736 



SCIENTIFIC RECREATIONS. 



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), 



Suppose we try the 3 3 game, which consists in filling ev^ry hole with 

 the exception of the centre one, and in "taking" all the balls, leaving one 



Fig. 867. Solitaire. 



solitary in the centre at the last. Suppose an inexperienced player arrives 

 at an impossible solution of five balls in 4, 1 1, 15, 28, and 30. 



To render the problem soluble, and to win his game, I will replace 

 No. II by two equivalents, 9 and 10, the ball 28 by two others, 23 and 

 1 6, 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. 



There are in reality three terminations possible to the problem the 

 single ball, the coiiple, and the tierce ; that is, you may have only one 

 left, or two placed diagonally, such as 9-17, 25-29, or a system of three in 



