You have a deck of 52 cards - for convenience, number them 1 through 52. You cut the cards into two equal halves and shuffle them perfectly. That is, the cards were in the order
1,2,3,...,52
and now they are
1,27,2,28,...,26,52. Let's call this a perfect in-shuffle.

If you repeat this in-shuffling process, how many in-shuffles will it take for the deck to return to its initial ordering (taking for granted that the cards will eventually do so)?
________________________

How does the solution change if you have a deck of 64 cards, or 10, or in general, n cards? For odd integer values of n, in-shuffling will take 1,2,3,...,n to 1,(n+3)/2,2,(n+5)/2,...,n,(n+1)/2. For example, when n=5, the first in-shuffle yields 1,4,2,5,3.

(In reply to there's a pattern there. I just know it! by GOM)

For powers of 6 I get

6             4
36            12
216           28
1296          36
7776          620
46656         420
279936        111972
1679616       648
10077696      34524
60466176      9300
362797056     3154756

Using

DEFLNG A-Z
CLS
sz1 = 6
DO
  size = sz1 - 1
  lg = 1: pw = 2
  DO
   lg = lg + 1: pw = (pw * 2) MOD size
  LOOP UNTIL pw MOD size = 1
  shCt = lg
  n = (size - 3) / 2
  col = n \ 40
  row = n MOD 40 + 1
  PRINT sz1, shCt
  sz1 = sz1 * 6
LOOP

Posted by Charlie on 2004-05-25 11:46:31