Clock solitaire is played with a standard deck of 52 cards. The cards are dealt out, face down, into 13 piles. We will assume that 12 of these piles are arranged in a circle, at positions corresponding to the hour markings on a clock, and that the 13th pile is in the center of the circle.
Play proceeds by repeatedly taking a card from some pile. The first card is taken from the top of the central pile. The value of this card specifies the pile from which the next card is drawn, and so on. Thus, if the first card was a 3, the next card is drawn from the top of the pile at 3'oclock. If that card is a 7, the next card is drawn from the top of the pile at 7 o'clock. The Jack counts as the pile at 11 o'clock, the Queen the pile at 12 o'clock, and the King as the center pile.
Play stops when the last card is drawn (success), or when the next specified pile is actually empty although some cards remain in other piles (failure).
You can easily imagine a situation in which failure will occur - for instance, put all four kings in the center pile. Just as surely, you can imagine a situation in which all the cards are picked up: just have four aces in the king pile, four 2's in the ace pile, four 3's in the 2's pile, and so on, to four kings in the queens pile. Then play will sweep around the clock four times, ending perfectly at the 12 o'clock pile.
If you play this game, you will find that it's hard, but not impossible, to win. The question is, can you determine the exact probability of winning a randomly dealt game of clock solitaire?
I give up, show me the solution.