Solitaire

[Dae314]

I was playing solitare on Windows (yes I was THAT bored) and I won two games in a row. As I was playing it dawned on me that there are only a finite number of deck configurations (and therefore only a finite number of solitaire configurations) that can be dealt. I decided to see if there was any information online about what probability there was to have a solvable game (this means that there does exist a certain pattern of putting down cards that follows the rules of solitaire where you win). I was also curious to see if there even existed unsolvable solitaire games (games where no matter what order you do the card movements in you cannot win).

Wikipedia had my answer for me (quoted below):

For a "standard" game of Klondike (of the form: Draw 3, Re-Deal Infinite, Win 52) the number of solvable games (assuming all cards are known) is between 82-91.5%.[1] The number of unplayable games is 0.25%[2] and the number of games that cannot be won is between 8.5-18%.[1]

"Unplayable" means that no cards can be moved anywhere, even at the start of the game. This should not be confused with a game in which some cards are moved but later the game is "lost" because not all 52 cards can be moved to the foundations. A game that has been "won", in this case, has 52 cards placed to the foundations. So there are unplayable lost games, playable lost games, and won games.[2]

A modified version of the game called "Thoughtful Solitaire", in which the identity of all 52 cards is known, has a known solution strategy that works 82% of the time but requires significant computing power. Because the only difference between the two games (Klondike and Thoughtful) is the knowledge of card location, all Thoughtful games with solutions will also have solutions in Klondike. Similarly, all dead-ends in Thoughtful will be dead ends in Klondike.[1] However, the theoretical odds of winning a standard game of non-Thoughtful Klondike are currently unknown. It has been said that the inability for theoreticians to calculate these odds is "one of the embarrassments of applied probability".

I had to lol at the end about statisticians being unable to figure out the exact probability of getting a winnable/unwinnable solitaire hand. However I am curious (the source link on wikipedia only gave me abstracts) about what the problem is with calculating the exact probability of a winning/losing-playable/losing-unplayable game of solitaire is. If anyone wants to help me find the research (I haven't really looked around yet) or wants to try running the numbers themselves (I don't have much probability background so I can't do this) and tell us the results, I think it would be interesting :3.

[lemmingllama]

The thing about running the numbers for proper Solitaire is that you dont know where all the cards are, and neither does the person playing. This means that they could screw up a winnable game simply by making dumb moves. The only way to estimate this is by taking the win/loss ratios of, let say, all windows users who play solitaire on their computers. This would only give an estimate though, as it would vary from person to person and their relative IQ/motivation. So that is why it is impossible to get one number for the probability

[Dae314]

I think you can take personal skill out of it though and say that a certain deck configuration is winnable if there exists a sequence of plays that wins the game. You can run numbers on that and get the pure probability of even dealing out a "winnable" solitaire game.

[lemmingllama]

But thats what you have above. Really its just that they havent bothered to sit down and figure out each card configuration because it would take too long and there isnt a good algorithm that could quickly go through configurations. So unless you can make a way to let the computer auto-solve multiple configurations per second, then we wont know anytime soon