Is mahjong solitaire always solvable?

What we measured

We took every layout in Meow-Jong's board files: the original 120 boards and 40 additional letter-and-symbol boards, 160 in all (these groups describe the October 4 study; itch.io version 3.2.2 now includes all 160), from 28 to 160 tiles. For each layout we generated 40 naive random deals, meaning the faces were shuffled onto the positions with no regard for solvability, the way a simple implementation would do it. Each deal was then checked by an exact search that tries every sequence of matches (with the usual safe shortcuts, such as clearing all four copies of a face when all four are free). The search had a budget of 200,000 positions per deal; deals it could not settle inside the budget are reported as unresolved, not guessed.

We also recorded two more things. First, on each solvable deal we let a hint-free player make random legal matches to see how often it got stuck anyway. Second, we ran the game's own reverse-dealing method 50 times per layout and counted how many attempts it needed to produce a deal.

Results

By board size

Size is not the main driver. Shape is. Wide, flat boards with many free tiles tolerate a random deal; narrow or tall boards, and boards made of thin strokes like letters, do not. 11 layouts produced no solvable random deal at all in our sample, and 51 produced nothing but solvable deals.

Most forgiving layouts

Least forgiving layouts

The classic Turtle

On the 144-tile Turtle, 18 of 18 resolved random deals were solvable (100%), with 22 of 40 deals left unresolved by the search budget. Random hint-free play got stuck on 17 of the 18 solvable deals it played.

Why reverse dealing fixes it

Meow-Jong never deals at random. It starts from the full layout, picks two free positions, gives them the same picture, removes them, and repeats until nothing is left. Playing those pairs back in the order they were chosen clears the board, so a solution always exists. The method can fail on a layout where fewer than two positions are free at some step, in which case the game tries again, up to 50 times, before falling back to a flat layout. In our 50 trials per layout it needed on average 1.20 attempts and at most 8; it failed all 50 attempts on 0 of 8,000 trials.

A guaranteed solution is not the same as a guaranteed win. Random hint-free play still got stuck on 78% of the solvable deals it played, because matching the wrong two of three free copies can strand the fourth. That is the honest case for undo and hint, and for a reshuffle that re-deals what is left rather than ending the game.

Method details and limits

• Rules: a tile is free when no tile occupies any of the nine cells around and above it on the next layer up, and it is not flanked on both the left and the right (two cells away, within one cell vertically) on its own layer. Tiles match when the faces are identical; there are no flower or season groups. These are the rules in Meow-Jong's code.
• Faces: each deal uses the game's own face distribution (44 pictures; pairs drawn from a pool so most faces appear four times and a few appear twice on smaller boards).
• Random deals: faces shuffled uniformly onto positions. This models a naive implementation, not any specific other game. Games that deal differently will see different rates.
• Solver: depth-first search with a table of dead positions, forced clearing of fully free faces, and move ordering by how many tiles a match frees. Budget 200,000 positions per deal; over budget is reported as unresolved. Unresolved deals are excluded from the solvable percentage and shown in their own column. Hard-to-prove deals are more often unsolvable than solvable, so the true solvable share is probably slightly lower than the figure shown.
• Sample: 40 random deals per layout, 6,400 in total; 50 reverse-deal trials per layout. Seeds are fixed, so the run is reproducible.
• Scope: solvability of the initial deal only. Scoring, time and reshuffles are not modelled.

Data: meowjong-solvability-2026-10.csv (one row per layout). Solver source: solvability_study.py (Python 3, no dependencies; the layout files are the game's own board data and are not included). Cite as: EPWH LLC, "Is mahjong solitaire always solvable?", meowjong.com, October 2026. Questions: meow@meowjong.com.