# Can every game of Klondike-Solitaire be solved?

I play a lot of Solitaire games on my Android phone and love to keep an eye out for the statistics.

Given that the Solitaire version lets you restart the game endlessly, I usually play until I solve it. But I never managed to solve more than 80% of the games played (1000+).

So now I wonder, is every Solitaire game solvable?

• I presume you mean Klondike solitaire? – McKay Jul 7 '10 at 20:56
• I have played thousands of solitaire games both on pc and the old fashioned way (yes with real cards) and have deduced that in order to find a solution to every game you play is to cheat. – TheX Dec 1 '11 at 3:37
• What a solitaire addict! Forever alone :) – kokbira Jun 25 '12 at 18:36

No. Example: If all of your cards face up on the board are red, and the cards that come up every third card are also red, and none of them are aces. You lose. Do not pass go, do not collect \$200.

• As a fact, I have come up with almost this exact setup on the computer version of Solitaire (but one card was black, just completely impossible to place anywhere). – Grace Note Jul 7 '10 at 21:02
• Another example that just happened to me: All cards shown are even. – McKay Jul 14 '10 at 15:48
• Even simplier: all aces are on the same column and 2 is above them. – SteeveDroz Dec 1 '11 at 10:50
• @Oltarus Aces in same column and 2 above them is still winnable. Its annoying and probably a loss but doable. – MaskedPlant Dec 1 '11 at 14:33

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%.

• Then I was actually doing a great job nearing it to 80% – Ivo Flipse Jul 21 '10 at 20:44

Literally just played a game in which one of the stacks (the one containing 4 cards) was lead by the 9 of diamonds, and the cards inside of it were the King of Spades, the 5 of diamonds, the 10 of spades, and the 10 of clubs (I know this because I had the entire field solved except for this stack and used process of elimination). As far as I can see this makes the game impossible. I have a 9 of diamonds in which can never be moved, as the two 10s that it's eligible to rest upon are trapped underneath it in the stack face down. Attempting to get rid of the 9 by moving it to the diamond stack would also be fruitless, as the 5 of diamonds is stuck underneath it too. Unless someone can tell me some way that this could be solved, I'm pretty darned certain that if a card that is leading a stack is covering a stack that contains the two cards it is capable of resting on, and a lower number of it's own suit, then the game is made impossible right from the get-go.

Solitaire is a game that precedes its computer version, and that means that all the cards are truly shuffled, without the computer peeking in to verify the game is solvable.

And like McKay mentioned, with a random shuffle you can definitely end up with an unsolvable game.

I'm sure it is possible to design a Solitaire variant in which each game is solvable, though.

• Would need a LOT of calculation, basically the computer would have to play through an entire game to make sure there's a solution, unless there's some kind of algorithm I'm missing. – Arda Xi Jul 21 '10 at 15:42
• @Arda, there are some conditions that could be easily tested - for example, a card other than a King can be played on only three other cards in the deck (the next-lowest card in its suit, or the foundation for an Ace, and the next-higher cards of the opposite color). If all three of those cards are face down below that card on a pile, the game isn't winnable. Unfortunately I think that's a small percentage, and testing for other conditions might require a ton of recursion. – Dave DuPlantis May 25 '11 at 21:05
• @DaveDuPlantis True, but you will have to test for all of those conditions that exist. I'm not sure whether we even know all of them. – Arda Xi May 26 '11 at 5:12
• @Arda - that's true, that's what I was thinking with respect to recursion. Without some way to demonstrate that any given position is unwinnable, you'd essentially have to play a certain series of cards until you were blocked, back up to the last decision point, and repeat ... it's an intriguing concept, but I've never seen a solitaire program do that. – Dave DuPlantis May 26 '11 at 12:31
• @Arda Could simply work backwards from the solution, randomly moving cards into the deck and onto the board from the four suit piles, always using the reverse of a legal play. Probably won't have the same probability distribution as shuffling and checking for winnability, but I doubt that matters to most players. – warbaker Jun 5 '12 at 16:52

However, if you started a list and enumerated the initial conditions -- I feel like I've seen this on a linux version of Solitare: the numbering of deck order, that is -- and you definitively decide a certain one is un-winnable, you then could compare notes across nodes (share with friends) and VOILA: a list of un-winnable starting deck stacks.

I've been starting to think the Windows 7 version has the un-winnable decks removed, ... I don't know, it's a little heavy-handed and smug about the statistics.

• With 52! starting shuffles, you'll need an... inconveniently long... time before you have a good list. Even after you solve the problem of determining unwinnable definitively. – Tynam May 24 '12 at 7:59
• 52 factorial = roughly 8 followed by 67 zeroes. That's a lot of combinations. A 1TB hard drive would store about a trillion of these, and you'd need trillions of terabytes to store even a decent fraction. Not very practical unfortunately, just because of the astronomical number of probabilities involved. Probably easier to just store a certain number of demonstrably winnable games. – doppelgreener Feb 24 '14 at 7:54
• @JonathanHobbs Not all of them have to be stored to make the calculation. `for 1 to 52! getdeck, try solving game, add to statistics` at each point only one deck need be stored, and the statistics can be quite small. – McKay Feb 24 '14 at 13:36
• @McKay You have to store quite a bit to develop a decent list, though. (I'm not sure which calculation you speak of.) As an aside also regarding the answer: the Windows 7 version actually just stores a few dozen thousand decks, and you're randomly given one each game. It might be they just picked a few dozen thousand decks known to be winnable. – doppelgreener Feb 24 '14 at 13:43
• @JonathanHobbs No, all you have to store is which deck you're looking at (which would need to go up to 52!, which means we'd need about 226 bits), and you'd need to store how many of them were solvable (another 226 bits, or less), and then one game of solitaire (which windows 3.1 was apparently able to store just fine), and the algorithm to actually solve the game. The data storage mechanisms do not need to be very much in order to do a full set of statistics on solvability. We're talking less than 1k of storage. Sure it would take a long time to do all these calculations. But not storage. – McKay Feb 24 '14 at 13:57

No. Eric Sink decided that he would start a micro-ISV to create a version of solitaire that is always winnable. This was mostly just an experiment to see what it would be like running a software company with one person, but he eventually sold the product which is still available for purchase.

There have been some estimates about the number of Klondike Solitaire games that are unplayable (no moves possible, about 1 in 400), and several guesses about how many games are unwinnable, although this percentage varies wildly from 30%-10%.

The difficulty of this problem stems from the sheer number of initial deals 54! that would need to be evaluated to determine which were winnable and which were not.

• would the number of initial deals be `52!` ? (unless you expect the jokers to be dealt, too) – warren Dec 10 '14 at 18:17
• Fortunately, one does not need to use the brute force method (look at all possible deals) to compute the odds of winning (since that computation would take longer than the age of the universe - 8x10 to the 68th power decks). An analysis of ways to fail provides an analytical line of attack. As already noted there are clear ways that a single stack may fail. The needed cards might also be unreachable within two stacks, three stacks or four stacks. Once the conformations of cards for locking up the needed cards are known, their individual odds may be computed and combined to get an answer. – M Willey Dec 9 '15 at 15:05

To add to the other great answers, this link has a nice explanation of how a deal is un-winnable.

Reasons for Getting Stuck in Klondike Solitaire