The most common questions that we get on the topic of Solitaire, are about the winnability of the game. Is every game winnable? If not, what percentage are winnable? How much does game play influence the outcome? I’m sure we’ve all asked ourselves at some point:
Can I definitely solve this hand if I play well enough?
Every solitaire player at some point
The answer in the case of Klondike Solitaire, is no. Not all games are winnable. Even if the player has perfect knowledge of the value of all of the cards (including those not yet flipped over), there are combinations of cards that simply cannot be solved under the rules of standard (i.e. turn-3) Klondike Solitaire.
Not every game of (Klondike) Solitaire is winnable.
As one example, imagine if all of the face up cards (including those available in the stock) are red and none of them are aces. In this situation, there is no move that can be played.
What percentage of Solitaire games are winnable?
Interestingly, no-one knows the exact percentage of Solitaire games that are winnable.
The sheer number of possible hands makes an exhaustive calculation infeasible. How many hands are possible? Well there are 52 cards in a deck. So…
We can choose any as the first card, giving 52 possible choices:
= 52
Now there are 51 cards remaining in the deck, so there are 51 possible choices for the second card:
= 52 x 51
And so on….
= 52 x 51 x ... x 2 x 1
= 51!
= 8.066+67
= 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
This is the number of unique ways to shuffle a 52-card deck of cards, and is a very, very, very big number.
Even if it was possible to check each possible hand, no-one has yet devised a mathematical solution to solve a winnable hand. As such, it wouldn’t be possible know if a particular hand was winnable without iterating through possible sets of moves.
Estimating the percentage of winnable games
Instead of checking all combinations, statistical techniques can be employed to evaluate a subset of the all possible hands, and thus estimate the percentage of winnable games. Using such techniques, it appears that 82% of Solitaire games are winnable.
This number is reported by several papers:
We report the winnability of Klondike as 81.956% +/- 0.096% (in the ’thoughtful’ variant where the player knows the location of all cards)
The Winnability of Klondike Solitaire and Many Other Patience Games (2019) - Charlie Blake, Ian P. Gent 1
We empirically demonstrate that no less than 82% and no more than 91.44% of instances of Thoughtful Solitaire (and therefore Klondike Solitaire) have winning solutions.
Searching Solitaire in Real Time (2007) - Ronald Bjarnason, Prasad Tadepalli, Alan Fern 2
What factors determine whether a winnable Solitaire hand is actually won?
Just because a hand is technically “winnable”, this does not mean that an average player will win when playing.
For a particular hand, there may be only one path that leads to a win, or there may be many. Along with the skill of the player, the number of possible winning paths determines how easy or difficult a particular game of Solitaire is to win.
How can I know if a particular Solitaire game is winnable?
Unfortunately there is no way to know if a particular Solitaire game is winnable until you play the game. It may become clear early that a particular game is likely not winnable, but since only 18% of games can’t be won, and many cards remain hidden at the start of the game, it takes a very experienced player - and time - to determine this.