Fig. 863.—Magic square formed by terms of geometrical progression.
We have so far considered only the first part of the puzzle. We may now examine the problem to which specially it has given rise. We are quite in accord with M. Piarron de Mondesir, who has been so good as to enlighten us upon the subject, which is really much more difficult than it appears.
A French paper once proposed to give a prize of 500 francs to any individual who would solve the following problem:—
Throw the numbers out of the box, replace them at hazard, then in arranging them place them in the following order (A fig. 864).
Fig. 864.—The Sixteen Puzzle.
Now nobody solved this problem, because in nine cases out of ten it is impossible to do so. The first twelve numbers will come correctly into their places, and even 13 can be put in its place without much trouble; but, instead of getting the last row right we shall find it will come out like B, viz., 14, 15, 13, in the large majority of instances. So any case can be solved in one of the two results given above, and we can tell in advance, without displacing a number, in which way the puzzle will eventuate.
Fig. 865.—Example 1.