Parity
In the endgame, the empty squares split into separate regions. Whoever plays last in a region leaves the opponent no square there to answer from.
Start with just three empty squares
White to move. Should you start at a5 or f8 to win? Choose a dotted square on the board.
The initial empties form two regions: a5 on its own, and f8–g8 together.
Only three moves remain. See what changes when you change their order.
Compare both move orders
| White’s first move | Best continuation (White → Black → White) | Final result |
|---|---|---|
| a5 | a5 → g8 → f8 | White wins by 6 |
| f8 | f8 → g8 → a5 | White loses by 2 |
Let your opponent enter the two-square region first. That is the starting point of parity.
These are fully solved results for this position. Passes and regions your opponent cannot enter can change the picture: playing in an odd region first is not a universal winning rule.
Try a longer endgame next
These follow-up puzzles have 11–12 empties. If they feel difficult, read the explanation below first.
Every legal move and its continuation in this three-empty exercise has been verified by an exact endgame search.
Why the last move matters
If you fill the last empty square of a region, your opponent has no square left there to answer from, so the discs you just flipped cannot be taken back from that region (a move elsewhere can still flip them). If your opponent moves last in a region, you lose that chance instead.
Play first into odd regions
Without passes, turns alternate. If you play first into a region with an odd number of empties, you also play its last square. In an even region, playing first lets your opponent play last. So in the endgame, play into odd regions first and let your opponent open even ones. This is parity.
Across the whole board, 60 squares are filled in turn, so without passes White makes the last move. Black needs a pass or a change of order to get it back.
Black to move. f8 ends -10 (a loss by 10); the best move f3 ends +2 (a win by 2).
- f8 is in an even region (2 empties); f3 is in an odd one (5).
- Opponent's moves: 8 after this move, 6 after f3.
Best line: f3 b7 a8 b6 c7 b8 d8 c8
Press a square with a dot to see the result of that move.
Checking in a game
- From about 14–20 empties, count the empties in each region.
- If you have a move in an odd region, consider it first.
- Keep regions where only you can move: they let you hand the turn back.
Parity is a principle. A corner-giving move or a region your opponent cannot enter can change it. With about 10 empties left, confirm by reading.
Parity in numbers
Solving 3,600 positions exactly: in the 1,740 with moves in both odd and even regions, the best move was in an odd region 72% of the time. Picking any odd-region move gained almost nothing. How we measured, and the results →
Endgame puzzles on this theme
- No. 2○ White · 11 empty
- No. 3● Black · 12 empty
- No. 4● Black · 12 empty
- No. 5○ White · 13 empty
- No. 6○ White · 13 empty
- No. 7● Black · 14 empty
- No. 10○ White · 15 empty
- No. 12○ White · 11 empty
- No. 41● Black · 10 empty
- No. 42○ White · 11 empty
- No. 43● Black · 12 empty
- No. 44○ White · 13 empty
- No. 45● Black · 14 empty
Play the dojo and get every move graded →