Reversi

The board starts with four discs in the middle, set as a cross with each colour on one diagonal, and Dark moves first. A legal move puts a disc on an empty square so that, in at least one of the eight directions, an unbroken run of your opponent's discs leads from the new disc to another disc of your own. Every disc in every run you close that way flips to your colour.

Two things trip people up. A placement that flips nothing is not a move — sitting a disc next to an enemy disc is not enough, the far end of the run has to be closed by a disc you already own. And flipping does not cascade: the discs that turn over are exactly the runs measured before you placed, so a disc that changes colour during your move never opens a second run in the same move.

Because every disc you own is a disc your opponent can flip, and a square you can no longer play on. What decides a game of reversi is not how many discs you hold but how many moves you have left. A player with a fat midgame lead is usually sitting on one solid block in the centre with very few legal replies, and sooner or later the only squares left are the ones beside a corner.

The opponent on this page is built on that idea rather than merely told about it. Disc difference counts for nothing at all in its evaluation until fewer than 20 empty squares remain, and only reaches its full weight on the last placement; mobility — the difference between the number of legal replies each side has — does the reverse, and is at full weight for the whole opening and midgame.

To check that this is a real effect and not a slogan, the test suite plays 24 complete games between two opponents searching the same four plies and differing only in how they score a position: 12 games with the positional evaluation playing Dark and 12 with it playing Light, each from its own seed, so neither definition gets the first move for free. The positional evaluation won all 24 of those games, by an average final margin of 45 discs. Same search, same depth — only the definition of a good position changed.

Twenty-eight of the 64 squares sit on the edge — four corners, eight C-squares and sixteen safe edge squares. Add the four X-squares just inside the corners and exactly half the board is named. The other half is the open middle, where discs change hands constantly and almost nothing is permanent.

The evaluation here penalises an X-square or a C-square only while the corner beside it is still empty. That condition is the whole point: once you own the corner, the square diagonally inside it is not a liability at all — it is frequently a disc that can never be flipped again. A fixed weight table that always punishes b2 has the rest of the game backwards.

They are skipped, and it is compulsory in both directions: you may never decline a turn you could take, and you never lose your turn for good. The opponent simply moves again, and the game carries on until neither side has a legal move. This board does it for you — there is no pass button, because a pass is never a choice.

It is not an edge case either. Across the 24 self-play games in the test suite a player was skipped 128 times, a little over five times a game, and 7 of those 24 games ended with empty squares still on the board because both sides ran out of moves first. Every one of those 128 skips is a point at which an implementation that ended the game the moment one player got stuck would have stopped early and called the wrong winner.

Alpha-beta minimax over the position, with no random numbers anywhere: even the choice between two moves it rates exactly equally is drawn from the game's own number rather than from the clock, so a game replays move for move from that number on any device. The three levels differ in how far they look and, on Easy, in what they are looking for.

"Solves exactly" is meant literally. Inside that window the search budget covers every disc still to be placed, so it is playing the rest of the game out to the final count rather than guessing at a cut-off — a pass costs no depth, because it places no disc. Beat Easy comfortably, then lose to Medium, and you will have learned the most useful thing about reversi in about ten minutes.

Reversi appeared in England in 1883. Both Lewis Waterman and John W. Mollett claimed to have invented it and each accused the other in print of copying; Mollett had already published a nearly identical game called The Game of Annexation. The dispute was never settled, and the name has been public property ever since — which is why this game is called Reversi here and nothing else.

The original rules differed in one detail worth knowing. The opening cross was not fixed: the first four discs, two from each player, went into the four central squares in whatever arrangement the players chose, so a game could begin with the colours side by side rather than diagonal. The fixed diagonal opening used on this page is the standardised form, and it is what every modern ruleset and every serious study assumes.

In 2023 Hiroki Takizawa published a computational proof that the 8 × 8 game is a draw under perfect play from that standard opening — a result of the same kind as the 1988 solution of Four in a Row, and about as much use in a real game between two imperfect players.