This card shuffler, instead of randomizing the positions of cards in the deck, always rearranges any given set of cards into the same new order.

If for example one puts all the hearts into the shuffler, in the order A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, and runs that through the shuffler first once, and then, without rearranging the cards, runs it through again, the resulting sequence after the second shuffle is always:

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

The question is:

What was the order of the cards after the first run through this non-randomizing shuffler?

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From the Skeptics' Guide to the Universe podcast episode of June 22, 2013 (episode 414).

I haven't proven it, but I think Steve Herman's method of subcycle determination can be taken to a solution for any set of subcycles just  by multiplying all the subcycle lengths.

 

In his example of a subcycles of 4 and 9, 36 repetitions of the 2-shuffle pattern led to the correct starting position, and the next one was what one would expect after a single shuffle.

I'm not seeing how even length cycles or more than one cycle of length 1 are a problem

Posted by Larry on 2013-07-28 15:03:18