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 Faulty Card Shuffler (Posted on 2013-07-24)
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.)
 Brians solution | Comment 5 of 10 |

I agree with Brian's result.

Position 0 is  A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. For the computer-illiterate like me, set this up as  row 0 in Excel. Enter 10, 9, Q, 8, K, 3, 4, A, 5, J, 6, 2, 7 in row 1, then cross reference each entry in row 2 to row 0 so that row 2 matches row 1. Row 1 can now be deleted.

Now the 'formulae' in row 2 can be copied and pasted in the even rows until row 0 repeats at row 26. In particular, row 12 is 2,8,7,3,10,K,6,Q,A,9,5,4,J, and row 14 is 9,A,4,Q,J,7,3,2,10,5,K,8,6

Assume that row 13 repeats row 0. We can achieve this simply by cross referencing in row 13 each entry in row 12 to produce the order in row 0.

If this assumption is correct, then using the same row 13 'formula' again will produce a copy of row 14. In fact, it does:

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

So the 'formula' to turn row 0 into row 1 is the same, and row 1 = row 14. The complete table (0-13) is:

0   A 2 3 4 5 6 7 8 9 10 J Q K
1   9 A 4 Q J 7 3 2 10 5 K 8 6
2   10 9 Q 8 K 3 4 A 5 J 6 2 7
3   5 10 8 2 6 4 Q 9 J K 7 A 3
4   J 5 2 A 7 Q 8 10 K 6 3 9 4
5   K J A 9 3 8 2 5 6 7 4 10 Q
6   6 K 9 10 4 2 A J 7 3 Q 5 8
7   7 6 10 5 Q A 9 K 3 4 8 J 2
8   3 7 5 J 8 9 10 6 4 Q 2 K A
9   4 3 J K 2 10 5 7 Q 8 A 6 9
10 Q 4 K 6 A 5 J 3 8 2 9 7 10
11 8 Q 6 7 9 J K 4 2 A 10 3 5
12 2 8 7 3 10 K 6 Q A 9 5 4 J
13 A 2 3 4 5 6 7 8 9 10 J Q K

Edited on July 25, 2013, 3:32 am
 Posted by broll on 2013-07-25 03:09:11

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