A dealer shuffles six cards, numbered 1 to 6 in order, by rolling a die six times: on the i-th roll, the...

probability/card/die.shuffle New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A dealer shuffles six cards, numbered 1 to 6 in order, by rolling a die
six times: on the i-th roll, the card in position i is switched with the
card in the position shown on the die.  Which arrangement of the cards is
least likely to result?

[ Editor: problem statement reconstructed from the postings, which give
only the answer and the remark that a roll may switch a card with
itself; the exact rule and question are not in the collected postings. ]

Show solution

carl@aoainc.com (Carl Witthoft) wrote:
> I think (a little :=) ) that there must be a couple ways to order
> your cards so that your sequence can't be reached by any even (e.g.,
> six) number of permutations of 1,2,3,4,5,6 .

Except that some of the dealer's permutations may be null - e.g. on the
first roll he rolls a 1, thus card #1 is "switched" with itself.

Allowing such non-moves on the dealer's part, independent computation
confirms Glenn Rhoads's answer of 612345 (32 ways out of 6^6 arrive at
this arrangement).

-- Ed Murphy <zvyezda@ix.netcom.com>