A device which shuffles cards always rearranges them in the same way relative to the order in which they were placed into it.

combinatorics/shuffling.machine New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A device which shuffles cards always rearranges them in the same way
relative to the order in which they were placed into it.  The Ace
through the King of hearts are arranged in order with the Ace on the
top and the King on the bottom.  After 2 shuffles the order of the
cards is:

10,9,Q,8,K,3,4,A,5,J,6,2,7

Determine the order of the cards after the first shuffle.

-- Frank Pinto <rolls@cis.umassd.edu>, November 1993

Show solution

9,A,4,Q,J,7,3,2,10,5,K,8,6 [ Editor: the last card is "7" in the
original; it must be the 6, which is otherwise missing, as the cycle
below shows. ]

First verify the double shuffle is a cycle.  This implies a single
shuffle is also a cycle.  By noting what positions go where you can
write the double shuffle as the cycle (10 A 8 4 7 K 5 9 2 Q 3 6 J).  By
filling every other position of the single shuffle cycle with an
element of the double cycle shuffle you get the single shuffle cycle
(10 9 A 2 8 Q 4 3 7 6 K J 5).  Applying this cycle to the original card
order gives the above order after the first shuffle.

-- Bruno Wolff III <bruno@cerberus.csd.uwm.edu>