Again, 45 may be subtracted from 45 in such a manner as to leave 45 for a remainder. Arrange the following figures, add the rows together, and each row will be 45; subtract the bottom row from the top row, and the sum of the result added together will also be 45.
| 9+8+7+6+5+4+3+2+1=45 |
| 1+2+3+4+5+6+7+8+9=45 |
| 8+6+4+1+9+7+5+3+2=45 |
THE COSTERMONGER'S PUZZLE.
A costermonger bought 120 oranges at two for a penny, and 120 more at three for a penny, and mixed the oranges all together in a basket. He sold them out, hoping to receive back his money again, at the rate of five for twopence; but on counting his money he found that he had sold the oranges for fourpence less than they had cost him. How this happened will be seen by following the accompanying figures. The first forty purchasers of the oranges would take 200 out of the 240 oranges, and taking it for granted that the fruit was equally mixed, would receive for their money 100 of the oranges originally bought at two a penny and 100 of those at three a penny, and would pay for them the sum of 6s. 8d. The forty remaining oranges would bring in, at the same rate, 1s. 4d. only, making 8s. in all. The cost of the fruit was, for the first 120, 5s., and for the second 120, 3s. 4d., or 8s. 4d. in all, making the loss of 4d. on the lot. To more fully explain the matter, we will suppose the oranges not mixed, but standing in separate baskets, from which, for each purchaser, the costermonger takes two of the two a penny oranges and three of the three a penny oranges, disposing of them in that way for twopence; it will then be clearly seen that the basket containing the three a penny oranges will be first exhausted, for the first forty purchasers, each having three oranges from one basket, will take all the 120 oranges purchased at three a penny, but will require only 80 oranges from the other basket, thus leaving 40 of the two a penny oranges to be sold at five for twopence, or a loss of fourpence on the last 40 sold.
THE PROGRESSION OF NUMBERS.
An illustration of the progression of numbers may be gathered from the description given of the American puzzle of "15," at the commencement of this section on Arithmetical Amusements. It is there stated that the different number of combinations or different arrangements of the fifteen cubes that can be made are 1,307,674,318,000. The reader may prove this for himself in the following manner:—The number of combinations that can be made with two cubes is 2, of three cubes 6, of four cubes 24, of five cubes 120, of six cubes 720, and so on, multiplying the result each time by one number higher than the previous result was multiplied by, until the amazing total quoted is reached; the arrangement of the cubes in rows and columns introducing additional variations of combinations. There are numerous instances on record in which it is stated that advantage has been taken of the known progression that ensues upon a repeated doubling of a given result. The Horse-dealer's Bargain is frequently quoted. A horse-dealer having a horse to dispose of, to which a gentleman had taken a great fancy, was asked to name any price he thought fit. Wishing at the first blush to appear generous, he offered to sell his horse, calculating its price according to the number of nails that were used to fasten on the four shoes, a farthing being allowed for the first nail, a halfpenny for the second, a penny for the third, twopence for the fourth, and so on. Upon examination it was found that it took six nails to fasten on each of the shoes, making in all twenty-four nails. The amount arrived at by repeatedly doubling the amount until the twenty-fourth nail had been allowed for was £8,738 2s. 8d.
The story of the Sovereign and the Sage gives a still more wonderful result. A king once, anxious to reward one of his subjects for valuable services performed to the State, asked in what way the subject would take his recompense. The king and the subject were both sixty-four years of age, and the wise man asked that he might be granted a kernel of wheat for the first year of their lives, two for the second, four for the third, eight for the fourth, sixteen for the fifth, and so on. By continuing the calculation until the result has been doubled for the sixty-fourth time, the astounding number of 9,223,372,036,854,775,808 will appear. It is generally conceded that the average number of wheat kernels in a pint is 9,216, which will give 18,432 for a quart, 73,728 for a peck, and 589,824 for a bushel, or 31,274,997,411,298 bushels of grain as the courtier's reward for his services, a larger amount than the whole world would produce in several years.
The Pin in the Hold of the "Great Eastern" Steamship is comparatively a modern calculation, based on this principle. It is calculated that 200 pins go to the ounce, and that if for the fifty-two weeks in the year one pin were dropped into the hold during the first week, two in the second, four in the third, and so on, that by the end of the year the weight of the whole would be no less than 628,292,358 tons of pins. As the Great Eastern steamship was built to carry 22,500 tons only, it follows that to carry all the pins there would be required 27,924 ships of the size of the Great Eastern.
As a last illustration of this subject we will instance the feat of counting a billion, which all boys know is a million millions. Allowing that so many as 200, which is an outside number, could be counted in a minute, it would, excluding the 366th day in leap years, take one person upwards of 9,512 years before the task would be completed. It is not, therefore, probable that any one person has yet counted a billion.
We next proceed to give a few of the rules showing