CH. i] ARITHMETICAL RECREATIONS 17 



the number of cards in that pile will be 13 — a;. A similar 

 argument holds for each pile. Also there are 52 cards in the 

 pack ; and this must be equal to the sum of the cards in the 

 p piles and the r cards left over. 



.\ (13 - 0,) + (13 - x 3 ) + . .. + (13 - x p ) + r = 52, 



.'. 13p - (x 1 + x 2 + ... + x p ) + r = 52, 



.: x 1 + x i + ... + x p = lSp- 52 + r 



= 13 (p_4)+r. 



More generally, if a pack of n cards is taken, and if in each 

 pile the sum of the pips on the bottom card and the number of 

 cards put on it is equal to m, then the sum of the pips on the 

 bottom cards of the piles will be (m + l)p + r — n. In an eearte" 

 pack w= 32, and it is convenient to take ra= 15. 



Fifth Example. It may be noticed that cutting a pack 

 of cards never alters the relative position of the cards provided 

 that, if necessary, we regard the top card as following im- 

 mediately after the bottom card in the pack. This is used in 

 the following trick*. Take a pack, and deal the cards face 

 upwards on the table, calling them one, two, three, &c. as you 

 put them down, and noting in your own mind the card first 

 dealt. Ask some one to select a card and recollect its number. 

 Turn the pack over, and let it be cut (not shuffled) as often as 

 you like. Enquire what was the number of the card chosen. 

 Then, if you deal, and as soon as you come to the original first 

 card begin (silently) to count, reckoning this as one, the 

 selected card will appear at the number mentioned. Of course, 

 if all the cards are dealt before reaching this number, you 

 must turn the cards over and go on counting continuously. 



Sixth Example. Here is another simple question of this 

 class. Remove the court cards from a pack. Arrange the 

 remaining 40 cards, faces upwards, in suits, in four lines thus. 

 In the first line, the 1, 2, ... 10, of suit A; in the second line, the 

 10, 1, 2, ... 9, of suit B; in the third line, the 9, 10, 1, ... 8, of 

 suit 0; in the last line, the 8, 9, 10, 1, ... 7, of suit D. Next 

 take up, face upwards, the first-card of line 1, put below it the 



* Baoliet, problem xix, p. 152. 



