Nine Men's Morris Strategy: Mills, Moves and the 50-Action Draw (2026)

1. The board is 24 points and 16 mills — and every point sits in two mills

The build generates the topology from the point coordinates instead of copying a table, so the numbers can be recounted rather than trusted:

• 24 points placed on a 7×7 grid: three concentric squares (outer, middle, inner) plus four spokes.
• 32 edges: a point is joined to its neighbours along a square side or along a spoke. Nothing crosses the empty centre.
• 16 mills: eight horizontal lines and eight vertical lines of three points. That is the 12 sides of the three squares plus the four spoke halves.
• Every point belongs to exactly two mills — one horizontal, one vertical. There is no such thing as a "double-mill point" that belongs to more; on this board all 24 are intersections.

What does differ is how many exits a point has. The coordinates (0,0) to (6,6) make the middle of each square side look like a junction, and the engine agrees:

Two consequences follow, and both matter later:

1. The four degree-4 points are the pivots of the four spoke mills. The spoke mills are (3,0)-(3,1)-(3,2), (3,4)-(3,5)-(3,6), (0,3)-(1,3)-(2,3) and (4,3)-(5,3)-(6,3). Their middle points are exactly the four points with four exits. A piece there can leave in four directions; a corner can leave in two.
2. The centre of the drawing, (3,3), is not a point. The horizontal and vertical spokes stop at the inner square's edge midpoints. So the two long spokes are each split into two separate three-point mills, and no piece ever passes through the middle.

The page draws the board with a pad of 80 units and a step of 90 units inside a 700×700 viewBox; pieces have a radius of 26 and the tap targets a radius of 38, which is why the points stay tappable on a phone stage even though the board is not a dense grid.

2. Three phases, three different games

A game is not one game. The build switches the legal-move generator twice, and each phase rewards a different habit:

The build's own constants are 9 pieces per side and a 50-action draw (section 9). There is no pass, no skip and no "place two pieces" turn: one action per turn, plus the removal that follows a mill. That is why the arithmetic in the next sections is so short — every advantage has to be bought with a single action.

Two boundary rules are worth knowing before the strategy starts, because both are easy to misread:

• Flying is exactly three pieces, not "three or fewer". With four pieces you slide normally; with two pieces you have already lost. The probe confirms canFly is true at three and false at four.
• A blockade is a loss, not a pass. When the side to move has no legal move, the game ends immediately in favour of the other side. In the 30 hard games the probe played, two ended this way, both after the placing phase.

3. A mill is a receipt for exactly one removal

Three of your pieces in a line form a mill, and forming one lets you remove one opponent piece. That is the whole scoring system: there are no points, no captures on the board while placing, and no bonus for a second mill in the same move.

The removal has one restriction, and it is the rule most beginners get backwards:

You may not remove a piece that is inside an enemy mill while the enemy still has a piece outside a mill.

In other words, a mill does not make all of its pieces permanently safe. It makes them safe only while at least one free enemy piece exists. When every enemy piece is inside a mill, all of them become removable.

This is not a rare corner case. In the probe's 140 games, the restriction was live at 637 of 967 removals at easy (65.9%), 258 of 380 at medium (67.9%) and 32 of 150 at hard (21.3%) — the number falls at hard because hard games form far fewer mills in the first place (4.87 per game against 11.72 at easy), so there is often no standing enemy mill at the moment of a capture. Practically: if you form a mill while the opponent has a free piece, you cannot take the piece you most want; you have to take a free one. If you can form the mill at a moment when the opponent's other pieces are all milled, the choice opens up.

A double mill still buys one removal. One placement can complete two mills at once — the probe builds the position with white on (3,0), (1,1), (5,1), (3,2) and a piece in hand, then places on (3,1): the row mill (1,1)-(3,1)-(5,1) and the column mill (3,0)-(3,1)-(3,2) both complete. The engine marks one pendingRemoval flag, offers the legal removals once, and after one removal the turn passes. There is no second removal and no extra turn. In the 80 easy games this happened three times; in the medium and hard samples it never happened, because those AIs do not leave the shape on the board.

So the value of a mill is exactly one enemy piece. That is the unit to count with. A mill that costs you two pieces of position to build is a bad trade; a mill that removes the piece holding the opponent's own mill together is a good one.

4. The placement phase is a threat auction

A threat is an empty point that would complete a mill if you placed there. With two pieces in a line, that line has exactly one threat. With one piece in a line, it has none. So the smallest useful unit in the placing phase is a pair of pieces in the same mill.

The arithmetic is simple and unforgiving:

• Each point lies in two mills. Placing your first piece therefore creates four partner points that would form a two-in-a-row (two in each of its mills).
• Two pieces that share a mill create one threat point (the third point of that line).
• A mill line has three points, so any single enemy piece on any of those three points blocks the whole line. There is no way to complete a mill through an occupied point.

That is why "build a mill" is not a plan by itself. The opponent can block a line by occupying one of its three points, and the cheapest block is usually the point you were about to use. The real decision in the placing phase is which of several two-in-a-rows to keep alive, and how many of the opponent's threats you can kill with the same piece.

The probe's games show how quickly this converts. The first capture arrives at a median of ply 7 at easy, ply 11.5 at medium and ply 20 at hard; the placing phase is 18 actions, so at hard the first capture usually lands after it. And the side that formed the first mill won 59 of 78 decisive easy games (75.6%), 21 of 30 medium games (70.0%) and 9 of 10 hard games (90%). That is a correlation, not a proof — the side that forms the first mill is usually the side that was already ahead — but it is a strong enough correlation to make the first mill the thing to plan for.

5. Opening points: the measurements do not support an opening book

Every point belongs to two mills, so advice like "grab the intersections" describes the whole board and decides nothing. The probe tested the other popular claim — that the four degree-4 points are the best opening — by pinning White's first placement to each of the 24 points and letting the easy AI play both sides for 20 games per point (480 games):

The degree-4 opening did not win more often; the degree-3 class did best, and the spread between individual points (from 45% to 95% over 20 games) is wider than the spread between classes. That is what noise looks like at this sample size, and it is the honest reading: the first placement is not where the game is decided.

The hard AI's own behaviour says the same thing. Over 60 seeds its first placement landed on 23 of the 24 points, never more than 5 times on any single point, and (3,4) was never chosen first. It is not hoarding a "best" square; it is choosing among many near-equal moves with a seeded tie-break.

What the opening does decide is the shape of your first two pieces:

• Put your first two pieces in the same mill line. That is the only way two pieces create a threat; two pieces in different lines create none.
• Expect the reply to block one of the three points of that line. The block is cheap, so do not treat an unblocked two-in-a-row as a promise.
• The four degree-4 points are worth remembering for the moving phase, not the placing phase: they are the only points with four exits, and they are also the pivots of the four spoke mills.

6. Which piece to take: break the threat, not the corner

When a mill gives you a removal, the candidates are the opponent's free pieces (or all of them, if every piece is milled). The tempting choice is the corner or the edge piece that looks important. The better choice is almost always the piece that is part of the opponent's two-in-a-row, because removing it deletes a threat instead of merely reducing a count.

The build's AI follows that idea, but not perfectly, and the probe measured the gap. For every removal the probe counted how many two-in-a-rows each legal target sat in, took the maximum, and recorded what the AI actually chose:

Two readings, both useful:

• The AI takes the most threatening piece about two times out of three, so copying its removal is a decent default.
• It takes a harmless piece roughly one time in three even when a threat is available. If you are playing the AI, that is free equity: after its removal, check whether your threat survived, because quite often it did.

For a human, the checklist is short. Before tapping a piece, ask:

1. Which of these pieces sits in the most two-in-a-rows? Take that one.
2. If the opponent has a mill, are all their pieces milled? If not, the milled ones are untouchable; do not waste the removal on a free piece that breaks nothing.
3. Does removing this piece open an exit that the opponent needs? Taking a piece that was blocking one of your lines can be worth more than taking a threat.

7. The moving phase: pieces beat exits

When the last piece is placed, the game changes. Now each turn is one slide along a line, and the side with no legal move loses. The natural instinct is to count exits — how many empty neighbours your pieces have — and to assume more exits is better. The probe measured that assumption and it does not hold in a straight line:

The exit count flips direction between levels (well below half at easy and medium, well above at hard), so it is not a stable signal. The piece count does not flip: the side that ends the placing phase with more pieces wins about three quarters of the decisive games at every level. The reason the two numbers disagree is mechanical: a crowded board has fewer exits. The side with more pieces is often the side with less room, so "more exits" is frequently a symptom of having fewer pieces, not of having a better position.

What that means for play:

• Material first. A capture is worth more than a prettier shape. The 12–13 captures per game at easy and medium are the game; at hard the average falls to 5 because both sides refuse to open.
• Then block. A blockade wins without a capture. In the 30 hard games, two ended by blockade; the engine treats "no legal move" as an immediate loss for the side to move.
• Degree-4 points are escape hatches, not trophies. A piece on (3,1), (1,3), (5,3) or (3,5) has four ways out, so it is harder to freeze. A corner has two, and a corner piece that is also blocked on one side is nearly dead. Use the four-way points when you need a piece that cannot be pinned; do not collect them for their own sake.

8. The flying endgame: three pieces is both freedom and danger

With exactly three pieces and an empty hand, you may fly: any of your pieces can go to any empty point. The probe verified the boundary — three pieces fly, four do not — and the mobility that comes with it is real. In a position where both sides are down to three, there are 18 empty points and therefore up to 54 legal moves for the flying side, compared with two or three for a blocked sliding piece.

Flying is not a rare curiosity in this build. It appeared in 78 of 80 easy games, all 30 medium games and 20 of 30 hard games. The reason is structural: the win condition is "opponent has fewer than three pieces", so every game that is not a draw passes through a three-piece position on the losing side, and often both sides reach three before the end.

The tension is that three pieces is also one capture away from losing. A flying side can escape any block, but it cannot survive a mill: if the opponent forms a mill and you have no fourth piece, the removal takes you to two and the game is over. So in the flying endgame:

• Keep your three pieces out of a single line. Three pieces in one mill looks strong, but it also means all three are committed to one line; a single enemy mill elsewhere decides the game.
• Fly to the point that creates a threat, not the point that looks safest. With nothing to block, tempo is everything; a threat forces the opponent to answer.
• Watch the 50-action counter (next section). A flying side that only shuffles can drift into a draw even when it is "winning" on shape.

9. The 50-action draw, and why the hard AI draws most

This build calls a draw after 50 actions without a capture. The probe pinned the boundary exactly: a position with 49 quiet actions is still live, and the 50th ends the game with winner 0 and reason "fifty". A removal resets the counter; placements and slides increment it. The count is actions, not turns, so a mill plus its removal counts as two actions, and the removal resets the clock.

That single rule explains the biggest difference between the three levels:

At hard, 19 of 30 games (63.3%) ended in the 50-action draw, and the average game had only 5 captures instead of 12–13. The mechanism is not a bug: the depth-4 AI values a mill against it so heavily that it refuses to open lines, and two such AIs can shuffle for 50 actions without either side giving a mill. The easy AI, by contrast, walks into mills and the games finish by piece count.

For a human player this has three practical consequences:

1. A draw is a real result, not a failure state. If you are playing the hard AI and neither side opens, the page will call it a draw. You have not "lost"; you have failed to create a forced mill.
2. Against a passive opponent, force contact. Trade pieces when you are ahead in material, and use your own mill threats to make the opponent spend actions. Every block is an action that does not reset the counter; only a capture does.
3. Do not chase the draw when you are behind. The draw rule is symmetric, so a losing side can sometimes hide behind it. If you need a win, you have to accept risk: open a line, offer a mill, and take the capture race.

One more measurement is worth keeping in mind: the side that formed the first mill won 90% of the decisive hard games (9 of 10). At hard the games are few and long, but the first mill is still the event that decides them.

10. Controls, levels and what the numbers on the page mean

The shell is deliberately small, and each control maps to a measurable behaviour:

• Undo steps back over one of your actions together with the AI's reply. The history stores the position before each action, so undo always returns the turn to you; it does not let you undo only the AI's move.
• Hint asks the hard AI (depth 4, 140k-node budget) for its move, highlights it and increments the hint count shown on the win card. It does not change the level; a medium game can still get a hard hint.
• The level chip cycles Easy / Medium / Hard and restarts the game. The levels change search depth (2 / 3 / 4) and the node budget (20k / 60k / 140k), not the rules. Easy also picks among moves within 30 evaluation points of the best (top four), which is why it walks into mills; medium and hard break ties within 12 points, hard within 0.
• The colour chip swaps sides and restarts. As White you move first; as Black the AI opens. The opening audit above was run with White moving first, which is why the white win rate is the one quoted.
• ?level=easy, ?side=black, ?seed=7 pin a game. The seed drives the AI's tie-breaking (a mulberry32 stream mixed with the ply count), so the same seed and the same moves reproduce the same replies. It does not seed the hint (the hint uses seed + 977 and the hard level).
• The record (win–loss–draw per level) and the move/hint counters stay in this browser's localStorage; nothing is sent anywhere. The analytics bridge fires game_load, game_start on the first human action, game_60s, game_restart and game_win only when you win.

A short checklist for the next game:

1. First two pieces in the same line — that is your first threat.
2. Count the opponent's threats before every placement; a block that also builds your line is the best move on the board.
3. When a mill lands, take the piece in the most two-in-a-rows, not the nearest corner.
4. If the opponent has a mill and a free piece, the milled pieces are safe; spend the removal elsewhere.
5. In the moving phase, count pieces first, exits second.
6. At three pieces, fly to a threat, not to safety.
7. If the game goes quiet, remember the 50-action clock and force a capture.

FAQ

How do you play Nine Men's Morris?

Each side has nine pieces. In the placing phase you alternate putting one piece on an empty point. Three of your pieces in a line form a mill, and a mill lets you remove one opponent piece. When both hands are empty you slide one piece per turn along a line to an adjacent empty point. With exactly three pieces left you may fly to any empty point. You win when the opponent has fewer than three pieces or no legal move. This build draws after 50 actions without a capture.

How many points and mills does the board have?

The build generates 24 points, 32 edges and 16 mills. Every point belongs to exactly two mills (one horizontal, one vertical). The 16 mills are the 12 sides of the three squares plus the four spoke halves; the centre (3,3) is not a point, so the two long spokes are split into separate mills.

Can you take a piece from a mill?

Not while the opponent still has a piece outside a mill. A piece inside an enemy mill is protected only while at least one free enemy piece exists. If every enemy piece is inside a mill, all of them become removable. In the measured games the restriction was live at 65.9% of easy removals, 67.9% of medium removals and 21.3% of hard removals.

What happens when you have three pieces left?

With exactly three pieces and no pieces in hand you may fly: any of your pieces can move to any empty point. With four pieces you slide normally; with two you have already lost. Flying appeared in 78 of 80 easy games, all 30 medium games and 20 of 30 hard games.

What is the best first move?

The measurements do not support a best first move. Pinning White's first placement to each of the 24 points and playing 480 games gave win rates of 58.8% for the four degree-4 points, 66.3% for the degree-3 points and 59.2% for the corners — a spread consistent with noise. The hard AI's first placement landed on 23 of the 24 points across 60 seeds. Put your first two pieces in the same line and play the threats.

Why did my game end in a draw?

This build calls a draw after 50 actions without a capture. A removal resets the counter; every placement and slide adds one. At hard, 19 of 30 measured games ended this way, because two depth-4 AIs refuse to open lines. The draw is a rule of this build, not a bug and not a loss.

Why can't I take the piece I want after forming a mill?

Because the opponent still has a piece outside a mill. The rule protects milled pieces while a free piece exists, so the legal targets are the free ones. If you want the milled piece, you must first remove every free piece — which usually takes more than one mill.

What do the AI levels change?

Only the search, not the rules. Easy searches 2 plies with a 20,000-node budget and picks among near-equal moves (within 30 points, top four); Medium searches 3 plies with 60,000 nodes and breaks ties within 12 points; Hard searches 4 plies with 140,000 nodes and breaks ties only at equality. That is why Easy walks into mills and Hard can hold a draw.

Is Nine Men's Morris solved?

Not by this page and not by this build. The page ships no opening table and no endgame tablebase; the AI is a depth-4 alpha-beta search with a node cap. Statements about perfect play belong to game theory, not to this build's measurements, and we do not make them here.

Does undo or hint cost anything?

Undo has no cost and always returns the turn to you (it restores the position before your action, including the AI's reply). Hint counts itself: the win card shows how many hints you used, but the game does not deduct points or block the hint at any level.

Can the AI be pinned to reproduce a game?

Yes. ?seed= fixes the tie-breaking stream, ?level= fixes the search depth and ?side=black lets the AI open. The same seed and the same moves reproduce the same replies; the hint uses its own seed offset (seed + 977), so a pinned game does not pin the hint.