236 MISCELLANEOUS PROBLEMS [CH. XI 



are numbered from the bottom of the original pack. Doing this, 

 we can show that if after s shuffles a card is in the rth place 

 from the bottom, its original number from the bottom was the 

 difference between 2'xr and the nearest multiple of 4p + 1. 

 Hence, if m shuffles are required to restore the original order, 

 m is the least number for which 2 m + 1 or 2 m — 1 is divisible by 

 4p + 1. The number for a pack of 2p + 1 cards is the same as 

 that for a pack of 1p cards. With an ecarte pack of 32 cards, 

 six shuffles are sufficient ; with a pack of 2™ cards, n + 1 shuffles 

 are sufficient ; with a full pack of 52 cards, twelve shuffles are 

 sufficient; with a pack of 13 cards ten shuffles are sufficient; 

 while with a pack of 50 cards fifty shuffles are required ; and 

 so on. 



W. H. H. Hudson* has also shown that, whatever is the 

 law of shuffling, yet if it is repeated again and again on a pack 

 of n cards, the cards will ultimately fall into their initial posi- 

 tions after a number of shufflings not greater than the greatest 

 possible L.C.M. of all numbers whose sum is n. 



For suppose that any particular position is occupied after 

 the 1st, 2nd, ..., pth shuffles by the cards A u A t , ..., A p re- 

 spectively, and that initially the position is occupied by the 

 card A e . Suppose further that after the pth shuffle A 9 returns 

 to its initial position, therefore A„ = A P . Then at the second 

 shuffling A t succeeds A x by the same law by which A x succeeded 

 A,, at the first; hence it follows that previous to the second 

 shuffling A 3 must have been in the place occupied by A± pre- 

 vious to the first. Thus the cards which after the successive 

 shuffles take the place initially occupied by A r are A 2 , A a , ..., 

 A p , Ai ; that is, after the pth shuffle A 1 has returned to the 

 place initially occupied by it: and so for all the other cards 



■"» -"8> •••> -0-J7— 1- 



Hence the cards i„ A a , ..., A p form a cycle of p cards, one 

 or other of which is always in one or other of p positions in the 

 pack, and which go through all their changes in p shufflings. 

 Let the number n of the pack be divided into p, q,r, ... such 

 cycles, whose sum is n; then the L.C.M. of p, q,r, ... is the 

 * Educational Times Beprints, Loudon, 1868, vol. ii, p. 105. 



