Patience Game - 1

Hi,

Does anyone know (or work out) the probability of the patience game called clock working out?

If you are not familiar with this game then the rules can be found at:
http://www.solitaire-card-game.com/rules/clock.htm

Cheers in advance.

Yes, exactly 1 in 13. A discussion can be found in Martin Gardner's book _Mathematical_Magic_Show_ (chapter 17, pp. 240-250), from his Scientific American column.

Michael Keller Solitaire Laboratory <http://home.earthlink.net/~fomalhaut/solitlab.html>

I know how to figure out the probability.

Theorem: You will lose the game if and only if there is a cycle among the 12 cards on the bottom of the 12 'hour' piles.

Exercise 1: Prove this theorem.

Exercise 2: Calculate the probability of not getting a cycle among the first 12 cards in the deck (this is the probability of winning).

Here's my method of working out the probability:

Consider a variant of the game where you start by playing to the existing rules. When you get stuck, you log the fact that this game was lost, but continue playing by picking the top card from the lowest numbered pile that's got cards stuck in it (E.g. if there are any cards stuck at One O'clock, pick from there). Any other rule that uniquely specifies how to continue after you get stuck will do. Continue until all 52 cards have been played, using the continuation rule as often as necessary.

I claim that the sequence in which the 52 cards get turned over is random.

The game works out if and only if the 52nd card in this sequence is a king. The probability of that is clearly 1 in 13.