Solitaire looks easy. You've got 52 cards and a few piles. So why do you lose so much? The answer hides in the math. Once you see the numbers, the game makes a lot more sense, and losing stops feeling like bad luck.
- A single deck can be shuffled into more orders than there are atoms in the Sun. That's why no two games feel the same.
- Hidden face-down cards create "missing information," which is the main reason solitaire is hard.
- Some deals simply can't be won, no matter how well you play.
- Math even helps computers figure out how many deals are solvable.
The huge number of possible deals
Start with the deck. A standard deck has 52 cards. The number of ways to order them is 52 times 51 times 50, all the way down to 1. That's written as 52! and read as "52 factorial."
The result is a number with 68 digits. It's so big it's hard to picture. If every person on Earth dealt a new random shuffle every second since the start of the universe, we still would not have seen them all. Not even close.
What does that mean for you? Every deal you play is almost certainly one that has never existed before and never will again. That's part of why Klondike and its cousins stay fresh no matter how much you play.
Why hidden cards matter so much
The huge number of deals isn't what makes solitaire hard, though. Chess has a giant number of games too, and you can still see the whole board.
The real trouble is hidden information. In Klondike, 21 of the 28 tableau cards start face-down. You can't see them. So every plan you make is a guess about cards you can't check yet.
This is why FreeCell is so much easier. Every card sits face-up from the start. There's no guessing. You have all the facts, so a careful player can win almost every deal. Take away the hidden cards, and you take away most of the difficulty.
Deals that can't be won
Here's a hard truth. Some deals are impossible. No move order wins them. It's not your fault, and it's not a bug.
Think about a simple trap. Two aces of the same color might block cards that each need the other to move first. If the layout locks them in a loop, the deal is dead from the start. You just can't tell until you dig in.
The share of impossible deals changes by game. In Klondike it's a real chunk. In FreeCell it's tiny. Our post on whether all solitaire games are winnable looks at these dead deals in more detail.
Choices that split into more choices
Now add your decisions. On many turns you have several legal moves. Each move opens up new moves after it. This is called branching.
Picture a tree. One choice at the top splits into three. Each of those splits into three more. After only a few turns, you're staring at hundreds of possible paths. Most lead to a loss. A few lead to a win. Your job is to find one of the good paths while blind to half the deck.
| Turns ahead | Rough paths to check (3 choices each) |
|---|---|
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 5 | 243 |
| 8 | over 6,000 |
No person can hold thousands of paths in their head. That's why good players use rules of thumb instead, like "free the face-down cards first." Simple rules cut the tree down to a size a human can handle.
How math counts winnable games
So how do we know FreeCell is about 99% solvable while Klondike is lower? Math and computers do the counting.
Researchers run programs that try to solve many random deals. The computer searches the branching tree, backs up when it hits a dead end, and tries another path. This is called a solver. Run it on thousands of deals and you get a solid estimate of the solvable rate.
FreeCell was famously tested this way. Out of the first 32,000 numbered deals in one popular version, only a single one turned out to be unsolvable. That's how we got the near-99% figure. Klondike is trickier because those face-down cards mean the solver sometimes can't be fully sure, so its number stays a little fuzzy.
The math behind why solitaire is hard to win
Put the pieces together and the picture is clear. The mountain of possible deals keeps the game fresh. The hidden cards force you to plan with missing facts. Branching turns each turn into a fork with many wrong roads. And a slice of deals can't be won at all.
None of that means you should feel bad about losing. It means solitaire is a real puzzle with real math inside it. Once you know that hidden cards and dead deals are working against you, you can focus on what you control: freeing face-down cards, keeping options open, and playing slow. Learn the moves in our Klondike strategy guide, and the math starts to tilt your way. That's the whole story behind why solitaire is hard to win.