Skill vs. Chance in Solitaire?

October 12, 2007 3:22 PM

Is any given deal of standard Solitaire, perfectly played, "winnable"?

Or is there an element of chance in each lay-out of the cards that makes it impossible to "win" some deals?

I discovered today that at one time there was gambling on Solitaire (pay for a deck, get money back for each playable card).

So I was puzzling out whether there was a house advantage, etc.

Math whizzes? Statisticians? Card Sharks? Game Theorists? Please ante up.

posted by penciltopper to sports, hobbies, & recreation (21 comments total)

I assume you mean the game of Klondike. I think the answer is "no". I can construct a deal where it's impossible to play a card.
posted by Steven C. Den Beste at 3:29 PM on October 12, 2007

Let me expand on that. In Klondike after the initial deal there are certain cards available to the player, and certain cards not available. Available cards are the seven which are face up on the table, and every third card through the remaining deck, another 8. 21 cards face down on the table plus 16 in the deck, are unavailable. So 15 cards are available, 37 are unavailable.

You can divide the deck into 24 "even" cards (2, 4, 6, 8, 10, Q) and 28 "odd" cards (A, 3, 5, 7, 9, J, K). So you stack the deck so that all 15 of the "available" cards come from the 24 "even" cards. The 37 unavailable cards are all 28 "odd" cards plus the other 9 "even" cards.

That means every face up card and every card you turn over on the deck will be "even", and there won't be any plays.
posted by Steven C. Den Beste at 3:39 PM on October 12, 2007

I'm also assuming you mean Klondike.

I can't vouch for the complete content of this page, but it does describe starting configurations that have zero moves. So there are certainly some games that can't be won.

There are many situations that are unsolvable, but I'm not sure if its known whether or not their unsolvable situations can be avoided in an individual game. So there may be games that can last some arbitrary number of moves before they are guarenteed to end badly, but I'm pretty sure nobody has proved they exist.

Personally, I just cheat a lot.
posted by jacobbarssbailey at 3:39 PM on October 12, 2007

I'm going to skip the stats and say that not every game is winnable based on memories of my grandfather who played all day long. Every once in a while we'd hear a "Son of a BITCH. Stuck!" and a slap of the cards down as he abandoned a game.
posted by nonmyopicdave at 3:44 PM on October 12, 2007

The proportion of games which are winnable in Klondike has not been calculated, but it's not 100%. This Monte Carlo estimation of unplayable (no legal moves at start) games came out at 0.25% of all games, which the author estimates (with no explicit rationale) to imply 2.5% to 10% of all games are strictly unsolvable. You'd have to know where all the cards were to hit that rate, but whatever percent are unsolvable sets a bare minimum. I don't know what is known about the success rate of perfect play with no special knowledge, particularly since the perfect strategy is probably not known either, but it can't imaginably be better than, I dunno, 75%?.
posted by abcde at 3:45 PM on October 12, 2007

Spider Solitaire is also vulnerable. If the last ten cards in the deck are all even or all odd, the game is over. (There are other card combinations among the last ten which are also unplayable.)

It's been claimed that every Freecell game can be won, but no one has proved it.
posted by Steven C. Den Beste at 3:46 PM on October 12, 2007

The page JBB links to makes a cognitive error: it conflates "games in which at least one move is possible" with "games which can be won". What they found with their Monte Carlo test was that one game in 400 didn't permit any moves. That doesn't mean the other 399 are winnable; it just means that at least one card can be played in them.
posted by Steven C. Den Beste at 3:50 PM on October 12, 2007

If you're playing on a computer, many applications have a predefined list of games determined to be winnable, rather than just a random assortment of cards.
posted by scottreynen at 3:54 PM on October 12, 2007

Any deal with no aces on top and everything on top the same color is unplayable, nevermind winnable.

There's lots that can't be won.
posted by aubilenon at 3:59 PM on October 12, 2007

Winnable Solitaire
posted by zanni at 4:08 PM on October 12, 2007

SCDB: There's at least one unwinnable Freecell game. Check out the Internet Free Cell Project as detailed on the Wikipedia page:

http://en.wikipedia.org/wiki/FreeCell

The result is interesting--apparently #11,982 is unwinnable--but the process is what's really intriguing to me. None of us can play Free Cell as well as all of us can play Free Cell.

Scroll down a bit on the Wikipedia page...supposedly there are two more unwinnable games that are included as Easter eggs, #-1 and -2.

(I hope you don't consider this off-topic, PencilTopper. I'm only adding it to the discussion because I thought that if you were interested in the mathematics of Klondike, this Freecell info might interest you as well.)
posted by Ian A.T. at 5:01 PM on October 12, 2007

"I discovered today that at one time there was gambling on Solitaire"

I was in Vegas for the first time earlier this year, and every casino I walked through had tables and/or machines where you could gamble on solitaire. So, no past tense needed.
posted by ewagoner at 6:40 PM on October 12, 2007

I have an lcd solitaire game (klondike) that I keep in the bathroom for idle amusement there. To avoid frustration, I don't try to play through any game where I can't make at least four moves off the bat (before flipping any cards from the deck). This has cut my aggravation considerably.

Sometimes I let one go that has only three, which I refer to as a "scholarship student". Ok, weird analogy but that's what sticks in my head. Occasionally they win and it's worth the gamble.
posted by marble at 9:32 PM on October 12, 2007

I definetly believe the claim that every Freecell game is winnable. In 12th grade I had a stretch on one machine of 68 wins in a row.
posted by Riemann at 10:22 PM on October 12, 2007

Gamblers generally consider Michael Shackleford, an actuary and Adjunct Professor of Casino Math at UNLV, to be the leading expert in these sorts of questions. He is known as the Wizard of Odds and runs a web site of the same name. In 2001, he was asked the odds of winning in Klondike and replied "I'm afraid I don't know. In the early days of Vegas they used to offer solitaire, I think Klondike, as a form of gambling but I don't know the exact betting rules. The odds on this would be very hard to figure out." He has also been asking anyone who finds the answer to tell him for years and has never got one. I don't think anyone knows.

There is an online casino (Cryptologic) that still allows wagering on Klondike and they probably know the rate at which real people win, but that doesn't answer your question. There was an article in Science magazine, but I don't think it answered your question either.
posted by Lame_username at 1:59 AM on October 13, 2007

The problem with many deals in Solitaire is that "perfectly played" is not always the route to winning, if "perfectly played" means playing every available card as soon as possible. Sometimes the only way to win is to skip one available play in favor of a later play that will free up more of the cards you need. Occasionally I realize "Thank god I missed that 4 of diamonds sitting in front of me all along, since playing the 4 of hearts finally got me the Ace of spades." It's all dumb luck on my part, but there must be strategy in there somewhere.
posted by junkbox at 7:29 AM on October 13, 2007

One of the variants of solitaire that was supposedly very lucrative for casino owners was (American) Canfield. It is very common for the game to become obviously unwinnable after only 5 or 6 plays.
posted by ijoshua at 7:50 AM on October 13, 2007

And again on FreeCell, here's an unwinable game.
posted by Neiltupper at 7:57 AM on October 13, 2007

In a standard game of Solitaire, just imaging that all the aces are deepest in the piles, and all the threes are right above them, followed by the twos.

You won't be able to move the twos away because the threes are hidden and the aces are below. Ta-da, an unwinnable game.
posted by Kickstart70 at 9:38 AM on October 13, 2007

The problem with many deals in Solitaire is that "perfectly played" is not always the route to winning, if "perfectly played" means playing every available card as soon as possible.

Well, "perfectly played" doesn't mean that. To increase your odds, you need to memorize the cards that are and aren't available to you. You do this by cycling through the deck once at the beginning. Generally speaking, it's a bad idea to play cards on top of aces (2,3,4) if those cards are early on in the deck. Since these cards can be played anytime (provided that you have the aces), it's best to save them. Then, when you need to access cards that are unavailable to you deep in the deck, you play one of those cards to mix up the order, which makes the cards you need available. Likewise, it's generally best to clear as much of the table as possible before playing cards in the deck.
posted by smorange at 1:24 PM on October 13, 2007

I definetly believe the claim that every Freecell game is winnable. In 12th grade I had a stretch on one machine of 68 wins in a row.

I think it's given that most are winnable - once you get the hang of Freecell, you only lose if you fuck up, and a game usually takes about 3 minutes - but that doesn't mean that it's logically impossible to create a set up which could not be won, as those links claim to have. It's just extremely unlikely you'd come across it on a random deal.
posted by mdn at 4:37 PM on October 13, 2007