Three Men’s Morris — Solved
Play Three Men’s Morris against a perfect engine, and see why one of its four rule sets is a first-player win while the other three are draws.
One of the four rule sets is broken. It is the one everybody plays.
Two house rules vary from table to table: whether the board has diagonals drawn on it, and whether a piece may move anywhere or only to an adjacent point. That is four games, and all four are small enough to solve exactly — 3,360 movement positions each, plus a full search of the placement phase.
| Rules | Result with perfect play | Opening values |
|---|---|---|
| Diagonals, adjacent moves | first player wins | DLDLWLDLD |
| Diagonals, move anywhere | draw | DDDDDDDDD |
| No diagonals, adjacent moves | draw | DDDDDDDDD |
| No diagonals, move anywhere | draw | DDDDDDDDD |
Exactly 1 of the four is decided, and it is the standard board with the standard movement rule. Loosening either rule fixes it — rub out the diagonals, or let pieces move anywhere, and the game becomes a draw. That is the opposite of the usual intuition: here the restriction is what creates the win, because it lets the first player build a position the second cannot slide out of.
And in that variant the opening move is the whole game
Values of the nine opening placements, from the first player’s side:
The centre wins. The four corners draw. All four edges lose outright — open on an edge and, against perfect play, you have handed the game away before a piece has moved. In every other rule set all nine openings draw, so this table is the whole difference between a fair game and a broken one.
How it is played
- Placing. Take turns putting your three pieces on empty points. Three in a row wins immediately, so this half is mostly about blocking.
- Moving. Once all six are down, take turns sliding one piece to an empty point. Under the standard rule that means an adjacent point along a drawn line — never jumping over another piece.
- Winning. Three of your pieces in a line. If you have no legal move at all, you lose — which does happen, and only under the adjacent-move rule.
- The engine plays perfectly, from a solved table rather than a search depth, so it will not slip. If it says the position is lost, it is lost.
How to use
- Pick a rule set: with or without diagonals, adjacent moves or move anywhere.
- Place your three pieces, then slide them to make three in a row.
- The engine plays from a solved table, so it never slips.
- Compare the four rule sets in the table — only one of them is unfair.
Frequently asked questions
Is Three Men’s Morris a draw?
In three of its four common rule sets, yes. The exception is the standard one — diagonals drawn on the board and pieces moving only to an adjacent point — which is a first-player win. Removing the diagonals or allowing moves to any empty point both turn it back into a draw.
What is the best opening move in Three Men’s Morris?
The centre, and in the standard rule set it is the only move that wins. The four corners draw and all four edges lose outright against perfect play. In every other rule set all nine openings draw, so the opening only matters in the variant that is broken.
Why does restricting movement make the game unfair?
Because the first player can build a position the second cannot slide out of. When pieces may move anywhere, any threat can be answered from across the board, so the extra freedom hands the defender enough resources to hold a draw. It is the opposite of the usual intuition that a freer game is a less fair one.
How is the game solved?
Exactly, by retrograde analysis over all 3,360 movement positions plus a full search of the placement phase — small enough to settle every position rather than search to a depth. The result is checked against a separate depth-limited search that works forwards.
What are the rules of Three Men’s Morris?
Each player has three pieces. Take turns placing them on empty points, then take turns sliding one piece per move. Three in a line wins. Under the standard rule a piece may only slide to an adjacent point along a drawn line, and a player with no legal move loses.
Is it the same as noughts and crosses?
Only the first half. Placing three pieces each is a shorter game of noughts and crosses, but nothing is ever full — once all six pieces are down the game continues with sliding, which is where all the strategy lives.
Can the game go on forever?
Yes, if neither side can force a mill, which is exactly what a drawn position means here. The solver treats positions from which no win can be forced as draws, and roughly a sixth of all movement positions are drawn under the standard rule.
Does this send anything anywhere?
No. The solver runs in your browser and nothing is stored or uploaded.
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