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, a.s.suring 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 quant.i.ty 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 quant.i.ty 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.

CHAPTER XXI

ONE HUNDRED CONUNDRUMS

WITTY QUESTIONS-FACETIOUS PUZZLES--READY ANSWERS--ENTERTAINING PLAY UPON WORDS

ONE HUNDRED CONUNDRUMS

He loved her. She hated him, but womanlike, she would have him, and she was the death of him. Who was he? Answer: A flea.

Why is life the greatest of riddles? Because we must all give it up.

If a church be on fire, why has the organ the smallest chance of escape?

Because the organ cannot play on it.

Why should a sailor be the best authority as to what goes on in the moon? Because he has been to see (sea).

What does a cat have that no other animal has? Kittens.

When is a man behind the times? When he's a weak (week) back. What is the difference between a baby and a pair of boots? One I was and the other I wear.

Use me well, and I'm everybody; scratch my back and I'm n.o.body. A looking gla.s.s.

What word becomes shorter by adding a syllable to it? Short.

If a stupid fellow was going up for a compet.i.tive examination, why should he study the letter P? Because P makes a.s.s Pa.s.s.

Why is b.u.t.termilk like something that never happened? Because it hasn't a curd (occurred).

Why is the letter O the noisiest of all the vowels? Because the rest are in audible.

Why is a Member of Parliament like a shrimp? Because he has M. P. at the end of his name.

Why is a pig a paradox? Because it is killed first and cured afterward.

Why is a bad half-dollar like something said in a whisper? Because it is uttered, but not allowed (aloud).

Why do black sheep eat less than white ones? Because there are fewer of them.

Why is a barn-door fowl sitting on a gate like a half-penny? Because its head is on one side and its tail on the other.

Why is a man searching for the Philosopher's Stone like Neptune? Because he is a-seeking (sea-king) what never was.

Why is the nose placed in the middle of the face? Because it's the scenter (cen-ter).

What is most like a hen stealing? A c.o.c.k robbing (c.o.c.k robin).

What is worse than "raining cats and dogs"? Hailing omnibuses. When is b.u.t.ter like Irish children? When it is made into little pats. Why is a chronometer like thingumbob? Because it's a watch-you-may-call-it.

Of what color is gra.s.s when covered with snow? Invisible green.

Name in two letters the destiny of all earthly things? D. K.

What is even better than presence of mind in a railway accident? Absence of body. What word contains all the vowels in due order? Facetiously.

Why is a caterpillar like a hot roll? Because its the grub that makes the b.u.t.terfly. What is that which occurs twice in a moment, once in a minute, and not once in a thousand years? The letter M.

What is that which will give a cold, cure a cold, and pay the doctor's bill? A draught (draft).

What is that which is neither flesh nor bone, yet has four fingers and a thumb? A glove.