CH. XI] MISCELLANEOUS PROBLEMS 241 



the problem the cards are held in the hand face upwards. The 

 result can be modified to cover any other way of dealing. 



Request a spectator to note a card, and remember in which 

 pile it is. After finishing the deal, ask in which pile the card 

 is. Take up the three piles, placing that pile between the 

 other two. Deal again as before, and repeat the question as 

 to which pile contains the given card. Take up the three 

 piles again, placing the pile which now contains the selected 

 card between the other two. Deal again as before, but in 

 dealing note the middle card of each pile. Ask again for the 

 third time in which pile the card lies, and you will know that 

 the card was the one which you noted as being the middle 

 card of that pile. The trick can be finished then in any way 

 that you like. The usual method — but a very clumsy one — is 

 to take up the three piles once more, placing the named pile 

 between the other two as before, when the selected card will be 

 the middle one in the pack, that is, if 27 cards are used it will 

 be the 14th card. 



The trick is often performed with 15 cards or with 21 cards, 

 in either of which cases the same rule holds. 



Gergonne's Generalization. The general theory for a pack 

 of m m cards was given by M. Gergonne*. Suppose the pack 

 is arranged in m piles, each containing m™ -1 cards, and that, 

 after the first deal, the pile indicated as containing the selected 

 card is taken up ath ; after the second deal, is taken up 6th ; 

 and so on, and finally after the mth deal, the pile containing 

 the card is taken up kth. Then when the cards are collected 

 after the mth deal the selected card will be nth from the top 

 where 



if m is even, n = km™' 1 — jm™^ 1 + . . . + bm — a + 1, 

 if m is odd, n = km™- 1 — jm™-* + ... — bm + a. 



For example, if a pack of 256 cards (i.e. m= 4) was given, 

 and anyone selected a card out of it, the card could be de- 

 termined by making four successive deals into four piles of 

 64 cards each, and after each deal asking in which pile the 

 * Gergonne's Annates de Mathimatiquea, Nismee, 1813-4, vol. iv, pp. 276 — 

 283. 



B. B. 16 



