cards of one colour, when placed alternately, can be brought together in four moves, two at a time, thus--
┌───┐ ┌───┐ ┌───┐ ┌───┐ ┌───┐ ┌───┐ ┌───┐ ┌───┐ │ A │ │ 2 │ │ 3 │ │ 4 │ │ 5 │ │ 6 │ │ 7 │ │ 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ♠ │ │ ♡ │ │ ♣ │ │ ♢ │ │ ♣ │ │ ♢ │ │ ♣ │ │ ♡ │ └───┘ └───┘ └───┘ └───┘ └───┘ └───┘ └───┘ └───┘
Place two and three beyond eight; Place five and six between one and four; Place eight and two between four and seven; Place one and five between seven and three.
No. LXVIII
You can in a moment tell the number chosen on these cards, when you are told on which of them it appears,
╔════════════════════════════════════════════════════════════════════╗ ║┌───────────────┐┌───────────────┐┌───────────────┐┌───────────────┐║ ║│ I. ││ II. ││ III. ││ IV. │║ ║│ ││ ││ ││ │║ ║│ 1 33 65 97││ 2 34 66 98││ 4 36 68 100││ 8 40 72 104│║ ║│ 3 35 67 99││ 3 35 67 99││ 5 37 69 101││ 9 41 73 105│║ ║│ 5 37 69 101││ 6 38 70 102││ 6 38 70 102││ 10 42 74 106│║ ║│ 7 39 71 103││ 7 39 71 103││ 7 39 71 103││ 11 43 75 107│║ ║│ 9 41 73 105││ 10 42 74 106││ 12 44 76 108││ 12 44 76 108│║ ║│ 11 43 75 107││ 11 43 75 107││ 13 45 77 109││ 13 45 77 109│║ ║│ 13 45 77 109││ 14 46 78 110││ 14 46 78 110││ 14 46 78 110│║ ║│ 15 47 79 111││ 15 47 79 111││ 15 47 79 111││ 15 47 79 111│║ ║│ 17 49 81 113││ 18 50 82 114││ 20 52 84 116││ 24 56 88 120│║ ║│ 19 51 83 115││ 19 51 83 115││ 21 53 85 117││ 25 57 89 121│║ ║│ 21 53 85 117││ 22 54 86 118││ 22 54 86 118││ 26 58 90 122│║ ║│ 23 55 87 119││ 23 55 87 119││ 23 55 87 119││ 27 59 91 123│║ ║│ 25 57 89 121││ 26 58 90 122││ 28 60 92 124││ 28 60 92 124│║ ║│ 27 59 91 123││ 27 59 91 123││ 29 61 93 125││ 29 61 93 125│║ ║│ 29 61 93 125││ 30 62 94 126││ 30 62 94 126││ 30 62 94 126│║ ║│ 31 63 95 127││ 31 63 95 127││ 31 63 95 127││ 31 63 95 127│║ ║└───────────────┘└───────────────┘└───────────────┘└───────────────┘║ ║ ┌───────────────┐┌───────────────┐┌───────────────┐ ║ ║ │ V. ││ VI. ││ VII. │ ║ ║ │ ││ ││ │ ║ ║ │ 16 48 80 112││ 32 48 96 112││ 64 80 96 112│ ║ ║ │ 17 49 81 113││ 33 49 97 113││ 65 81 97 113│ ║ ║ │ 18 50 82 114││ 34 50 98 114││ 66 82 98 114│ ║ ║ │ 19 51 83 115││ 35 51 99 115││ 67 83 99 115│ ║ ║ │ 20 52 84 116││ 36 52 100 116││ 68 84 100 116│ ║ ║ │ 21 53 85 117││ 37 53 101 117││ 69 85 101 117│ ║ ║ │ 22 54 86 118││ 38 54 102 118││ 70 86 102 118│ ║ ║ │ 23 55 87 119││ 39 55 103 119││ 71 87 103 119│ ║ ║ │ 24 56 88 120││ 40 56 104 120││ 72 88 104 120│ ║ ║ │ 25 57 89 121││ 41 57 105 121││ 73 89 105 121│ ║ ║ │ 26 58 90 122││ 42 58 106 122││ 74 90 106 122│ ║ ║ │ 27 59 91 123││ 43 59 107 123││ 75 91 107 123│ ║ ║ │ 28 60 92 124││ 44 60 108 124││ 76 92 108 124│ ║ ║ │ 29 61 93 125││ 45 61 109 125││ 77 93 109 125│ ║ ║ │ 30 62 94 126││ 46 62 110 126││ 78 94 110 126│ ║ ║ │ 31 63 95 127││ 47 63 111 127││ 79 95 111 127│ ║ ║ └───────────────┘└───────────────┘└───────────────┘ ║ ╚════════════════════════════════════════════════════════════════════╝
by adding together the numbers at the top left-hand corner of these.
No. LXIX
This diagram shows that the postman can take a course which involves fewer turnings than that indicated, when he had to pass round eighteen corners.
●┅┅┅●┅┅┅●┅┅┅●┅┅┅●┅┅┅●┅┅┅●┅┅┅● ┇ ┇ ● ●┅┅┅●┅┅┅●┅┅┅●┅┅┅●┅┅┅● ● ┇ ┇ ┇ ┇ ● ● ●┅┅┅●┅┅┅●┅┅┅● ● ● ┇ ┇ ┇ ┇ ┇ ┇ ● ● ● ●┅┅┅○ ● ● ● ┇ ┇ ┇ ┇ ┇ ┇ ┇ ┇ ● ● ● ● ● ● ● ● ┇ ┇ ┇ ┇ ┇ ┇ ┇ ┇ ● ● ● ● ● ● ● ● ┇ ┇ ┇ ┇ ┇ ┇ ┇ ┇ ● ● ● ● ● ●┅┅┅● ● ┇ ┇ ┇ ┇ ┇ ┇ ●┅┅┅● ●┅┅┅● ●┅┅┅●┅┅┅●┅┅┅●
It will be seen that he has to turn only fifteen times.
No. LXX
This shows how a square can be divided into three parts, so that these can be reunited to form No. 2 and No. 3 of the diagram.
[Illustration]
[Illustration: 1.]
[Illustration: 2.]
[Illustration: 3.]
Try it with scissors and paper or cardboard.
No. LXXI
┏━━━┯━━━┯━━━┯━━━┯━━━┯━━━┯━━━┯━━━┓ ┃ │###│ │###│ │###│ │###┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃###│ │###│ │###│ │###│ ┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃ │###│ │###│ │###│ │###┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃###│ │#♘#│ ♖ │#♚#│ ♖ │###│ ┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃ │###│ │###│ │###│ │###┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃###│ │###│ │###│ │###│ ┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃ │###│ │###│ │###│ │###┃ ┠───┼───┼───┼───┼───┼───┼───┼───┨ ┃###│ │###│ │###│ │# #│###┃ ┗━━━┷━━━┷━━━┷━━━┷━━━┷━━━┷━━━┷━━━┛
This position fulfils the conditions of the puzzle. Obviously it could not occur in actual play.
No. LXXV
The dotted lines in this diagram show where the flag with a cross taken out from its centre must be cut, so that the two pieces can be rejoined to form a perfect flag.
[Illustration]
The piece on the right is moved upward, and to the left.
No. LXXVI
This is a way in which the eleven parts can be readjusted to form a square:--
[Illustration]
No. LXXVIII
This shows the shortest course--
[Illustration]
This track takes him completely round every block, passing only once round four of them.
No. LXXIX
Here is a very simple and symmetrical arrangement, by which on a board of 36 squares twelve counters are so placed that there are two, and two only, on each line, column, and diagonal.
╔═══╤═══╤═══╤═══╤═══╤═══╗ ║ ◎ │ │ │ │ │ ◎ ║ ╟───┼───┼───┼───┼───┼───╢ ║ │ │ ◎ │ ◎ │ │ ║ ╟───┼───┼───┼───┼───┼───╢ ║ │ ◎ │ │ │ ◎ │ ║ ╟───┼───┼───┼───┼───┼───╢ ║ │ ◎ │ │ │ ◎ │ ║ ╟───┼───┼───┼───┼───┼───╢ ║ │ │ ◎ │ ◎ │ │ ║ ╟───┼───┼───┼───┼───┼───╢ ║ ◎ │ │ │ │ │ ◎ ║ ╚═══╧═══╧═══╧═══╧═══╧═══╝
There are other arrangements which fulfil the conditions.
No. LXXXI
In this nest of triangles of five tiers there are 1196 separate triangles, or nearly double the number (653) of a similar nest of four tiers.
[Illustration]
In such a figure with 10,000 tiers there would be 6,992,965,420,332 different triangles!
No. LXXXII
The match puzzle, in which eight matches set in a row are to be rearranged in four pairs, by passing one match over two four times--
║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║ ║
is solved, if the matches are numbered 1, 2, 3, 4, 5, 6, 7, 8, by moving 4 to 7, 6 to 2, 1 to 3, and 5 to 8.
No. LXXXIII