Is Mahjong Solitaire Always Winnable? We Solved 10,000 Deals

No, Mahjong Solitaire is not always winnable. When the 144 tiles are simply shuffled onto the classic Turtle layout, about 3 deals in 100 can't be cleared at all, even by a player who plays perfectly and can see every tile. We solved 10,000 random deals exactly and found 300 that are impossible.

Every deal in Poozee's Mahjong Solitaire can be won, because each one is built backwards from a solution. So when you get stuck there, the deal wasn't the problem: a move you made was, usually about 20 moves earlier. Here is what we measured, why the impossible deals fail, and what the Shuffle button can and can't fix.

What "winnable" means

A deal is winnable if at least one sequence of legal moves clears all 144 tiles. Whether you find that sequence is a separate question.

The question also assumes you know every tile, including the ones buried under others. Researchers call that "peeking". In a real game you can't see the hidden tiles, so a winnable deal can still beat a careful player. Every result below is for the peeking version: if a deal fails there, nobody could ever clear it.

The rules are the standard ones: a tile is free when nothing lies on it and its left or right side is open, and matching free tiles come off in pairs. Any flower matches any flower and any season any season (our tiles guide shows every kind).

How often a random deal can't be won

We dealt 10,000 games the simplest way: shuffle the 144 tiles and lay them on the Turtle in order, with no check that the result can be cleared. Then a computer solved each deal exactly. It either found a way to clear the board, which we replayed move by move with the game's own rules, or proved that none exists. It settled all 10,000; none was left undecided.

That agrees with the largest study we know of. Michiel de Bondt, who designed the solving method our solver follows, scanned 10,000,000 random boards of the Turtle (the default layout of the program xmahjongg) and found that 2.95% cannot be won even with peeking (de Bondt, "Solving Mahjong Solitaire boards with peeking", section 3.4; the same figure is on his solver's page). xmahjongg's default layout file lists the same 144 tile positions as Poozee's Turtle.

How the solver works, briefly: once you decide which tile of each kind pairs with which, the order of play hardly matters, so it searches over those pairings (three ways to split each kind's four tiles) instead of over moves, and throws out any branch that couldn't clear even under looser rules.

Why the impossible deals fail

We sorted the 300 impossible deals by what dooms them:

One kind is stacked on itself. There are four tiles of each kind, and they must come off as two pairs. If three of them sit in one stack, one on the next, every way of pairing them puts a tile with the tile under it, and a tile can never be matched with the one it covers. That was the case in 152 of these 155 deals. In the other 3, the four tiles were spread over a few stacks so that every pairing still put one tile on top of its partner.

Only the side rule makes it impossible. These deals would clear if tiles were blocked only from above. What kills them is that a tile needs an open left or right side, so a row can only be emptied from its ends. The simplest case is a row of four tiles reading A B A B, holding the last two tiles of two kinds: only the two end tiles are free, they don't match, and neither can come off to free the tiles inside.

Stacks of different kinds pin each other. In a handful of deals, no kind is stacked on itself, yet the stacks cross: freeing kind A's last pair means removing kind B first, and freeing kind B means removing kind A. That loop rules out a win even before the side rule counts.

You can't see buried tiles while you play, so an impossible deal looks like any other at the start.

How Poozee makes every deal winnable

Poozee doesn't shuffle and hope. It plays a random game backwards: starting from an empty Turtle, it repeatedly picks two spots that would be free and gives them a matching pair, until all 144 tiles are placed. Removing those pairs in the same order clears the board, so a solution exists before you make your first move. We replayed that hidden solution on 10,000 deals with the game's rules, and it cleared all 10,000.

That method does more than rule out impossible deals. Built deals turn out to be friendlier to play, too:

(The last two rows are 3,000 deals each, played by a computer with no undo and no shuffle.) A random deal is almost always winnable, yet the random-pair player cleared random deals only about a seventh as often as built ones.

Why you can't win a deal that can be won

Since every Poozee deal can be cleared, a lost game is always lost by a move. The frustrating part is when. We had a computer play 3,000 deals by taking free pairs at random, then used the exact solver to find the move after which each lost game could no longer be cleared.

• It lost 1,794 of the 3,000 games.
• The losing move came a median of 22 moves before it ran out of matches. Only 4 games were lost on the very last move.
• In 1,515 of the lost games (84%), the damage was done 10 or more moves before the end.
• At the losing move there were typically 84 tiles left. When it got stuck, there were 36 on average.

So the pair you took just before getting stuck is almost never the one that cost you the game. What did?

"Lost only to the side rule" means the position after that move could still have been cleared if tiles were blocked only from above. The player using the strategy guide's rules never stranded a kind, because that is its rule 1. It still lost 1,159 games, almost all of them that way, and its losing move came just as early: a median of 23 moves before it got stuck.

Pro tip: When you're stuck, don't just undo one move. The mistake is usually 20 or more moves back. Undo in big steps and press Hint after each: it searches for a way to clear the board from there.

What Shuffle does

Poozee's Shuffle button rearranges the tiles still on the board, on the same spots, into a position that can be cleared, using the same backwards method as a new deal. It can't help when the spots themselves can't be emptied two free tiles at a time, for example when the last two tiles lie one on top of the other. Then the game tells you no shuffle can clear them.

We replayed the same 3,000 games with one change: each time the player got stuck, it pressed Shuffle and kept going.

Of the random player's 1,794 stuck games, 100 were beyond any shuffle the first time they got stuck. The other 1,694 could shuffle, and 1,687 of those went on to be cleared. For the better player it worked almost every time. The trade-off: the win screen counts your shuffles, and a shuffle won't show you which move went wrong.

What to do when you're stuck

1. Press Hint before you're out of moves. If it says the tiles can no longer all be cleared, you already made the losing move. Its search has a time limit, so it can't always tell: in one of our test games it still offered a pair right after the losing move.
2. Undo in big steps. The losing move was usually around 20 moves back. Undo a handful of moves at a time, pressing Hint after each, until it stops saying the tiles can't all be cleared.
3. Replay that stretch differently. Avoid stranding a kind, and watch the long rows: don't leave the last tiles of a kind walled in the middle of one.
4. Shuffle if you'd rather keep going. It almost always turns a stuck game into a winnable one.
5. Start a new game only if you want a fresh challenge. Every Poozee deal can be cleared.

Keep playing

Put this to the test in Mahjong Solitaire, with the three tested rules from our Mahjong Solitaire strategy guide. If you're new to the game, start with how to play Mahjong Solitaire. For the same question asked about cards, see are all solitaire games winnable.