242 MISCELLANEOUS PROBLEMS [CH. XI 



selected card lay. The reason is that after the first deal you 

 know it is one of sixty-four cards. In the next deal these 

 sixty-four cards are distributed equally over the four piles, and 

 therefore, if you know in which pile it is, you will know that it 

 is one of sixteen cards. After the third deal you know it is 

 one of four cards. After the fourth deal you know which card 

 it is. 



Moreover, if the pack of 256 cards is used, it is immaterial 

 in what order the pile containing the selected card is taken up 

 after a deal. For, if after the first deal it is taken up ath, 

 after the second 6th, after the third cth, and after the fourth 

 dth, the card will be the (64eZ - 16c + 46 — a + l)th from the 

 top of the pack, and thus will be known. We need not take 

 up the cards after the fourth deal, for the same argument will 

 show that it is the (64 — 16c + 46 — a + l)th in the pile then 

 indicated as containing it. Thus if a = 3, 6 = 4, c = 1, d = 2, 

 it will be the 62nd card in the pile indicated after the fourth 

 deal as containing it and will be the 126th card in the pack 

 as then collected. 



In exactly the same way a pack of twenty-seven cards may 

 be used, and three successive deals, each into three piles of 

 nine cards, will suffice to determine the card. If after the 

 deals the pile indicated as containing the given card is taken 

 up ath, 6th, and cth respectively, then the card will be the 

 (9c — 36 + a)th in the pack or will be the (9 — 36 + a)th card in 

 the pile indicated after the third deal as containing it. 



The method of proof will be illustrated sufficiently by 

 considering the usual case of a pack of twenty-seven cards, for 

 which m = 3, which are dealt into three piles each of nine cards. 



Suppose that, after the first deal, the pile containing the 

 selected card is taken up ath : then (i) at the top of the pack 

 there are a — 1 piles each containing nine cards ; (ii) next there 

 are 9 cards, of which one is the selected card ; and (iii) lastly 

 there are the remaining cards of the pack. The cards are dealt 

 out now for the second time : in each pile the bottom 3 (a — 1) 

 cards will be taken from (i), the next 3 cards from (ii), and the 

 remaining 9 — 3a cards from (iii). 



