Mathematical_Recreations-Kraitchik-2e

Games 321
up after the fifth deal in this way, the order of the cards willbe
reversed, and any card whose original order was a multiple
of three still has that property. (Marceil)
This table gives the results of the five deals:
1 2 3 4 5
1, 2, 3, 4 29,25,21,17 16,32,15,31 2, 4, 6, 8 25,17, 9, 1
5, 6, 7, 8 13, 9, 5, 1 14,30,13,29 10,12,14,16 26,18,10, 2
9,10,11,12 30,26,22,18 12,28,11,27 18,20,22,24 27,19,11, 3
13,14,15,16 14,10, 6, 2 10,26, 9,25 26,28,30,32 28,20,12, 4
17,18,19,20 31,27,23,19 8,24, 7,23 1, 3, 5, 7 29,21,13, 5
21,22,23,24 15,11, 7, 3 6,22, 5,21 9,11,13,15 30,22,14, 6
25,26,27,28 32,28,24,20 4,20, 3,19 17,19,21,23 31,23,15, 7
29,30,31,32 16,12, 8, 4 2,18, 1,17 25,27,29,31 32,24,16, 8
14. MONGE'S SHUFFLE. Monge investigated the follow~
ing method of shuffling cards. Form the deck by placing the
second card above the first, the third underneath the pair,
the fourth above the first three, the fifth below the first four,
and so on. With a pack of 52 cards, for example, one gets
the following arrangement: 52, 50, 48, ... , 6, 4, 2, 1, 3,
5, ... , 47, 49, 51.
If the pack contains 6k + 4 cards, the (2k + 2)th card occupies
the same position after shuffling as before. In a pack
of 10h + 2 cards the (2h + l)th and the (6h + 2)th cards are
interchanged. Hence on the ordinary deck of 52 cards, the
18th remains fixed and the 11th and 32nd are interchanged.
More generally, with a pack of 2p cards the final position
Xl of the card whose initial position was Xo is
Xl = t(2p + Xo + 1) if Xo is odd,
Xl = t(2p - Xo + 2) if Xo is even.