A simple question without a simple answer
Patience is the British name for what many people call solitaire, a family of one-player card games. A natural question is what share of deals can be won. It sounds like a counting problem, but for the best-known versions the exact answer is not known, and several different numbers get mixed together under that one question.
The first thing to separate is winnable from won. A deal is winnable if some sequence of legal moves clears it. A player’s win rate is how often a real player actually wins, which depends on skill, patience and, in some games, on how much information is hidden. The two can be far apart.
Why exact counts are out of reach
A 52-card deck has about 8.07 x 10^67 possible orderings. Each ordering is a possible deal, and the number is far too large to check them all. Researchers instead estimate winnability by sampling, which is the same idea as a Monte Carlo simulation: run many random deals through a solver and report the share solved.
Even one deal can be hard to check. A solver has to search through the possible move sequences, and in many solitaire games the number of positions explodes. Clever pruning helps, but a solver may still run out of time without a verdict. That leaves three outcomes for a deal: shown to be winnable, shown to be unwinnable, or undecided. Reported figures often come with a range for this reason.
Open information versus hidden cards
Whether the player can see all the cards is the biggest factor.
In games like FreeCell, all cards are dealt face up. There is no hidden information, so winnability is a pure puzzle with a definite answer for each deal. Players with good judgment can solve a very large share of them. A well-known fact is that among the first 32,000 numbered deals in the version Microsoft popularised, all but one (deal number 11982) are solvable. That result concerns that specific set of numbered deals, not every possible deal, though the share of unsolvable deals in general is also believed to be very small.
In Klondike, the version many people mean by “solitaire”, some cards are face down. A player must make moves without knowing what is underneath. That raises two different questions. One asks how often a deal would be winnable if all the cards were turned face up and a perfect player played it. The other asks how often a deal can be won by a player who only sees what the rules show. Research with computer solvers has focused on the first, which gives an upper bound for the second. Estimates for the face-up version suggest that a large majority of deals are winnable, while the share winnable by playing blind to hidden cards is lower. No single figure should be treated as settled, because published estimates differ with the rules used.
Rules that change the answer
Small rule differences matter a lot:
- Turning over one card from the stock at a time makes a game much easier than turning over three.
- Allowing unlimited passes through the stock is more forgiving than a limited number.
- Whether cards can be moved back from the foundations changes the options available.
- Scoring versions of the game that reward partial progress have a different goal from simply clearing the board.
Any quoted win rate should be read alongside which of these rules applied. A statement that “most deals can be won” in one setup says little about another.
The Ulam story
A story often told in this context credits the mathematician Stanislaw Ulam. While recovering from an illness, he is said to have wondered about the chance of winning a game of solitaire (often given as Canfield) and realised that working it out by pure combinatorics was too hard. He then reasoned that playing out many random games and counting the wins would give a practical estimate. That idea of replacing an intractable calculation by repeated random trials is commonly linked to the origins of Monte Carlo methods. The details of the story are retold in many versions, so treat it as a useful illustration rather than a precise record.
What this means for a player
A few careful points follow from all this:
- A losing game does not show that the deal was unwinnable. Most lost games of Klondike are lost because of choices, hidden cards or unlucky order, and some of those deals could have been won.
- A deal that looks hopeless may still be winnable, and this is more common in open-information games.
- Because the shuffle decides the deal and each game is independent, a run of lost games does not make the next one more likely to be won. This is the same error as expecting a coin to even out, and it is related to the idea of random chance not having memory.
- Solitaire played for fun carries no money risk. If it is played for stakes, the usual cautions about gambling apply, and losses can add up.
Takeaway
The question of how often solitaire can be won splits into several smaller ones, and honest answers are partly known and partly estimated. For FreeCell, nearly every tested deal is solvable. For Klondike, the exact winnable share is unsettled and depends on the rules and on what the player can see. The most reliable summary is that the share of winnable deals can be large, while the share a typical player actually wins is smaller.