You have just won $6,000,000 at a casino.
probability/die.shuffle New, unsolved
New entry, edited from material collected for the archive and not yet in a released version.
Problem
You have just won $6,000,000 at a casino. The manager arranges six cards, numbered 1 to 6, in order: 1 2 3 4 5 6. You also have six cards, numbered similarly, which you have to arrange in any order you choose. The manager then shuffles his cards by throwing a die 6 times. The first time, he exchanges his 1st card with the card whose position is shown by the die (if the die shows 1, he leaves them as they were). The 2nd time, he exchanges his 2nd card with the card whose position is shown on the die.... and so on ... When his shuffle is complete, if his cards match yours, you lose all your winnings :-( How would you arrange your cards? What are the odds that you will lose?
Show solution
No solution was found in the collected postings. In article <51c38o$ngo@gaia.aoainc.com>, carl@aoainc.com (Carl Witthoft) writes: >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 . But the manager's procedure COULD result in an ODD number of transpositions! E.g. suppose the manager rolls 1,2,3,4,5,1. Then the manager's arrangement is 6 2 3 4 5 1 which is achieved by ONE transposition. ALL permutations are reachable following this procedure. But what I found surprising is that there is a UNIQUE arrangement of cards that minimizes your chances of losing (equivalently, maximizes your chances of winning). It is very difficult to find this permutation (and its probability of losing) without a computer. I already posted the answer to this problem but for now will simply let you work on it. Does anybody have an easily understandable solution that doesn't use a computer? -- Glenn Rhoads <rhoads@george.rutgers.edu> [ Editor: Rhoads's earlier posting of the answer is not in the collected material. ]