Probability of "clock patience" going out Here is a question I've often wondered about, but have never figured out a satisfactory answer for. Here are the rules for the solitaire game "clock patience." Deal out 12 piles of 4 cards each with an extra 4 card draw pile. (From a standard 52 card deck.) Turn over the first card in the draw pile, and place it under the pile corresponding to that card's number 1-12 interpreted as Ace through Queen. Whenever you get a king you place that on the side and draw another card from the pile. The game goes out if you turn over every card in the 12 piles, and the game ends if you get four kings before this happens. My question is what is the probability that this game goes out?
One thought I had is that the answer could be one in thirteen, the chances that the last card of a 52 card sequence is a king. Although this seems plausible, I doubt it's correct, mainly because I've played the game probably dozens of times since I was a kid, and have never gone out!
Any light that people could shed on this problem would be appreciated!

The name clock patience (solitaire in the US) is appropriate, not just because of the shape of the layout but because it is about cycles in the permutation.  As you start with the King pile, a cycle ends when you find a King.  If four cycles (one for each King) include all 52 cards, you win.  You lose if the bottom card on any non-King pile matches its position, as that would be a one-card cycle in the permutation.  You also lose if the bottom card in the Ace pile is a Two and the bottom card in the Two pile is an Ace.  I'm trying to figure out the impact of the fact that suits are ignored.  Maybe you can give each card its particular destination (always put the spade ace on top, for example) and ask for a single cycle of the 52 cards.  In that case, it would be 1/52.  To make a single cycle, the first card cannot go to itself (51/52).  The card the first card goes to cannot go to the first (50/51).  Then the next card in the chain cannot go to the first (49/50) and so on.
