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        riddles >> easy >> Clock solitaire
        
(Message started by: visitor on Oct 25^th, 2003, 11:53am)

Title: Clock solitaire
Post by visitor on Oct 25^th, 2003, 11:53am

The insane passenger reminded me of an old game of solitaire I used to play. Take a deck of cards and deal them out in the shape of a clock, one pile of four cards ending up at each numeral position of a clock face, and the thirteenth pile in the center of the clock. You then pick up one card from the center. If it's a 4, you next pick up a card from the 4 o'clock pile. If that's a queen, you take your next card from the 12 o'clock pile. A king means you take another card from the center. The goal is to pick up the entire deck of cards before you hit a dead end by picking up the fourth king (and there are no more cards in the center to continue).
I used to play that game until I came to a realization that took all the fun out of it. What did I realize?

Title: Re: Clock solitaire
Post by Icarus on Oct 25^th, 2003, 5:24pm

Well, I stopped playing it when I finally decided that this was a lot of work just to discover something that was completely decided by the initial shuffle of the deck.

Title: Re: Clock solitaire
Post by SWF on Oct 25^th, 2003, 6:45pm

on 10/25/03 at 11:53:58, visitor wrote:

The goal is to pick up the entire deck of cards before you hit a dead end by picking up the fourth king

Since you can't pick up the entire deck before taking the fourth king, you can't win. :-)

In that game, I think there is another rule you left out: move on to the next pile if you exhaust a pile. For example (and this partially of gives away the answer):[hide] if the last card in the four pile is a four you have might finish that pile any of the ways you list for ending the game occur. Although this does not guarantee you will win, I think victory is impossible unless the bottom card in the king's pile is a king and there is king at the bottom of some other pile.[/hide]

Icarus, why don't you feel the same way about mathematics: just a lot of work to discover something over which you have no control.

Title: Re: Clock solitaire
Post by Icarus on Oct 25^th, 2003, 7:50pm

The interest in a discovery is related to its worth and its beauty. Tell me what is worthwhile or beautiful about discovering that the shuffling of the deck happened to meet the conditions needed for this process to work out?

You can never need to take a card from an exhausted pile except the King pile - and that is the losing condition, so SWF's "exhausted pile" rule is unnecessary.

You have lost if there is not a king on the bottom of one of other 12 piles.

You have also lost if the bottom card of any of the piles is a card for that pile. This includes the King pile, so definitely do not want to have a king at the bottom of it

More generally, you have also lost if there is a "closed loop" of cards on the bottom which does not include a king. For example, in the following situation,


        BOTTOM
PILE     CARD
  A        4
  2        3
  3        5
  4        6
  5        7
  6        A
  7        2
  8        9
  9       10
 10        J
  J        Q
  Q        K
  K        8

you will never turn over the bottom cards of the A through 7 piles. Why? because, in order to turn over the bottom of the Ace pile, you need all 4 aces. One of those aces is on the bottom of the 6 pile. To get it, you need all 4 sixes. One is on the bottom of the 4 pile. To get it you need all 4 fours. One is on the bottom of the Ace pile. So to get the last Ace pile card turned over, you need to turn the last Ace pile card over first. :P I.e. you're out of luck!

If there is a King in the loop, then it's okay, because you only need the third king to turn over the bottom king pile card.

I do not know that a lack of such loops is sufficient to produce a win, however. Only that their presence definitely means a loss!

Title: Re: Clock solitaire
Post by visitor on Oct 25^th, 2003, 8:01pm

Let me rephrase that. The goal is to pick up all 52 cards with the fourth king as the final card.
You'll never run into a dead end by running out of 4's or queens, or anything but the final king.
And btw the bottom card on the king's pile is irrelevant to winning or losing. If you've never played clock solitaire, try it a couple times. What is unusual about what it takes to win this game? [hide]Icarus was actually on the right track, but it's more than that[/hide]

Title: Re: Clock solitaire
Post by visitor on Oct 25^th, 2003, 8:06pm

Ah, you got it while I was responding. The entire outcome of the game is decided by the first 12 cards you deal. If every bottom card loops to a king, you win. If any card or cards create a kingless loop you lose.
Now can you tell me the probability of winning? (There's a problem that goes far beyond my own probability skills).

Title: Re: Clock solitaire
Post by Icarus on Oct 25^th, 2003, 8:10pm

You're right - even if the bottom of the King pile is a king, you can still win, for the same reason a loop as I described is workable, as long as a king is part of it.

Essentially, this game sets up a permutation of the deck. The game is a win iff the permutation consists of a single cycle.

This does not seem like enough to me to explain the sudden loss of interest, however.

I post this and see that you've responded in the meantime. What I've said does not go that far. My example does not include the case of multiple of one card on bottom. I am not sure what you can tell if this is the case. The presence of any kingless loops is a killer, but I don't see that the lack of them necessarily means that the game is winnable.

Title: Re: Clock solitaire
Post by visitor on Oct 25^th, 2003, 8:10pm

And proving that you win if there are no kingless loops is as easy as proving that a kingless loop loses.

Title: Re: Clock solitaire
Post by visitor on Oct 25^th, 2003, 8:12pm

The loss of interest? Why waste time dealing out 52 cards when the game is over after the 12th card hits the table?

Title: Re: Clock solitaire
Post by SWF on Oct 25^th, 2003, 10:56pm

Yes, my memory of how this game works was obviously mistaken. I was thinking the game was over when there are no more face down cards on the king pile. It has been a long time, but I can recall a realization similar to vistor's. I also remember looking under the piles and redealing upon seeing a game was a waste of time (as if playing solitaire, was not already a waste of time). Since the game involves no decisions, peeking is not cheating, but it would be if the cards were rearranged to win. :-)

Icarus is correct about not needing the additional rule I mentioned, but it is interesting that I have a book containing the rules which says,

Quote:

Should a card turn up on the pile of its own number, as a 6 on the pile at six o'clock, simply put it under and turn up the next card. If there is no next card--the pile now comprising all four cards of the pile number--take instead the top card of the next higher pile.

Maybe my memory isn't so bad after all. Funny that this rule should be suggested when the situation is not possible. Maybe that takes care of what to do when the starting configuration was misdealt.

As for computing the probability of winning:[hide]Imagine adding that uneeded rule mentioned above and also do not stop when the last king is turned over. These rule changes allow all cards to eventually be turned over, and treat all ranks of cards equally. However if the last card turned over with the modfied rules was a king, then it would have been a win using normal rules of clock solitaire. Probability of last card being king equals probability of winning clock solitaire = 1/13[/hide].