CH. XI] MISCELLANEOUS PROBLEMS 235 



Shuffling a Pack. Any system of shuffling a pack of 

 cards, if carried out consistently, leads to an arrangement which 

 can be calculated ; but tricks that depend on it generally require 

 considerable technical skill. 



Suppose for instance that a pack of n cards is shuffled, as 

 is not unusual, by placing the second card on the first, the third 

 below these, the fourth above them, and so on. The theory of 

 this system of shuffling is due to Monge*. The following are 

 some of the results and are not difficult to prove directly. 



One shuffle of a pack of 2p cards will move the card which 

 was in the «„th place to the a^th place, where x^ = \ (2p + x + 1) 

 if x„ is odd, and x 1 = £ (2p - x + 2) if x„ is even. For instance, 

 if a complete pack of 52 cards is shuffled as described above, 

 the 18th card will remain the 18th card. If an ^cart6 pack of 

 32 cards is so shuffled, the 7th and the 20th cards will change 

 places. 



By repeated applications of the above formulae we can show 

 that the effect of m such shuffles is to move the card which was 

 initially in the a? th place to the « m th place, where 



2 m+1 x m = (4>p + 1) (2 m ~ 1 ± 2™~* ± . . . ± 2 + 1) + 2x + 2 m ± 1, 

 the sign ± representing an ambiguity of sign. 



Again, in any pack of n cards after a certain number of 



shufflings, not greater than n, the cards will return to their 



primitive order. This will always be the case as soon as the 



original top card occupies that position again. To determine 



the number of shuffles required for a pack of 2p cards, it is 



sufficient to put x m = x and find the smallest value of m which 



satisfies the resulting equation for all values of x from 1 to 2p. 



The result can however be obtained more easily if the cards 



* Monge's investigations are printed in the MSmoires de VAcadimie des 

 Sciences, Paris, 1773, pp. 390 — 412. Among those who have studied the eubjeot 

 afresh I may in particular mention V. Bouniakowski, Bulletin physico-mathe'ma- 

 tique de St Pitersbourg, 1857, vol. xv, pp. 202—205, summarised in the Nouvelles 

 annates de matMmatiques, 1858, Bulletin, pp. 66—67 ; T. de St Laurent, Memoires 

 de V Academic de Gard, 1865 ; L. Tanner, Educational Times Reprints, 1880, 

 vol. xxxin, pp. 73 — 75 ; M. J. Bonrget, Liouville's Journal, 1882, pp. 413 — 431; 

 H. F. Baker, Transactions of the British Association for 1910, pp. 526—528 ; and 

 P. H. Cowell, The Field, 2 April, 1921, p. 444. 



