Reversi (Othello) has an estimated game tree of ~10^28 possible game sequences — far smaller than chess (~10^123) but vastly beyond exhaustive solution with current hardware. Every game lasts exactly 60 moves when counting both disc placements and potential passes. The maximum possible score is 64–0. Understanding these numbers clarifies why computers dominate the endgame but cannot yet solve the full game. For the question of whether the game has been solved, see Is Reversi solved?
The Game Tree: How Big Is Reversi?
Total Possible Games
The game tree counts the number of distinct sequences of moves from the starting position to any possible ending. For Reversi:
- Estimated game tree size: ~10^28 possible games (as estimated in Victor Allis’s 1994 doctoral dissertation Searching for Solutions in Games and Artificial Intelligence, University of Limburg)
- Average branching factor: ~10 legal moves per position (varies: typically 4–12 in the opening, up to 20+ in the midgame, then decreasing)
- Game length: 60 moves (fixed, as each move fills one empty square; passes are moves too)
For perspective:
| Game | Game Tree Size |
|---|---|
| Tic-tac-toe | ~10^5 |
| Connect Four | ~10^21 |
| Checkers | ~10^21 |
| Reversi (8×8) | ~10^28 |
| Chess | ~10^123 |
| Go (19×19) | ~10^360 |
Reversi sits between checkers (which has been solved) and chess (which has not). Its game tree is roughly 10^7 times larger than checkers, explaining why Reversi has not yet been solved despite computers playing at superhuman strength.
Board Position Count
The number of legal board positions — distinct arrangements of discs that can occur during legal play — is estimated at approximately 10^28. This differs from the game tree (which counts sequences) because the same board position can theoretically be reached by different move sequences.
The theoretical maximum of disc arrangements on 64 squares (including illegal ones) is 3^64 ≈ 4.3 × 10^30 (each square can be empty, black, or white). The actual legal positions are a subset of this.
Fixed Game Length
One mathematically distinctive feature of Reversi: every game has a fixed length.
- The board has 64 squares
- 4 are filled at the start
- Each move fills exactly one empty square
- Therefore, every game has exactly 60 moves (including passes)
This is different from chess, where games can end at very different lengths through checkmate or resignation. In Reversi, the game always ends when all 60 empty squares have been filled (or the game ends early because neither player can move — which can happen before the board is full if all remaining empty squares are unreachable).
The implication: Every Reversi position has a defined game-phase. Move 10 is always the opening. Move 45 is always deep in the midgame. Move 55 is always the endgame. This predictability benefits systematic study — see board values for how square importance shifts across these phases.
Disc Count Mathematics
Starting and Ending States
Starting position: 2 Black discs + 2 White discs = 4 discs total, 60 empty squares
Ending state: Between 4 and 64 discs total, depending on how many squares are filled. In standard play (all 60 empty squares filled), exactly 64 discs are on the board.
Sum constraint: Black discs + White discs = 64 at the end of a full game.
Therefore: if Black has B discs at the end, White has (64 - B) discs. This means you can describe any final result with a single number — Black’s disc count. Typical competitive results range from around 30–34 (close games) to 45–19 or higher (dominant victories).
The 64–0 Result
Can one player win 64–0 (all discs)? Yes, theoretically. But it requires:
- A significant skill or knowledge gap
- The losing player making moves that systematically give away their discs
- Specific game trajectories where the winner’s final moves flip all remaining opponent discs
In practice, 64–0 results are extremely rare and require deliberate effort or a catastrophic collapse. Results like 60–4, 58–6, or 54–10 represent dominant but more typical victories.
Conservation of Disc Flips
Each move places one new disc and flips one or more existing discs. A key identity:
Black placed + White placed = 60 (total moves)
Net Black discs = Black placed - (Black discs flipped by White) + (White discs flipped by Black)
This means the final disc count isn’t just about how many moves each player made — it’s a complex interaction of placements and flips. A player who makes 30 moves but has their discs flipped repeatedly can end up with far fewer than 30 discs.
Endgame Solvability Mathematics
Why the Last 20 Moves Are Computable
With 20 empty squares remaining, the number of positions reachable from that point is bounded by the product of move counts at each step. Even with 10 options per move, 20 moves yields at most 10^20 positions — large, but tractable for modern computers with efficient algorithms.
With alpha-beta pruning and fastest-first move ordering, strong Reversi programs can solve positions with 25–30 empty squares in seconds to minutes. This is why endgame solving is standard in competitive AI.
The Branching Factor Declines
Reversi’s branching factor decreases as the game progresses:
- Opening (moves 1–20): 4–12 legal moves per position
- Midgame (moves 20–44): 4–15 legal moves per position
- Endgame (moves 44–60): 1–8 legal moves per position (rapidly decreasing)
This declining branching factor is what makes the endgame tractable. In the final 10 moves, the number of available moves per turn often drops to 2–4, making exact calculation straightforward for computers.
Probability and Outcome Statistics
Opening Symmetry
The starting position is perfectly symmetric — both players face an identical situation (with the exception of move order). The first player (Black) has four possible opening moves, but by symmetry, all four lead to equivalent positions (just rotated or reflected).
This means the true first move in Reversi occurs on the second move — when White responds to Black’s first move, the symmetry is broken for the first time.
Black vs White Win Rate
At equal skill levels, competitive data suggests:
- Black (first player) wins slightly more often at beginner to intermediate level
- At the highest level, this advantage is debated — White has the information advantage of responding to Black
- Computer analysis has not definitively established whether perfect play from the starting position is a Black win, White win, or draw
The opening symmetry means neither player has an obvious structural disadvantage — unlike, say, chess, where White is universally considered to have first-move advantage.
Score Distribution in Competitive Play
In high-level competitive play, scores cluster around the 32–36 disc range for the winner — games between strong players are often close. The distribution is roughly:
- 30–35 range: Most common for close competitive games
- 36–44 range: Decisive but not dominant victory
- 45–60+ range: One-sided game, often caused by early corner loss
In games against weaker opponents or AI at lower settings, high-scoring victories become more common as positional blunders multiply.
Mathematical Curiosities
Maximum Flips in a Single Move
A single Reversi move can flip discs in up to 8 directions simultaneously (all eight directions from a placed disc). The theoretical maximum number of discs flipped in one move is 24 (all remaining opponent discs in a symmetric configuration), though in practical play flipping 6–10 discs in one move is impressive.
Perfect Game Symmetry
If both players play symmetrically (mirroring each other’s moves across the board’s diagonal), the game proceeds in perfect symmetry until one player breaks it. This mathematical observation led to theoretical curiosities in early Reversi research — a “mirror strategy” for White that can sometimes guarantee at least a draw.
Parity and the Last Move
The player who makes the last move in any closed region of the board has a parity advantage. This is mathematically analogous to the “last player to move wins” principle in combinatorial game theory. Reversi’s endgame parity analysis is a direct application of combinatorial game theory principles — a connection between recreational mathematics and practical game strategy.