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Solved games

Where the answer is already known, and where nobody knows it yet

What "solved" means

A game is solved when the result of perfect play by both sides is known. That is a statement about the game, not about any program: once checkers was solved, no future engine can ever beat another perfect engine at it. Every game in the list below has been settled by a computation, most of them by years of one.

Being solved does not make a game boring for people — it makes it boring for machines. You will still lose to a perfect opponent, but you would have lost anyway; the difference is that now the loss is a theorem.

Three kinds of solution

Game Solution Who and when Perfect play ends in
Tic-tac-toe 3×3
on this site
strongknown long ago; the search is trivialdraw
Nim
on this site
strongCharles Bouton, 1901the side to move from a non-zero XOR wins
Connect Four 7×6
on this site
strongVictor Allis and James Allen, independently, 1988first player wins
Nine Men's Morris
on this site
strongRalph Gasser, 1993draw
Gomoku 15×15
on this site
weakVictor Allis, 1993first player wins (free-style rules)
Kalah 6×4 (mancala)
on this site
weakGeoffrey Irving, Jeroen Donkers and Jos Uiterwijk, 2000first player wins
Awari (mancala)strongJohn Romein and Henri Bal, 2002draw
Kuhn poker
on this site
strongHarold Kuhn, 1950 — solved on paper, no computer involvedthe second player wins 1/18 of an ante per hand; optimal play is randomised
Pig (dice)
on this site
strongTodd Neller and Clifton Presser, 2004 — value iterationthe win-maximising strategy is known; the dice still settle each game
English draughts 8×8
on this site
weakthe Chinook team, Jonathan Schaeffer, 2007 — eighteen years of computationdraw
Othello 8×8
on this site
weakHiroki Takizawa, 2023draw
Chomp 8×5
on this site
ultra-weakDavid Gale, 1974 — a strategy-stealing proof; small boards computedfirst player wins, though the theory names no moves
Sim (the Ramsey game)
on this site
strongproposed by Gustavus Simmons, 1969; solved by exhaustive searchsecond player wins — and a draw is impossible by Ramsey's theorem
Hex
on this site
ultra-weakJohn Nash, 1949 — a strategy-stealing proof; boards up to 9×9 computedfirst player wins, though on large boards nobody knows how
Dots and Boxes
on this site
partlyJoseph Barker and Richard Korf, 2012 — boards up to 4×5depends on the board; the general case is open
Gopartly5×5 — Erik van der Werf, 2002; 19×19 unsolved and likely to stay soon 5×5 the first player wins; on 19×19 nobody knows
Chesspartlyonly endgames up to seven pieces are fully computed (2012)unknown; a draw is considered most likely
Heads-up limit hold'emweakCepheus, Michael Bowling's group, 2015essentially solved: the gap is unmeasurable in a lifetime
Backgammon
on this site
openunsolved: the dice get in the wayno solution in the usual sense exists
Russian draughts
on this site
openunsolvedunknown — forced captures and flying kings widen the tree
Ataxx 7×7
on this site
openunsolvedunknown
Minesweeper
on this site
opennot a two-player game; Richard Kaye, 2000 — deciding a safe move is NP-completea fast way to always play safely most likely does not exist

Why the rest is still open

The bars are powers of ten: how many positions the game has. Each step to the right is ten times more than the step before, so the distance between checkers and chess is not a gap you close with a faster machine.

Tic-tac-toe 3×3103
Nine Men's Morris1010
Connect Four 7×61013
English draughts 8×81020
Backgammon1020
Othello 8×81028
Chess1044
Go10170
Published estimates, rounded. Red means the game is unsolved.

And the games on this site

Nine of ours are settled. Nine Men's Morris is a draw with perfect play, Gomoku on a free-style board is a first-player win, and Othello was shown to be a draw in 2023. Connect Four and Kalah 6×4 both go to whoever moves first. Kuhn poker was solved on paper back in 1950, and Pig in 2004 — solved in strategy, that is: the cards and the dice still decide each single game. English draughts is a draw too — but the game here is Russian draughts, which is not solved. In Chomp the first player wins and in Sim the second one does — both of those seats are handed to you. Hex stands apart: our board is proven to be a first-player win, and nobody knows the winning line.

None of this makes our opponents perfect. The engines on this site search a few plies and guess the rest with an evaluation function; a solved game means a perfect player exists, not that the one you are facing is it.

Where to go from here
To cite this page: “Solved games”, You vs Computer, https://youvscomputer.com/games/solved.html