Variants of the mill game

Board

Three squares, 24 points

Pieces

Nine each

The standard game, and the one played online here. Nine pieces each, placed then moved along the lines, three in a row takes an enemy piece, and you lose by dropping to two pieces or by having no legal move. It is solved: perfect play from both sides is a draw.

The one rule people genuinely disagree about is flying — whether a side down to three pieces may jump anywhere. It is on by default here, and you can turn it off in local games and against the computer. Turning it off makes the endgame far sharper, which is why serious play usually does.

ComputerPass and play

Board

A 3×3 grid, 9 points

Pieces

Three each

The smallest version, and much older than the nine-piece game. Each side places three pieces on a three-by-three grid, then slides them along the lines; a line of three wins outright rather than capturing. It is close to noughts and crosses with a movement phase bolted on, and like noughts and crosses it is a draw once both players see the trick.

Its value is as a first game for a child: the rules take thirty seconds and the idea of “three in a row, but the pieces move” is exactly the idea the nine-piece game is built on.

ComputerPass and play

Board

Two squares, 16 points

Pieces

Six each

The outer and middle squares only — the inner square is gone, and so are its four points. Six pieces each, and no flying: a side reduced to two pieces has simply lost. It was popular in medieval Europe and is a genuinely different game rather than a smaller one, because with sixteen points the board fills up fast and the placing phase decides much more.

ComputerPass and play

Board

Three squares plus four diagonals, 24 points

Pieces

Twelve each

The same twenty-four points, with four diagonal lines added joining the corners of the three squares. Those diagonals add four more possible mills and give the corner points — the weakest squares in the standard game — a third neighbour, which changes the placing phase completely.

With twelve pieces each, twenty-four of the twenty-four points are filled at the end of placement, so the game can end in a stalemate before anybody moves at all. Traditions differ on what that means: some score it a draw, some a loss for the player who cannot move.

ComputerPass and play

Board

Three squares plus four diagonals, 24 points

Pieces

Twelve each

The Southern African game, played on the twelve-piece diagonal board and taken seriously enough in South Africa and Lesotho to have organised league competition. The pieces are called cows, and the game has its own vocabulary and its own body of opening theory that does not translate straight across from the European game.

It is the most-played member of this family that is not the nine-piece game, and it is played here on the twelve-piece board, with flying at three cows.

ComputerPass and play

Board

Three squares, 24 points

Pieces

Ten each

Proposed by Emanuel Lasker, the world chess champion, and the sharpest variant of the lot. Ten pieces each, and — the real change — there are no separate phases. On any turn you may either place a new piece or move one you have already placed, for as long as you have pieces in hand.

That single change removes the thing that makes the standard game drawish: you can no longer count on your opponent being obliged to keep placing while you set something up. It has also been solved, and it too is a draw — but it is a much harder draw to hold.

ComputerPass and play