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Faulty Card Shuffler (Posted on 2013-07-24) Difficulty: 3 of 5
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).

See The Solution Submitted by Charlie    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(3): ? Solution for any set of subcycles? Comment 10 of 10 |
(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

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
  Posted by Steve Herman on 2013-07-29 23:37:59

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