Monge's Shuffle

A Shuffle in which Cards from the top of the deck in the left hand are alternatively moved to the bottom and top of the deck in the right hand. If the deck is shuffled $m$ times, the final position $x_m$ and initial position $x_0$ of a card are related by

$$2^{m+1}x_m=(4p+1)[2^{m-1}+(-1)^{m-1}(2^{m-2}+\ldots+2+1)]+(-1)^{m-1}2x_0+2^m+(-1)^{m-1}$$

for a deck of $2p$ cards (Kraitchik 1942).

See also Cards, Shuffle

References

Conway, J. H. and Guy, R. K. ``Fractions Cycle into Decimals.'' In The Book of Numbers. New York: Springer-Verlag, pp. 157-163, 1996.

Kraitchik, M. ``Monge's Shuffle.'' §12.2.14 in Mathematical Recreations. New York: W. W. Norton, pp. 321-323, 1942.