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Tracing Card Paths during Perfect Shuffles

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This Demonstration displays the path that each card takes through a deck over the course of a sequence of consecutive perfect shuffles. Each colored line traces the path of one card as a plot of position versus time. Use the controls to adjust the number of cards in the deck and whether the shuffle is an in-shuffle or an out-shuffle. The minimum number of shuffles required to return the deck to its original order is shown at the top.

Contributed by: Adam Rumpf (November 2017)
Open content licensed under CC BY-NC-SA

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A perfect shuffle is a type of riffle shuffle wherein a deck of cards is divided into two equal halves, and then the halves are interlaced with one another. There are two types of perfect shuffles: an out-shuffle, for which the original top card remains on top, and an in-shuffle, for which the original top card becomes the new second card [1].

Repeating a perfect shuffle on a finite deck must eventually return it to its original order. If the deck contains cards (where is even), then the number of out-shuffles required is the multiplicative order of modulo , while the number of in-shuffles required is the multiplicative order of modulo [2].

References

[1] P. Diaconis, R. L. Graham and W. M. Kantor, "The Mathematics of Perfect Shuffles," Advances in Applied Mathematics, 4(2), 1983 pp. 175–196. doi:10.1016/0196-8858(83)90009-X.

[2] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "Multiplicative Order of ." oeis.org/A002326.

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