Part 4
The above figures illustrate that when 1 is the number thought of there is only an addition of 1. When 2 is the figure, no addition is required to the first result; but the second result being 9, 1 is added and _two_ noted, which, of course, is the figure thought of. When 3 is thought of two additions are necessary, one to the 9 and one to the 15, making a total of _three_ to be remembered, which represents the original number. When 4 or any succeeding number is thought of the final result is always divisible by 9, and in your mental calculation each 9 must represent 4, to which you add the figures you have previously noted.
EXAMPLES.
Number thought of 4 × 3 = 12 ÷ 2 = 6 × 3 = 18 ÷ 2 = 9.
Here we have one 9, which represents 4, the number thought of.
Number thought of 7 × 3 = 21 + 1 = 22 ÷ 2 = 11 × 3 = 33 + 1 = 34 ÷ 2 = 17. From which is obtainable only one 9, which represents 4, to which you add 1 for the first addition of 1, and 2 for the second addition, making a total of 7, the number thought of.
Number thought of,
11 × 3 ---- 33 + 1 note 1 ---- ÷ 2 34 17 × 3 ---- 51 + 1 note 2 ÷ 2 52 ---- 26 two 9's = 8 = 11
HOW TO NAME A NUMBER WHICH HAS BEEN ERASED
Request a member of the company to write a row of figures, the number of which is immaterial, add them together and subtract the addition from the row. Then to cross out any figure from the result, add the remaining figures together and give you the total, when you will tell him which figure he has erased. Of course, you do not see his figures and can leave the room while he makes them.
EXAMPLE.
567219 = 30 - 30 -------- 567189
We will suppose he crosses out 7, which makes the addition of the row, minus that figure, 29. He gives you that result and you at once name the crossed off figure. There are two ways of arriving at the answer. The simplest and quickest way is to add the units in the result together until only one figure remains and deduct it from 9. For instance, we will take 29. Add the 2 and 9 together, which make 11; add 1 and 1 together and you have 2, which deduct from 9, leaving 7, the figure erased in the above example.
Supposing 1 was the figure erased, the addition of the remaining figures would then be 35; 3 + 5 = 8, 9 - 8 = 1, the figure crossed off.
The second method is to reckon the next multiple of 9 above the figures given you; for instance, supposing they are 29, the next multiple of 9 is 36. Deduct 29 from it and it leaves 7, the erased figure. If either 9 or 0 is erased the result is the same. You can get out of the difficulty, on being told you are wrong, by saying (in case you have given 9), "Yes, I see it is a nought; I thought it had a tail, so mistook it for a nine." If you have named 0 and it turns out to be 9, you can say, "Oh, I didn't notice the tail; of course I should have said nine."
A LESSON IN THE CORRECT FORMATION OF A FIGURE
Request a friend to write the following figures:--
1 2 3 4 5 6 7 9
Take the paper from him and, after pretending to scrutinise the row, ask him to point out which figure he considers most imperfectly made. If he should select the 1, say, "You had better practise making that figure. Oblige me by multiplying the row by nine." When he does so the result will be
1 1 1 1 1 1 1 1 1
Then say, "After this practice you will be able to make better ones in future."
If he selects the 4 request him to multiply by 36 and the result will be
4 4 4 4 4 4 4 4 4
Whichever figure he selects, mentally multiply it by 9 and request him to multiply the row by the result. If he thinks 9 the most imperfectly made figure, you, of course, tell him to multiply by 81 and the result will be all 9's.
FOUR NINES PROBLEM
How can four 9's be written so that they will make 100?
SOLUTION.
99 9/9
AN ANSWER TO A SUM GIVEN IN ADVANCE
Ask some one to start a sum in addition by writing the top line of four figures. We will suppose he writes 1912. You mentally subtract the 2 and place it before the 1, making 21,910, which figures write on a piece of paper, which you fold up and lay on the table. You then ask a second person to place four figures under the first line. Then add a line yourself, which must be a deduction of the second line from four 9's. Ask a third person to add four figures to those already written. Then add another line yourself, making it a deduction of the third person's figures from four 9's. Request a fourth person to add up the sum and tell him you have already done so, and he will find the answer on the table. The sum will appear something like this:--
1912 7234 2765 4891 5108 -------- 21,910
Which answer corresponds with the figures on the paper, which has been on the table the whole time. If you have in the company two friends upon whom you can rely as confederates, previously arrange with them to write the third and fifth lines, explaining to them that they must deduct the line immediately preceding theirs from 9's and make their lines the products. This adds greatly to the mystery of the trick.
AN ARITHMETICAL PUZZLE
Take 9 from 6; from 9 take 10, and from 40 take 50, and you will find 6 remains.
SOLUTION.
FROM SIX | FROM IX | FROM XL TAKE IX | TAKE X | TAKE L S | I | X
AN ARITHMETICAL MYSTERY
Thirteen commercial travellers arrived at an inn, and each desired a separate room. The landlady had but 12 vacant rooms, which may be represented thus:--
---------------------------------------------------- | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ----------------------------------------------------
But she promised to accommodate all according to their wishes. So she showed two of the travellers into room No. 1, asking them to remain a few minutes together. Traveller No. 3 she showed into room No. 2, traveller No. 4 she showed into room No. 3, traveller No. 5 into room No. 4, traveller No. 6 into room No. 5, and so on until she had put the twelfth traveller into Room No. 11. She then went back to where she had left the two travellers together, and asking the thirteenth traveller to follow her, led him to No. 12, the remaining room. Thus all were accommodated. Ask your friends to explain the mystery.
HOW TO TELL HER AGE
Girls of a marriageable age do not like to tell how old they are, but you can find out by following the subjoined instructions, the young lady doing the figuring: Tell her to put down the number of the month in which she was born, then to multiply it by 2, then to add 5, then to multiply it by 50, then to add her age, then to subtract 365, then to add 115, then tell her to tell you the amount she has left. The two figures to the right will tell you her age and the remainder the month of her birth. For example, the amount is 822, she is twenty-two years old and was born in the eighth month (August).
A RACE IN ADDITION
Tell a friend that you will race him in counting from 1 to 100, and guarantee to win, under the following conditions: You will allow him to start first, at any number from 1 to 10, and you are both to have the privilege of adding any figure up to 10 to the last number called. For instance, we will suppose he starts with 5. You call 15, having mentally added 10 to his number. He then calls 20, having added 5; and so on, until 100 is reached. Until he sees through the trick you will win every time, and even then you will win if you start first and commence at 1. In that case, as he can only add 10, his first call could not exceed 11, to which you immediately add 1 and call 12. If his next call is 22, you say 23. No matter what his additions may be, the numbers you must always reach first are 12, 23, 34, 45, 56, 67, 78, and 89. When you call the latter number, as he can only add 10 to it, your next call will, of course, be 100. By this you will observe that, although you can only add 10 to your opponent's last number, you in reality add 11 to your own. So you are, so to speak, always 1 ahead of him. If, when you suggest the trick, you see your friend is not familiar with it, you can give him the option of starting first, and you need not pick up the thread of your winning numbers until you reach 50, adding low numbers to his additions, which will help to puzzle him; but he will soon see that it is necessary to reach 89; then he will notice you strike 78 and 67. When you see he is getting on the right track, pick up the winning numbers earlier, and at last insist that you must now start first. In starting with a person who does not know the trick it is advisable, and more puzzling, to dodge about at first and not get on the track of the winning numbers until 56 or 67. But if your friend knows the trick and starts at 1 you cannot beat him. I have seen good accountants puzzle for hours over this little trick, which was invented by Mr. William Lawtey, a dear old friend of mine.
TO PREDICT THE HOUR YOUR FRIEND INTENDS TO RISE ON THE FOLLOWING MORNING
Request your friend to make up his mind as to the time he intends to rise on the following morning, and then to mention an entirely different hour to you. To the latter you mentally add twelve, and giving him the number of the total, request him to look at his watch, and starting at the hour preceding the one he has selected for rising, to count backwards until he reaches the number you have given him, beginning with the number which he previously gave you. Ask him to state the hour at which he stops, which he will find is the one he selected for rising. For instance; supposing your friend intends to rise at nine and gives you four. To four you mentally add twelve and request him to start at the hour before his getting-up time (which would be eight) and count sixteen backwards on the face of the watch, starting with the number he gave you--four--and when he reaches sixteen his finger or pencil will rest upon nine, the hour he selected for getting up.
MATCH PUZZLES
EXPERIMENT WITH TEN MATCHES
Lay ten matches side by side (Fig. 7) and request some one to lift each match singly, and passing it over two matches, cross a third match with it until there are five crosses on the table (Fig. 8). Two matches (and only two whether crossed or single) must be passed over at a time.
1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Fig. 7.
\ / \ / \ / \ / \ / \ \ \ \ \ / \ / \ / \ / \ / \ Fig. 8.
The secret is that No. 1 must be crossed first and No. 9 second, or the trick cannot be accomplished.
The following are the correct moves: 4 over 2 and 3 and crossed on 1; 6 over 7 and 8 and crossed on 9; 8 over 7 and 5, crossed on 3; 2 over the 3 and 5, crossed on 7; the 10 over the 9 and 7, crossed on 5.
THE MAGIC NINE
Make the figure 9 with a long tail with matches (Fig. 9) and tell a member of the company to think of a number, which must exceed the number of matches in the tail; and, commencing at the first match in the latter, count mentally round the figure, stop when he reaches the number thought of, and then, recommencing at the match he stopped at, count the reverse way, this time avoiding the tail, and continuing on the upper part of the 9 until he again reaches the number he selected, when you will point to the match he has stopped at. This you can do very easily, for if there are seven matches in the tail he will, of course, stop at the seventh match on the left from the tail, as will be seen by the numbering on the diagram, which assumes he thought of fifteen. Each time the puzzle is tried vary the length of the tail by taking some matches out of the latter and adding them to the upper part of the figure, or vice versa. If this is not done the stop will always be made at the same match, which will give the trick away.
[Illustration] Fig. 9.
TRIANGLES WITH MATCHES
Make three equilateral triangles with six matches. Of course, two can be made with five matches; but then there is one over, and how to make a third triangle with only one match is a puzzler. It is as easy as possible. Make a triangle with three matches, and stand the other three upon end inside the triangle in the form of a tripod (Fig. 10).
[Illustration] Fig. 10.
Here is another triangular puzzle. With five matches form two equilateral triangles. Tell the company they are to remove three matches; then add two and make two more equilateral triangles. This is only a "sell." You do not say where the two matches are to be added. You add them to the three removed, and form the same figure over again (Fig. 11).
/|\ / | \ / | \ \ | / \ | / \|/ Fig. 11.
MATCH SQUARES
Make nine squares with twenty-four matches (Fig. 12). Then request some one to remove eight matches, and without touching those left, to leave two perfect squares.
-- -- -- | | | | -- -- -- | | | | -- -- -- | | | | -- -- -- Fig. 12.
Fig. 13 shows the solution.
-- -- -- | | -- | | | | -- | | -- -- -- Fig. 13.
YOUR OPPONENT MUST TAKE THE LAST MATCH
Place twenty-five matches in a row on the table. Request some one to select one end of the row and to take one, two, or three matches from it, you having the same privilege at the other end; and you guarantee he will be compelled to take the last match no matter how he may vary the number he takes.
The secret is to remove four matches each time between you. For instance, if your opponent takes three you take one; if he takes two you take two; if he takes one you take three and so on. It is obvious if four matches are taken six times one match will be left on the table, which your opponent must take.
A SHAKESPEAREAN QUOTATION
Lay five matches on the table and request a member of the company to form a well-known quotation from Shakespeare by the addition of three more matches (Fig. 14). "But," some one will say, "how does KINI represent a Shakespearean quotation?" Your reply is obvious: "Can't you see KINI is 'a little more than kin, but rather less than kind'?"
| / | |\ | | |/ | | \ | | |\ | | \ | | | \ | | \| | Fig. 14.
NUMERAL
Place five matches on the table and challenge any one to make them into thirteen without breaking any of them, and then, without moving them, to make eight by the use of a card. The solution will be found in Fig. 15.
\ / | | | \ | | | / \ | | | Fig. 15.
To make eight, hide the lower half of the row from sight, and it of course shows viii.
SIX AND FIVE MAKE NINE
Place six matches on the table and request a person to add five more in such a manner as to make nine. The solution is shown in Fig. 16.
_____ |\ | | |\ | | | \ | | | \ | |_____ | \ | | | \ | | | \| | | \| |_____ Fig. 16.
THE ARTFUL SCHOOLBOYS
At a certain school were four long dormitories, built in the form of a square, in which thirty-two boys occupied beds, as shown by matches in Fig. 17.
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|||| |||| |||| Fig. 17.
By this arrangement the master, in going his rounds at night, counted twelve boys in each corridor. One night four boys absented themselves from the school, and the remaining boys rearranged themselves in such a manner that the master was still able to count twelve boys in each corridor, and the absence of their four comrades was not noticed. How they did it is shown in Fig. 18.
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||||| || ||||| Fig. 18.
The four absentees returned on the following night, accompanied by four friends; but the master was unable to notice the addition, for he again counted twelve boys in each dormitory. The new arrangement was as Fig. 19.
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||| |||||| ||| Fig. 19.
There were now thirty-six boys sleeping in the dormitories, and next night they were joined by four more, which brought the number up to forty, and yet the master only counted twelve in each dormitory on his rounds that night. How the new distribution was made is shown in Fig. 20.
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|| |||||||| || Fig. 20.
Next night four more chums popped in for a snooze, making a total of forty-four, and again the master was bamboozled by the following readjustment (Fig. 21).
[Illustration] | |||||||||| |
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| |||||||||| | Fig. 21.
History is silent upon the subject of the arrangement at the breakfast-tables.
The proper way to present this puzzle to your friends is to lay forty-four matches on the table, and after showing the initial arrangement, allow them to work the rest out for themselves.
WHAT ARE MATCHES MADE OF?
Arrange fourteen matches as in Fig. 22, and tell your friends to take away any three matches they may select without disturbing the others, and replace one in any position they may choose in such a way as to show what matches are made of. They will endeavour to form the word "wood"; but Fig. 23 gives the correct solution.
----- ----- ----- | | | | \ / | | | | | | \ / | | | | | | \/ | | ----- ----- ----- Fig. 22.
----- ----- | | | \ / | | | | \ / |----- | | | \/ | ----- ----- ----- Fig. 23.
A SHEEP PEN
Arrange eight matches as shown in Fig. 24, and state that this enclosure, formed by eight hurdles, is supposed to hold one hundred sheep. Ask your friends how many more hurdles would be required to enable the enclosure to contain two hundred sheep? The reply is generally eight more, and your friends will be surprised to learn that only two more hurdles are required--one at each end across the enclosure. Three hurdles being moved to admit of the introduction of the additional two, the pen will, of course, be doubled in size.
----- ----- ----- | | | | | | ----- ----- ----- Fig. 24.
POST AND RAIL PUZZLE
Put the following question to the company: Supposing there was a tunnel through a hill and a post and rail fence was constructed through it, and another fence was made exactly above it, over the hill, how many more posts would be required for the latter route, supposing they were the same distance apart by both routes?
After several calculations have been made you can astonish the company by telling them that exactly the same number of posts would be required for both routes, which you can prove by making a rough sketch of the diagram, Fig. 25, and placing matches on it to represent the posts.
[Illustration] Fig. 25.
SIMPLE MISCELLANEOUS TRICKS
A GOOD AFTER-DINNER TRICK
Procure an egg, an apple, an orange, and two dozen nuts. Place the latter on a plate, and request three persons during your absence from the room to each pocket one of the three former, asserting that you will eventually state in whose pockets the different articles are to be found. On returning to the room present to one of the persons you have asked to assist you one nut, to a second person two nuts, and to the third three nuts, which will of course leave eighteen nuts on the plate. You must mentally name the person to whom you gave one nut "number one," to the person holding two nuts "number two," and the one who has three nuts "number three."
Announce your intention of again leaving the room, and request your three assistants to help themselves during your absence to nuts as follows--the one holding the apple to take the same number of nuts you presented him with, the one who has the egg to twice as many as you gave him, and the holder of the orange to four times as many as he originally received.
Impress on them that the number of nuts they take must be _in addition_ to those they already hold.
On returning to the room you glance at the nuts remaining in the plate and at once call for the egg, apple, and orange from their respective holders.
EXPLANATION.
You must memorise the following Latin words: Attento, Beato, Cantores, Erocat, Fortasse, Glossema, numbering them 1, 2, 3, 4, 5, and 7. The initials of these words, it will be observed, are the first six letters of the alphabet, omitting D, which is not required; A, of course, standing for Apple, E for Egg, and O for Orange.