CH. XI] 



MISCELLANEOUS PROBLEMS 



237 



utmost number of shufflings necessary before all the cards will 

 be brought back to their original places. In the case of a 

 pack of 52 cards, the greatest l.c.m. of numbers whose sum 

 is 52 is 180180. 



Arrangements by Rows and Columns. A not uncommon 

 trick, which rests on a species of shuffling, depends on the 

 obvious fact that if n* cards are arranged in the form of a 

 square of n rows, each containing n cards, then any card will 

 be denned if the row and the column in which it lies are 

 mentioned. 



This information is generally elicited by first asking in 

 which row the selected card lies, and noting the extreme left- 

 hand card of that row. The cards in each column are then 

 taken up, face upwards, one at a time beginning with the 

 lowest card of each column and taking the columns in their 

 order from right to left — each card taken up being placed on 

 the top of those previously taken up. The cards are then 

 dealt out again in rows, from left to right, beginning with the 

 top left-hand corner, and a question is put as to which row 

 contains the card. The selected card will be that card in the 

 row mentioned which is in the same vertical column as the 

 card which was originally noted. 



The trick is improved by allowing the pack to be cut as 

 often as is liked before the cards are re-dealt, and then giving 

 one cut at the end so as to make the top card in the pack 

 one of those originally in the top row. For instance, take the 



00 

 00 



00 

 00 



0000 

 0000 

 0000 



0000 



Figure l 



Figure ii. 



