Here is a very simple proof that the number must be 1:—
| Let | a + b | = | no. of sheep |
| then | a2 + b | = | b2 + a |
| a2 - b2 | = | a - b | |
| or | (a + b)(a - b) | = | a - b |
| therefore | a + b | = | 1. |
101.
A horse that carries a load can draw a greater weight up the shaft of a mine than a horse that bears no burden. The load holds him more firmly to the ground, and thus gives him greater power over the weight he is raising from below.
102.
In the six chests, of which two contained pence, two shillings, and two pounds, there must have been at least the value of 506 pence. This can be divided into 22 (or 19 + 3) shares of 23d. each, or 23 (19 + 4) shares of 22d. each. Evidently then the treasure can be divided so that 19 men have equal shares, while their captain has either 3 shares or 4 shares.