MONTMORT. Ill 



venu a la derniere de chaque tas, apres les avoir enleve toutes deux a 

 deux, auquel cas seulement il a gagne. 



The game is not entirely a game of pure chance, because the 

 l^layer may often have a choice of various methods of pjairing and 

 removing cards. In the description of the game forty cards are 

 supposed to be used, but Montmort proposes the problem for solu- 

 tion generally without limiting the cards to forty. He requires 

 the chance the player has of winning and also the most ad- 

 vantageous method of i^roceeding. He says the game was rarely 

 played for money, but intimates that it was in use aniong ladies. 



192. On his page 821 Montmort gives, without demonstration, 

 the result in a particular case of this problem, namely when the 

 cards consist of ?2 pairs, the two cards in each pair being numbered 

 alike ; the cards are supposed placed at random in n lots, each of 

 two cards. He says that the chance the player has of winning is 



92 — 1 



^ — -. On page 334^ Nicolas Bernoulli says that this formula is 



correct, but he wishes to know how it was found, because he him- 

 self can only find it by induction, by jDutting for n in succession 

 2, 3, ^,o, ...We may suppose this means that Nicolas Bernoulli veri- 

 fied by trial that the formula was correct in certain cases, but could 

 not give a general demonstration. Montmort seems to have 

 overlooked Nicolas Bernoulli's inquiry, for the problem is never 

 mentioned again in the course of the correspondence. As the result 

 is remarkable for its simplicity, and as Nicolas Bernoulli found the 

 problem difficult, it may be interesting to give a solution. It will 

 be observed that in this case the game is one of pure chance, as the 

 player never has any choice of courses open to him. 



193. The solution of the problem depends on our observing 

 the state of the cards at the epoch at which the player loses, that 

 is at the epoch at which he can make no more pairs among the 

 cards exj^osed to view ; the player may be thus arrested at the 

 very beginning of the game, or after he has already taken som^j 

 steps : at this epoch the player is left icitk some number of lots, 

 which are all unbroken, and the cards exposed to vieiu present no 

 pairs. This will be obvious on reflection. 



