Entertainments for Home, Church and School

by Frederica Seeger

HOW TO TELL ANY NUMBER THOUGHT OF

Ask any person to think of a number, say a certain number of dollars; tell him to borrow that sum of some one in the company, and add the number borrowed to the amount thought of. It will here be proper to name the person who lends him the money, and to beg the one who makes the calculation to do it with great care, as he may readily fall into an error, especially the first time. Then say to the person: "I do not lend you, but give you $10; add them to the former sum." Continue in this manner: "Give the half to the poor, and retain in your memory the other half." Then add: "Return to the gentleman, or lady, what you borrowed, and remember that the sum lent you was exactly equal to the number thought of." Ask the person if he knows exactly what remains; he will answer "Yes". You must then say: "And I know also the number that remains; it is equal to what I am going to conceal in my hand." Put into one of your hands 5 pieces of money, and desire the person to tell how many you have got. He will answer 5; upon which open your hand and show him the 5 pieces. You may then say: "I well knew that your result was 5; but if you had thought of a very large number, for example, two or three millions, the result would have been much greater, but my hand would not have held a number of pieces equal to the remainder." The person then supposing that the result of the calculation must be different, according to the difference of the number thought of, will imagine that it is necessary to know the last number in order to guess the result; but this idea is false, for, in the case which we have here supposed, whatever be the number thought of, the remainder must always be 5. The reason of this is as follows: The sum, the half of which is given to the poor, is nothing else than twice the number thought of, plus 10; and when the poor have received their part, there remains only the number thought of plus 5; but the number thought of is cut off when the sum borrowed is returned, and consequently there remains only 5. The result may be easily known, since it will be the half of the number given in the third part of the operation; for example, whatever be the number thought of, the remainder will be 36 or 25, according as 72 or 50 have been given. If this trick be performed several times successively, the number given in the third part of the operation must be always different; for if the result were several times the same, the deception might be discovered. When the five first parts of the calculation for obtaining a result are finished, it will be best not to name it at first, but to continue the operation, to render it more complex, by saying for example: "Double the remainder, deduct two, add three, take the fourth part," etc.; and the different steps of the calculation may be kept in mind, in order to know how much the first result has been increased or diminished. This irregular process never fails to confound those who attempt to follow it.

ANOTHER WAY

Tell the person to take 1 from the number thought of, and then double the remainder; desire him to take 1 from this double, and to add to it the number thought of, in the last place, ask him the number arising from this addition, and, if you add 3 to it, the third of the sum will be the number thought of. The application of this rule is so easy that it is needless to illustrate it by an example.

A THIRD WAY

Ask the person to add 1 to the triple of the number thought of, and to multiply the sum by three; then bid him add to this product the number thought of, and the result will be a sum from which if 3 be subtracted, the remainder will be ten times the number required; and if the cipher on the right be cut off from the remainder, the other figure will indicate the number sought.

Example—Let the number thought of be 6, the triple of which is 18; and if 1 be added, it makes 19; the triple of this last number is 57, and if 6 be added it makes 63, from which if 3 be subtracted, the remainder will be 60; now, if the cipher on the right be cut off, the remaining figure, 6, will be the number required.

A FOURTH WAY

Tell the person to multiply the number thought of by itself; then desire him to add 1 to the number thought of, and to multiply it also by itself; in the last place, ask him to tell the difference of these two products, which will certainly be an odd number, and the least half of it will be the number required.

Let the number thought of, for example, be 10; which, multiplied by itself, gives 100; in the next place, 10 increased by 1 is 11; which, multiplied by itself makes 121; and the difference of these two squares is 21, the least half of which, being 10, is the number thought of.

HOW TO TELL NUMBERS THOUGHT OF

If one or more numbers thought of be greater than 9, we must distinguish two cases; that in which the number or the numbers thought of is odd, and that in which it is even. In the first case, ask the sum of the first and second; of the second and third; the third and fourth; and so on to the last; and then the sum of the first and the last. Having written down all these sums in order, add together all those, the places of which are odd, as the first, the third, the fifth, etc.; make another sum of all those, the places of which are even, as the second, the fourth, the sixth, etc.; subtract this sum from the former, and the remainder will be the double of the first number. Let us suppose, for example, that the five following numbers are thought of: 3, 7, 13, 17, 20, which, when added two and two as above, give 10, 20, 30, 37, 23; the sum of the first, third, and fifth is 63, and that of the second and fourth is 57; if 57 be subtracted from 63, the remainder 6, will be the double of the first number, 3. Now, if 3 be taken from 10, the first of the sums, the remainder 7, will be the second number; and by proceeding in this manner, we may find all the rest.

In the second case, that is to say, if the number or the numbers thought of be even, you must ask and write down as above, the sum of the first and second; that of the second and third; and so on, as before; but instead of the sum of the first and the last, you must take that of the second and last; then add together those which stand in the even places, and form them into a new sum apart; add also those in the odd places, the first excepted, and subtract this sum from the former, the remainder will be double of the second number; and if the second number, thus found, be subtracted from the sum of the first and second, you will have the first number; if it be taken from that of the second and third, it will give the third; and so of the rest. Let the numbers thought of be, for example, 3, 7, 13, 17; the sums formed as above are 10, 20, 30, 24; the sum of the second and fourth is 44, from which if 30, the third, be subtracted, the remainder will be 14, the double of 7, the second number. The first therefore is 3, third 13, and the fourth 17.

When each of the numbers thought of does not exceed 9, they may be easily found in the following manner:

Having made the person add 1 to the double of the first number thought of, desire him to multiply the whole by 5, and to add to the product the second number. If there be a third, make him double this first sum, and add 1 to it, after which, desire him to multiply the new sum by 5, and to add to it the third number. If there be a fourth, proceed in the same manner, desiring him to double the preceding sum; to add to it 1; to multiply by 5; to add the fourth number; and so on.

Then ask the number arising from the addition of the last number thought of, and if there were two numbers, subtract 5 from it; if there were three, 55; if there were four, 555; and so on; for the remainder will be composed of figures, of which the first on the left will be the first number thought of, the next second, and so on.

Suppose the numbers thought of be 3, 4, 6; by adding 1 to 6, the double of the first, we shall have 7, which, being multiplied by 5, will give 35; if 4, the second number thought of, be then added, we shall have 39, which doubled gives 78; and, if we add 1, and multiply 79, the sum, by 5, the result will be 395. In the last place, if we add 6, the number thought of, the sum will be 401; and if 55 be deducted from it, we shall have, for remainder, 346, the figures of which, 3, 4, 6, indicate in order the three numbers though of.

GOLD AND SILVER GAME

One of the party having in one hand a piece of gold and in the other a piece of silver, you may tell in which hand he has the gold and in which the silver, by the following method: Some value, represented by an even number, such as 8, must be assigned to the gold, and a value represented by an odd number, such as 3, must be assigned to the silver; after which, desire the person to multiply the number in the right hand by any even number whatever, such as 2; and that in the left hand by an odd number, as 3; then bid him add together the two products, and if the whole sum be odd, the gold will be in the right hand and the silver in the left; if the sum be even, the contrary will be the case.

To conceal the trick better, it will be sufficient to ask whether the sum of the two products can be halved without a remainder; for in that case the total will be even, and in the contrary case odd.

It may be readily seen, that the pieces, instead of being in the two hands of the same person, may be supposed to be in the hands of two persons, one of whom has the even number, or piece of gold, and the other the odd number, or piece of silver. The same operations may then be performed in regard to these two persons, as are performed in regard to the two hands of the same person, calling the one privately the right and the other the left.

THE NUMBER BAG

The plan is to let a person select several numbers out of a bag, and to tell him the number which shall exactly divide the sum of those he has chosen; provide a small bag, divided into two parts, into one of which put several tickets, numbered, 6, 9, 15, 36, 63, 120, 213, 309, etc.; and in the other part put as many other tickets marked number 3 only. Draw a handful of tickets from the first part, and, after showing them to the company, put them into the bag again, and, having opened it a second time, desire any one to take out as many tickets as he thinks proper; when he has done that, you open privately the other part of the bag, and tell him to take out of it one ticket only. You may safely pronounce that the ticket shall contain the number by which the amount of the other numbers is divisible; for, as each of these numbers can be multiplied by 3, their sum total must, evidently, be divisible by that number. An ingenious mind may easily diversify this exercise, by marking the tickets in one part of the bag with any numbers that are divisible by 9 only, the properties of both 9 and 3 being the same; and it should never be exhibited to the same company twice without being varied.

THE MYSTICAL NUMBER NINE

The discovery of remarkable properties of the number 9 was accidentally made, more than forty years since, though, we believe, it is not generally known.

The component figures of the product made by the multiplication of every digit into the number 9, when added together, make Nine.

The order of these component figures is reversed after the said number has been multiplied by 5.

The component figures of the amount of the multipliers (viz. 45), when added together, make Nine.

The amount of the several products or multiples of 9 (viz. 405), when divided by 9, gives far a quotient, 45; that is, 4 plus 5 = Nine.

The amount of the first product (viz. 9), when added to the other product, whose respective component figures make 9, is 81; which is the square of Nine.

The said number 81, when added to the above-mentioned amount of the several products, or multiples, of 9 (viz. 405), makes 486; which, if divided by 9, gives, for a quotient, 54; that is 5 plus 4 = Nine.

It is also observable, that the number of changes that may be rung on nine bells, is 362,880; which figures added together, make 27; that is, 2 plus 7 = Nine.

And the quotient of 362,880, divided by 9, will be 40,320; that is, 4 plus 0 plus 3 plus 2 plus 0 = Nine.

To add a figure to any given number, which shall render it divisible by Nine: Add the figures named; and the figure which must be added to the sum produced, in order to render it divisible by 9, is the one required. Thus

Suppose the given number to be 7521: Add these together, and 15 will be produced; now 15 requires 3 to render it divisible by 9; and that number 3, being added to 7521, causes the same divisibility; 7521 plus 3 gives 7524, and divided by 9, gives 836. This exercise may be diversified by your specifying, before the sum is named, the particular place where the figure shall be inserted, to make the number divisible by 9; for it is exactly the same thing whether the figure be put at the head of the number, or between any two of its digits.

THE MAGIC HUNDRED.

Two persons agree to take, alternately, numbers less than a given number, for example, 11 and to add them together till one of them has reached a certain sum, such as 100. By what means can one of them infallibly attain to that number before the other? The whole secret in this consists in immediately making choice of the numbers, 1, 12, 23, 34, and so on, or of a series which continually increases by 11, up to 100. Let us suppose, that the first person, who knows the game, makes choice of 1; it is evident that his adversary, as he must count less than 11, can, at most, reach 11 by adding 10 to it. The first will then take 1, which will make 12; and whatever number the second may add, the first will certainly win, provided he continually add the number which forms the complement of that of his adversary, to 11; that is to say, if the latter take 8, he must take 3; if 9, he must take 2; and so on. By following this method, he will infallibly attain to 89; and it will then be impossible for the second to prevent him from getting first to 100; for whatever number the second takes, he can attain only to 99; after which the first may say—"and 1 makes 100." If the second take 1 after 89, it would make 90, and his adversary would finish by saying—"and 10 makes 100." Between two persons who are equally acquainted with the game, he who begins must necessarily win.

TO GUESS THE MISSING FIGURE

To tell the figure a person has struck out of the sum of two given numbers: Arbitrarily command those numbers only, that are divisible by 9; such, for instance, as 36, 63, 81, 117, 126, 162, 261, 360, 315, and 432. Then let a person choose any two of these numbers; and, after adding them together in his mind, strike out from the sum any one of the figures he pleases. After he has so done, desire him to tell you the sum of the remaining figures; and it follows, that the number which you are obliged to add to this amount, in order to make it 9 or 18, is the one he struck out. Thus:—Suppose he chooses the numbers 162 and 261, making altogether 423, and that he strike out the center figure; the two other figures will, added together, make 7, which, to make nine, requires 2, the number struck out.

THE KING AND THE COUNSELLOR

A King being desirous to confer a liberal reward on one of his courtiers, who had performed some very important service, desired him to ask whatever he thought proper, assuring him it should be granted. The courtier, who was well acquainted with the science of numbers, only requested that the monarch would give him a quantity of wheat equal to that which would arise from one grain doubled sixty-three times successively. The value of the reward was immense; for it will be seen, by calculation, that the sixty-fourth of the double progression divided by 1: 2: 4: 8: 16: 32: etc., is 9223372036854775808. But the sum of all the terms of a double progression, beginning with 1, may be obtained by doubling the last term, and subtracting from it 1. The number of the grains of wheat, therefore, in the present case, will be 18446744073709551615. Now, if a pint contains 9216 grains of wheat, a gallon will contain 73728; and, as eight gallons make one bushel, if we divide the above result by eight times 73728, we shall have 31274997411295 for the number of the bushels of wheat equal to the above number of grains; a quantity greater than what the whole earth could produce in several years.

THE NAILS IN THE HORSE'S SHOE

A man took a fancy to a horse, which a dealer wished to dispose of at as high a price as he could; the latter, to induce the man to become a purchaser, offered to let him have the horse for the value of the twenty-fourth nail in his shoes, reckoning one farthing for the first nail, two for the second, four for the third, and so on to the twenty-fourth. The man, thinking he should have a good bargain, accepted the offer; the price of the horse was, therefore, necessarily great. By calculating as before, the twenty-fourth term of the progression 1:2:4:8: etc., will be found to be 8388608, equal to the number of farthings the purchaser gave for the horse; the price, therefore amounted to 8738 pounds 2s. 8d.

THE DINNER PARTY PUZZLE

A club of seven agreed to dine together every day successively as long as they could sit down to table in different order. How many dinners would be necessary for that purpose? It may be easily found, by the rules already given, that the club must dine together 5040 times, before they would exhaust all the arrangements possible, which would require about thirteen years.

BASKET AND STONES

If a hundred stones be placed in a straight line, at the distance of a yard from each other, the first being at the same distance from a basket, how many yards must the person walk who engages to pick them up, one by one, and put them into the basket? It is evident that, to pick up the first stone, and put it into the basket, the person must walk two yards; for the second, he must walk four; for the third, six; and so on, increasing by two, to the hundredth. The number of yards which the person must walk, will be equal to the sum of the progression, 2, 4, 6, etc., the last term of which is 200, (22). But the sum of the progression is equal to 202, the sum of the two extremes, multiplied by 50, or half the number of terms; that is to say, 10,000 yards, which makes more than 5 1/2 miles.