Probability of “clock patience” going out

Question

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!

Answer 1

Here's an explanation of why it's 1/13.

We are essentially dealing out a randomly ordered deck into piles of a kind of four, and we require that the pile of kings be the last pile completed. And so the probability of winning is 1/13.

Maybe this is more convincing. Imagine playing the game backwards with the deck facing you so that you can see the cards. Remove the top card, a seven, say, the next card facing you is a two, say, so you place the seven in the pile at the two o'clock position, the next card facing you is a five, so you place the two at the five o'clock position. You now take the five from the deck and see a king beneath it, so you place the five in the king pile, and so on.

You continue until you've placed out all the cards and, if you are going to win this game (when it's played in the correct direction), the last card must be placed in the king pile because that's where you take your first card from when you run the game in the right direction. Now the only way that can happen is if you placed a king down as your first card. And that's a 1/13 chance with a random ordered deck.

Answer 2

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.

Answer 3

If I understand you correctly, then the game is won if the $48$ cards on the regular piles contain fewer than $4$ kings. Let's try to calculate the probability that the game is lost instead. It is exactly $$\frac{\binom{48}{44}}{\binom{52}{48}} = \frac{38916}{54145} \approx 72\%.$$ There are $\binom{52}{48}$ possible choices of the regular piles, and out of these $\binom{48}{44}$ contain all four kings.