Nine Men's Morris

Only when there is nothing else to take. As long as the other side has at least one piece on the board outside a mill, the pieces in its mills are safe. When every piece it has stands in a mill, the protection is gone and you may take whichever you like.

On this board the pieces you may take have a chalk ring around them.

You take one piece. Rule books differ on this, and some give two. This board follows the rules that were used when the game was solved, where two mills closed by the same move still take a single piece.

Yes, and it counts every time. Move one of the three pieces away along a line, move it back on a later turn, and the mill is closed again and takes another piece.

The risk is the turn in between. While the mill is open its pieces are not safe, and the other side can put a piece on the point you left.

No. A mill is three points in a straight row along one painted line. Three pieces that follow a line around the corner of a square are not a mill, and nothing counts on a diagonal, because no line runs there.

When its side has exactly three pieces left. From then on, any of the three may go to any empty point. A side with four pieces or more never flies, not even when one of its pieces is shut in.

You lose. There is no sitting a turn out in this game. It only happens in the moving stage: while pieces are being placed there is always an empty point, and a side that flies can reach any of them.

It can happen with all nine pieces still on the board, which is why counting pieces does not tell you who is winning.

Yes, in two ways. The same position three times is a draw, and so is fifty moves by each side without a mill. Between two careful players a draw is the natural result: a game played perfectly by both sides ends in one.

Not if both sides play perfectly. Then the game is a draw whoever starts. Between people the first move is a small thing next to the first mistake.

Yes. Nine men's morris, mill, mills, the mill game, merels and merrills are all names for it, and in North America it has also been called cowboy checkers. The nine men are the nine pieces each side has.

There are smaller and larger relatives with three, six and twelve pieces a side. This page is about the game with nine.

Old enough that nobody knows who played it first. It goes back at least to the Roman Empire. In England, boards for it were cut into the cloister seats of the cathedrals at Canterbury, Gloucester, Norwich and Salisbury and at Westminster Abbey.

Shakespeare has Titania mention it in A Midsummer Night's Dream, where after a season of floods the board cut into the village green is "filled up with mud."

Yes. In November 1993 Ralph Gasser, at the Swiss Federal Institute of Technology in Zurich, announced that a game played perfectly by both sides is a draw.

His program worked backward through about ten billion positions of the moving and flying stages and stored the outcome of each. It then searched the placing stage, all eighteen pieces deep, down to those stored positions. The work was published in 1996.

A mill made while the pieces are going down wins one piece. Pieces put where they have nowhere to go can lose the whole game later, when the moving starts and they cannot.

So place with the second stage in mind: spread across the three squares, with empty points next to your pieces rather than white ones.

Four points on the board have four neighbors: the middle of each side of the middle square. Eight have three, the middles of the sides of the outer and the inner square. The twelve corners have two.

A piece on one of the four crossings has the most ways out and stands in the way of the most rows. A piece in a corner is the easiest one to shut in.

Two white pieces in a row with the third point empty is a mill on the next move. While pieces are being placed, the answer is nearly always to put your own piece on that third point.

One threat can be stopped. Two at once cannot, so the move to look for, and to keep the computer from making, is a piece that threatens two rows at the same time.

The strongest formation in the game is two rows that share one moving piece: a closed mill beside a row of two, with the gap one step away. Move the shared piece across and it closes the second row. Move it back and it closes the first again.

Every move takes a piece, and the other side can do nothing about it except close a mill of its own or get a piece into the gap.

A side that cannot move has lost with every piece it owns still on the board. In the moving stage, look at how many moves each side has left. Shutting in two or three white pieces is often worth more than taking one.

The engine here counts it the same way. It rates a side that cannot move a single piece as nearly a whole piece behind, before the game is formally lost.

A side with three pieces flies, and a piece that can land anywhere blocks any row you are about to close. Taking the computer down from four pieces to three can make your work harder.

Before you do it, have the next mill prepared twice over: two rows that each need one more move. A flying piece can block one of them, not both.

The Hint button asks the engine at full strength what it would play. When its move is not the one you had in mind, work out why before you play either. After a mill it also tells you which piece it would take.

The computer tries its moves, your answers and its answers to those, as many moves deep as the level allows, scores the positions at the end, and plays the move that leaves it best off if you reply as well as you can. A move that closes a mill counts as one move, the piece it takes included.

The score is counted in pieces. On top of the pieces themselves it adds a little for a closed mill and more for a row of two that can be completed on the next move. It compares how many moves each side has, and counts the pieces that are shut in.

When the side to move could close a mill just past the point where the search stops, the search follows those moves a little further, so that a piece about to be taken is not counted as safe. It knows both draw rules and treats a repeated position as a draw.

All of this happens on your own device, in your browser.

Looks at its own move and your answer, with a beginner's eye: it counts pieces, its own mills and its own rows of two, and sees neither your threats nor who is being shut in. About two moves in three it is careless and looks at its own move only, and that is when it lets you close a mill.

Thinks the same way, but is careless in only one move out of five.

Still looks two moves ahead, now with the full score: threats from both sides, and who is running out of room. It is careless in a little more than one move in four.

Looks five moves ahead and is never careless. Five moves is far enough to see a running mill being built while there is still time to stop it.

Gets one second for every move in place of a fixed depth. On the computer where it was measured, that second was about nine moves of look-ahead. Apart from a little variety in its first few pieces, so that games do not all begin the same way, it always plays the best move it has found.

Each level has played the level below it from an empty board, taking turns to go first. A win counts as one point and a draw as half.

• Easy against Beginner: 81.9 percent of the points, over 1,000 games.
• Medium against Easy: 90.4 percent, over 1,000 games.
• Hard against Medium: 98.3 percent, over 400 games.
• Expert against Hard: 93.8 percent, over 24 games.

Depth matters a great deal in this game, so the upper steps are large. The last figure rests on few games, because Expert uses its full second for every move.

The Hint button is the engine at full strength: one second, no fixed depth and never a careless move. That is Expert without the variety in its opening.

Using it does not make the level you are playing any stronger: every search starts from nothing. Hints are counted, and the count is shown with your record at the bottom of the page.