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.