2×3×4×5×6×7×8×9×10×11×12×13×14×15.
That is to say, 1,307,674,368,000 in all.
Solitaire.
This somewhat ancient amusement is well known, and the apparatus consists of a board with holes to receive pegs or cups to receive the balls, as in the illustrations (figs. 867 and 870.) The usual solitaire board contains thirty-seven pegs or balls, but thirty-three can also be played very well. Many scientific people have made quite a study of the game, and have published papers on the subject. M. Piarron de Mondesir has given two rules which will prove interesting.
The first is called that of equivalents, and supposes the game to be played out to a conclusion; the second, called the ring-game, admits of a calculation being made so that the prospects of success can be gauged beforehand.
The method of play is familiar, so we need not detail it. It is simply “taking” the balls by passing over them in a straight line. The method of “equivalents” consists in replacing one ball with two others, as we will proceed to explain by the diagram (fig. 868).
Fig. 867.—Solitaire.
Suppose we try the 33 game, which consists in filling every hole with the exception of the centre one, and in “taking” all the balls, leaving one solitary in the centre at the last. Suppose an inexperienced player arrives at an impossible solution of five balls in 4, 11, 15, 28, and 30.
To render the problem soluble, and to win his game, I will replace No. 11 by two equivalents, 9 and 10, the ball 28 by two others, 23 and 16, and the ball 30 by 25 and 18. These substitutions will not change the “taking off,” for I can take 10 with 9, 23 with 16, and 25 with 18. But by so doing I substitute for an irreducible solution of five balls a new system of eight (those shown with the line drawn through them in the diagram), which can easily be reduced to the desired conclusion, and the game will be achieved.