CH. XI] MISCELLANEOUS PROBLEMS 245 



taken up in order that after the nth. deal the selected card may 

 be rth from the top. 



The principle will be sufficiently illustrated by one example 

 treated in a manner analogous to the cases already discussed. 

 For' instance, suppose that an 6carte" pack of 32 cards is dealt 

 into four piles each of 8 cards, and that the pile which contains 

 some selected card is picked up ath. Suppose that on dealing 

 again into four piles, one pile is indicated as containing the 

 selected card, the selected card cannot be one of the bottom 

 2 (a — 1) cards, or of the top 8 — 2a cards, but must be one of 

 the intermediate 2 cards, and the trick can be finished in any 

 way, as for instance by the common conjuring ambiguity of 

 asking someone to choose one of them, leaving it doubtful 

 whether the one he takes is to be rejected or retained. 



The Mouse Trap. Teeize. I will conclude this chapter 

 with the bare mention of another game of cards, known as 

 the Mouse Trap, the discussion of which involves some rather 

 difficult algebraic analysis. 



It is played as follows. A set of cards, marked with the 

 numbers 1, 2, 3, . . . , n, is dealt in any order, face upwards, in 

 the form of a circle. The player begins at any card and counts 

 round the circle always in the same direction. If the kth card 

 has the number k on it — which event is called a hit — the 

 player takes up the card and begins counting afresh. According 

 to Cayley, the player wins if he thus takes up all the cards, and 

 the cards win if at any time the player counts up to n without 

 being able to take up a card. 



For example, if a pack of only four cards is used and these 

 cards come in the order 3214, then the player would obtain 

 the second card 2 as a hit, next he would obtain 1 as a hit, 

 but if he went on for ever he would not obtain another hit. 

 On the other hand, if the cards in the pack were initially in 

 the order 1423, the player would obtain successively all four 

 cards in the order 1, 2, 3, 4. 



The problem may be stated as the determination of what 

 hits and how many hits can be made with a given number of 



