Is every game of solitaire winnable?

No. Even a player who could see every card would lose about one Klondike deal in five when cards are turned three at a time, and about one in ten when they are turned one at a time. In real play, where most cards start face down, far more deals are lost — a strong player wins well under half of Turn 3 games.

I ran the numbers below on this site's own engine partly to settle it for myself: after a run of deals that went nowhere, I wanted to know whether I was playing badly or had simply been dealt losses.

That is not a flaw in any one game. It is how Klondike works: the deck is shuffled, and some shuffles put the cards you need under the cards that need them. You can try it for yourself in Klondike Solitaire — and if a deal stops making sense, it may simply be one of those.

How many deals can be won at all?

The best answer comes from a program called Solvitaire, built by Charlie Blake and Ian Gent at the University of St Andrews. It plays "thoughtful" Klondike, where the position of every card is known, and searches every line of play to the end. Run over a very large number of random deals, it found that 81.9% of Turn 3 deals can be won, and about 90.5% of Turn 1 deals.

Those are ceilings, not averages. Thoughtful Klondike is a much easier game than the one you play, because you can plan around the face-down cards instead of guessing at them. If a deal cannot be won with every card in view, it cannot be won by anyone.

The first row is Blake and Gent's result. The other two we measured on the engine that runs the game here, over 200,000 deals for the second row and 2,000 deals of each mode for the third.

How often do good players actually win?

Far less often than the ceiling. In a well-known study from Stanford, "Solitaire: Man Versus Machine", an expert mathematician played 2,000 games of thoughtful Turn 3 Klondike — with every card visible — taking about twenty minutes over each, and won 36.6% of them. The researchers' own program, which looked ahead by simulating thousands of continuations, won about 70%.

So even with perfect information, a careful human player won fewer than half of the deals a perfect player could win. With the cards face down, as in the real game, the gap is wider still. Losing a Turn 3 deal is the normal result, not a sign that you played it badly.

Why some deals cannot be won

A deal is lost when the card you need is buried under a card that needs it. The simplest case: the only red seven you could put a black six on is face down underneath that same black six. No order of moves gets the six off the seven, so neither can go anywhere.

Turn 3 makes this far more likely, because only every third card of the stock is ever on top of the waste on a given pass. A card can sit in the stock for the whole game without once becoming playable — that is why Turn 3 loses roughly twice as many deals to the ceiling as Turn 1 does. The difference between the two modes runs through everything about the game.

A few deals are lost before they begin. About one Turn 3 deal in 420 offers no legal move at all: no ace in view, nothing that fits on anything, and no playable card in the eight the stock will show. In 200,000 Turn 1 deals we did not find a single one, because Turn 1 lets you see every card in the stock.

How to tell early that a deal is going nowhere

• You have been through the stock twice and nothing new has come up. On the third pass the same cards will come up in the same order.
• Every face-up card you could move would only cover a face-down card you need.
• The only way to free a hidden card is to put a card home that something else still has to stand on.
• In Turn 3, the waste cycles through the same handful of cards no matter what you take from it.

When two or three of these are true at once, the deal is very probably lost. Undo is the honest way to check: go back to the last real choice you made and try the other branch. If that runs out as well, start a new deal.

What about Spider and FreeCell?

The other solitaires behave very differently. FreeCell deals every card face up, and almost every deal can be won — only about one in ninety thousand cannot. Spider depends on how many suits are in play, and the difference between one suit and four is the difference between two games.

So should you keep playing a hard deal?

If it is fun, yes — a deal that looks lost often turns on one move you have not tried, and the hint is there to show you one. If it has stopped being fun, a new deal is one click away, and statistically you were never meant to win them all. For the moves that improve your odds on the deals that can be won, read how to win more often, or go back to all the solitaire games.