How often is parity right?偶数理論は何割当たるか
The parity rule says: in the endgame, move first in a region with an odd number of empty squares. How often does that match the best move? We took 3,600 positions with 8, 10, 12 and 14 empty squares from games between the Itte Dojo computer players and solved every legal move to the end.
What we found
- In the 1,740 positions where you can move in both an odd and an even region, the best move was in an odd region 72% of the time: 76% with 8 empty squares, falling to 64% with 14.
- Narrowing by region clearly helps. The best odd-region move was on average 2.3 discs below the best move; the best even-region move was 6.5 below.
- But in about 28% of them only an even-region move was best, and in 15% every odd-region move lost 6 discs or more. Taking a corner, or not giving one away, can matter more (see the example below).
- Picking just any odd-region move cost 9.8 discs on average, only 1.4 better than picking any move at all (11.2). Parity tells you which region to play in; which square depends on corners and on how many moves you leave the opponent.
Results by number of empty squares
| Empty | Positions | With a choice | Best move in an odd region | Loss: best odd-region move | Loss: best even-region move | Loss: any odd-region move | Loss: any move | Best odd move loses 6+ |
|---|---|---|---|---|---|---|---|---|
| 8 | 1,000 | 412 | 76% | 2.1 | 6.8 | 8.5 | 9.9 | 14% |
| 10 | 1,000 | 487 | 74% | 2.2 | 6.6 | 9.5 | 11.2 | 15% |
| 12 | 1,000 | 518 | 73% | 2.3 | 6.5 | 10.4 | 11.8 | 15% |
| 14 | 600 | 323 | 64% | 2.9 | 5.8 | 10.9 | 11.8 | 19% |
"With a choice": positions with moves in both odd and even regions (in the rest, every move is in an odd region or every move is in an even one, so parity cannot choose). Losses are average final disc margins below the best move.
When parity is wrong
One of the positions where an even-region move is much better than any odd-region move. Tap a dotted square to see the final margin after that move with best play to the end.
Black to move, 10 empty. The best move a1 is in an even region and ends 20 discs better than g2, the best odd-region move.
- a1（even region）: +20 (a win by 20)
- b7（even region）: +6 (a win by 6)
- g2（odd region）: ±0 (a draw)
- g7（even region）: -4 (a loss by 4)
- g1（odd region）: -20 (a loss by 20)
- d1（odd region）: -22 (a loss by 22)
Press a square with a dot to see the result of that move.
How we measured
- The Itte Dojo computer played itself at the dojo's "Easy" and "Normal" levels. The first 6 moves were random legal moves, so the positions vary.
- We kept positions with 8, 10, 12 and 14 empty squares where the side to move had at least two legal moves (1,000 each; 600 at 14, which takes longer to solve).
- Empty squares were grouped into regions (connected horizontally, vertically or diagonally) and each region counted as odd or even.
- Every legal move was solved to the end. When several moves tie for best, the position counts as "best in an odd region" if any of them is.
The same engine runs on this site. node scripts/parity-study.mjs 1000 2026 <empties> reproduces each row (600 for 14).
Limits
- The positions come from computer games, not human or expert games, which reach different shapes. Parity may be right more (or less) often in strong human play.
- This is the plain form of the rule, "play in an odd region". Uses such as keeping a region the opponent cannot enter, or handing over the move, are not counted.
- Positions before 14 empty squares (the midgame) were not studied.