Othello Parity: Odd Regions, the Last Move and When It Matters

Othello parity is about who gets the last move. If nobody passes, the player to move gets the last move whenever an odd number of squares are empty, and the same is true inside each region of empty squares near the end. So the rule of thumb is: play in an odd region. With an exact solver on 1,965 real endgames, the best move was in an odd region 68.0% of the time it had the choice, but parity only paid off as a tie-breaker, after corners and mobility.

This guide goes deeper than section 5 of our Othello strategy guide, which shows how making your opponent pass wins endgames. If you need the rules first, including how passing works, start with how to play Othello.

How we tested

We had two copies of the Medium computer from Poozee's Reversi play 2,000 seeded games, and stopped each one with 10 to 14 squares empty (a different count for each game). That left 1,965 endgame positions; the other 35 games ended early. For each position, an exact search to the end of the game gave every legal move's final disc margin with perfect play by both sides. Then we played each position out twice: once with perfect play (following one best line each time), and once the way the two Medium players actually played it.

A region here is a group of empty squares that touch, along a side or at a corner. A region is odd or even by how many empty squares it has. In our positions there were 4.4 regions on average.

Who gets the last move

Count the empty squares. If the number is odd on your turn, and nobody passes from here, you will place the last disc. That's called having parity. The game starts with 60 empty squares and Black moves first, so White starts with parity: if nobody passes and the board fills up, White places the last disc.

Passes break that count. Every time a player has no legal move and passes, parity flips to the other side. And passes are common near the end:

So having parity on the count told you nothing about who would win: 49.9% is a coin flip. Actually making the last move went with winning: the player who made it finished 12.6 discs ahead on average. The games the two Medium players played out looked the same: whoever made the last move won 68.7% of them, and White made the last move in only 62.2%.

The lesson is to count passes, not just squares. A move that leaves your opponent with no legal reply gives you the next move and flips parity your way. In the 47 positions where such a move existed, it was a best move in 39.

Odd and even regions

Late in the game the empty squares split into regions, and moves in one region don't change the empty squares in another. Inside a region, players often take turns, so the player who moves first in an odd region also tends to move last in it: they fill the last square of that region and keep the move that follows.

That's why the parity rule says to play in an odd region and leave the even ones alone. Our positions back it up as a tendency:

• In 1,477 positions, the player to move could play in both an odd and an even region.
• Only 46.5% of their legal moves were in odd regions, but a best move was in an odd region 68.0% of the time.
• The best odd-region move averaged 4.1 discs better than the best even-region move.

Worked example: parity breaks the tie

White has five moves with 11 squares empty, split into regions of 3, 1, 2, 3 and 2 squares. b7 and b8 would let Black into the a8 corner, and both lose. That leaves b4, h2 and h7, and each of them leaves Black exactly 7 replies, so counting moves can't separate them.

Parity can. Of the three, only h2 is in an odd region (a region of one square), and it's the only winning move: White wins by 4 with best play. The even-region moves b4 and h7 lose by 36 and 32. After h2, Black is to move with 10 squares empty in regions of 3, 2, 3 and 2, and in the best line our solver follows from there nobody passes, so White places the last disc.

Pro tip: When two moves look equally good, count the empty squares in each one's region. If one region is odd and the other even, the odd one is the better bet.

Parity is the last rule, not the first

To see how much parity is worth next to other habits, we scored simple rules on the 1,898 positions where the player to move had more than one move. Each rule picks from the moves it likes, at random. "Best move" is how often it found a move as good as the exact solver's; "Discs lost" is how many discs it gave away per move compared with the best move; "Result thrown away" is how often it turned a win into a draw or loss, or a draw into a loss.

Parity on its own beats random play, but not by much. "Don't give a corner" did far more, and so did leaving your opponent few moves. The best rule we tried used parity last, only to choose between moves that were already equally safe and equally restrictive. Putting parity before the reply count instead gave 54.7% best moves, about the same as leaving it out (54.6%). We also tried preferring odd regions your opponent can't play in, and it found a best move only 28.7% of the time, slightly worse than plain odd regions.

The rule with parity last found the best move more often than Medium, which looks two moves ahead, and gave away 4.9 discs per move against Medium's 8.0.

Worked example: a corner beats parity

Ten squares are empty, in regions of 2, 2, 3 and 3. The parity rule points at b7 and b8, the only moves Black has in an odd region, but both open the a8 corner for White: b7 loses by 10 and b8 by 4. The h1 corner sits in an even region, and it wins by 32 with best play.

How to use parity in your games

1. Count early. Around 14 empty squares, count the squares and the regions. Note which regions are odd.
2. Corners come first. Take a corner when it's safe, and don't make a move that lets your opponent take one, whatever region it's in. Our X- and C-squares guide shows when a move next to a corner is safe.
3. Then count replies. Of the safe moves, prefer the ones that leave your opponent the fewest moves, and look hard for one that leaves none.
4. Then use parity. Between moves that are still equal, play in an odd region and leave even regions alone.
5. Watch for passes. Each pass flips who gets the last move, so recount after one.

If you want a feel for regions, play a few games against the Hard level of Poozee's Reversi, which plays the last 10 moves perfectly, and count the regions when about 14 squares are left. Hint shows the move the Expert computer would play, so you can check your counting. Odd-or-even counting decides another classic endgame, too: see the long chain rule in our dots and boxes strategy guide.

Keep playing

• Play Reversi against the computer or a friend
• Othello strategy: corners, mobility and the squares to avoid
• Othello openings for beginners: the first ten moves, tested
• How to play Othello: the rules, passing and scoring