Three in a row: how not to lose
Three in a row is a solved game: with correct play every opening ends in a draw. What the complete game tree means for a player.
Three in a row is a preset of the game Lines on a 3 × 3 board. It is so small that all of it can be searched: of 255,168 possible games, 131,184 are won by the first player, 77,904 by the second and 46,080 end in a draw.[1] With random play the first player therefore has the edge, but with correct play nobody wins.
What holds with correct play
- Every opening leads to a draw if the second player answers correctly.
- After an opening in the centre, only one of the four corners saves the draw.
- After an opening in a corner, the only answer is the centre; seven of the eight answers lose.
- After an opening on an edge, the centre, two adjacent corners or the opposite edge hold the draw.
- Every losing answer loses on the seventh move.
The fork
A fork is a move that makes two threats at once, and the opponent can block only one. The best-known trap: the first player plays a corner, the second the centre, the first the opposite corner. The second player must then play an edge, and an answer in a corner loses.
The order of eight rules: win, block the opponent, make a fork, stop the opponent's fork, centre, opposite corner, empty corner, empty edge. In a check of 762 positions it did not worsen the result in any of them.
Against the AI
All the numbers come from games of AI opponents; people were not measured.
| Way of playing against the Weak AI | Won as the first player | Won as the second player |
|---|---|---|
| random moves | 14.3% | 1.3% |
| blocking only | 68.4% | 29.6% |
| centre and corners | 74.9% | 41.0% |
| plus the fork | 81.1% | 41.0% |
| the eight rules | 86.1% | 41.0% |
| perfect play | 88.5% | 41.0% |
| best play against the Weak AI | 95.5% | 41.0% |
- As the second player the ceiling is 41.0%, because the Weak AI does not err often enough in the centre.
- Against the Weak AI the best opening is a corner: the Weak AI errs in 86.0% of the games and with the correct continuation 94.0% are won.
- The Normal and the Strong AI have not lost a single game (1,386 games against the complete tree); centre, corners and blocking always end in a draw against them.
- The Weak AI never misses a win in one move, plays at random in 8% of its moves and misses the opponent's threat in 40% of the rest.
The other presets of Lines (Five in a row, Gravity four) are larger and have no such clear trees; what was measured about them is on the page Lines.
Notes
- See (forthcoming article no. 25). ↩︎