After more thought, I cannot see any way to easily fix the error. I have updated the article to insert warnings that I have made a huge mistake.

Sorry for the mistake. Props to Jan for figuring this out; thank you! (And sorry for taking so long to update it.)

]]>I thought about the original problem, and I can’t see a good general solution (is there a group theorist in the house?) But there is some exploitable symmetry.

The better a sequence is for getting out forward (ABCDEF), the worse it is for getting out reverse (FEDCBA) – any two cards that are correctly ordered for forwards are incorrectly ordered for reverse. Since forward and reverse are symmetrical, the number of permutations that will work at each number of runs is also symmetrical (the more that work forward, the fewer work reverse).

In particular, the number of permutations that will work for n/2 runs forward is the same as the number that will work for n/2 runs backwards, and this must therefore be half the total number of runs.

So in the special case of n/2 runs through n cards, your 50% chance is not a coincidence.

]]>http://webpbn.com/survey/

and

http://politaire.com/article/captivequeens.html

By the way, I too spent several days thinking myself smart for noticing that the suits in Captive Queens were separable before I saw the flaw in that argument.

]]>Whoa! Good point. I need to give this some more thought.

]]>Alas, our answers don’t agree. We aren’t even close. I found 59% of games winnable.

I think you are overlooking the fact that, under the normal rules, after you play a card from the waste, you don’t have to deal a new card. You can play the next card from the waste. So if the cards were all backwards, FEDCBAFEDCBA, you could still win the game easily by playing all cards to the waste, and then playing them off the the waste in reverse order. In effect, you can use the waste to reverse any subsequence of the deck as you make a pass through the deck. This helps hugely.

But this brings up another problem. The suits aren’t separable once you allow playing multiple cards off the waste. FEDCBA of hearts is easily winnable without a redeal, but not if there is some other unplayable card of another suit between the B and the A of hearts. This loss of separability makes the analysis quite a bit more difficult. Too difficult for me anyway, which is why I fell back on doing the dim-witted Monte Carlo thing.

]]>