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?

From the

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

(In reply to

re(2): ? Solution for any set of subcycles? by Larry)

Yes, that is correct.

To put it differently,

IF A234 is 34A2 after two shuffles, there is no method than can possibly tell if the initial shuffle was 234A or 4A23, because both initial shuffles lead to 34A2.

Even worse, if A234 is back to A234 after two shuffles, then the initial shuffle could be

A234 or

2A34 or

32A4 or

423A or

A324 or

A432 or

A243 or

2A43 or

34A2 or

432A

That demonstrates the problem with having more than 1 cycle that appears to have length 1. It could be four cycles of length 1, or 2 cycles of length 2 (3 different ways), or 2 cycles of length 1 and one of length 2 (6 different ways), for a total of 10 different possibilities.

*Edited on ***December 28, 2013, 9:44 am**