Mathematical_Recreations-Kraitchik-2e

Mathematical Recreations
If the deck be shuffled m times in this way we find the following
relation connecting the final position Xm and the initial
position Xo of any card:
A pack of n cards will regain its original order after a certain
number m of such shuffiings. This number may be
found by sUbstituting Xo for Xm in the preceding formula and
solving for m. The value of m will depend upon the divisibility
of 4m - 1 by 4p + 1.
Here is a recreation based on these formulas. Arrange the
cards in the order ace to king in spades, then clubs, then
diamonds, then hearts. Shuffle the pack n times by Monge's
system and then cut them any number of times. (Instead
of cutting, one may perform a false shuffling, which consists
in appearing to shuffle while actually forming a succession of
cuts.) Then the cards are dealt out into 4 rows of 13 cards.
N ow we may either guess the position of a given card, or
tell what card occupies a certain position. To do this we
must cheat a little - just a peek at the bottom card. Suppose
that the bottom card is the 7 of diamonds, and that the
cards have been shuffled 3 times. The initial position of this
card was 2 ·13 + 7 = 33. As a result of the shufflings its
position was successively (52 + 34) + 2 = 43, (52 + 44) +
2 = 48, (52 - 46) + 2 = 3. Since after the cutting the card
appeared at the bottom, that is, in the 52nd place, the effect
of the cuts has been to shift every card back 49 places, or
forward 3 places.
Now we can answer any question of either sort. Suppose
we wish to know the location of the jack of hearts. Its initial
position was 11. Ills successive positions after the shufflings
were (52 + 12) + 2 = 32, (52 - 30) + 2 = 11, (52 +
12) + 2 = 32. Raising its position 3 places to take account