In May 2016, the Australian mathematician Mark Watkins published a result that the chess-variant community had been working toward for fifteen years: antichess from the standard starting position is a forced win for White with the move 1.e3. The proof was computational. It required a tree of game positions large enough to occupy several terabytes of compressed storage and a verification effort distributed across the Lichess community over several years. It is the largest game that has been completely solved to date by any group.

Antichess — also called Losing Chess or Giveaway Chess — is a variant in which the objective is reversed: you win by losing all your pieces or by being stalemated, and captures are compulsory whenever available. The compulsory-capture rule cuts the branching factor of the game dramatically; many positions admit only one or two legal moves; the game is far smaller in state-space terms than standard chess. Even so, the proof that 1.e3 wins took fifteen years of accumulated work and was greeted by computer-game theorists as a benchmark achievement. The full version of the antichess article on Caissly covers the rules and the proof in more depth.

The natural next question — can standard chess be solved? — is the question this piece is about. The short answer is no, not within any timeframe relevant to humans. The long answer is more interesting, because it reveals what solving a game actually means and where the boundaries of feasibility currently lie.

What “solved” means

In combinatorial game theory, a game is strongly solved when, for every reachable position, the game-theoretic value (win, draw, loss) and an optimal move are known. Weakly solved means the value from the standard starting position is known with an algorithm to play optimally from there. Ultra-weakly solved means the value is known but the optimal moves are not.

Antichess is weakly solved. We know 1.e3 wins for White; we know how to play optimally from the start; we do not yet have a tablebase covering every reachable position in the game. Strong solution is in principle possible for antichess but has not been completed.

Standard chess is none of the above. We do not know the value of the starting position. The consensus belief — based on engine analysis at depths far beyond human capacity — is that the starting position is a draw with best play. We have no proof. We have not solved a single one of the first ten moves to a terminal value. We have solved every position with seven pieces or fewer (the Lomonosov seven-piece tablebase), but seven pieces remain a long way from thirty-two.

Why chess is harder

The combinatorial gap between antichess and standard chess is not a matter of degree. Antichess has roughly 1040 distinct positions. Standard chess has roughly 1044 to 1047 legal positions and 10120 distinct games (the Shannon number, the standard order-of-magnitude estimate). The compulsory-capture rule in antichess collapses move trees that would, in standard chess, branch wide and remain branched for a hundred plies. A typical antichess game lasts twenty to thirty moves; a typical standard chess game lasts forty to sixty.

This is the central asymmetry. Solving a game by brute-force computation requires touching every position the algorithm cannot rule out by symmetry or transposition. For chess, the lower bound for the number of relevant positions is around 1040 after aggressive pruning, and the upper bound is far higher. No computational substrate available or projected has the storage or the operations-per-second budget to enumerate that space in any human-relevant timeframe.

The question is not even close. Even quantum computing, which provides speed-up on certain search problems, does not change the fundamental order-of-magnitude problem for chess. The number of legal positions is larger than the number of atoms in the observable universe by several orders of magnitude.

Endgame tablebases as partial solution

What we have solved is the endgame. The seven-piece Syzygy and Lomonosov tablebases cover every position on the board with seven or fewer total pieces (including kings). The Syzygy six-piece tablebase fits in about 150 GB; the seven-piece version is about 17 TB. Every position in these regions has its game-theoretic value computed and its optimal move identified.

This is genuinely useful. Endgame study at the highest level now consults tablebases routinely. Famous studies — the Lucena and Philidor positions, the Vancura drawing technique, the K+P vs K opposition — are all confirmed by tablebase analysis and have been for over a decade. The tablebases have produced new results too: certain seven-piece endings that grandmasters had judged drawn for a century are actually winning with deep play involving sequences of fifty or more moves that humans cannot reasonably find.

The next tablebase frontier is eight pieces. The compressed storage requirement for an eight-piece tablebase is estimated at around two petabytes, which is feasible but expensive. A nine-piece tablebase is, as of current projections, beyond any plausible single project but not beyond the global cloud-storage market. The ten-piece frontier is genuinely speculative.

These tablebases solve endgames. They do not, even in principle, scale up to solving the whole game. The middlegame branching factor and position count is too high for any tablebase-style enumeration.

What an unsolved game leaves us

The interesting consequence is that chess will remain unsolved for the foreseeable future and yet will continue to be played at a level engines call near-optimal. This is unusual. Most well-studied finite games have a clear gap between optimal play and human play — checkers (solved, draw with best play) is the largest cleanly-solved game, and the difference between the proof’s analysis and human play in checkers is enormous. Antichess is solved, and human play is still nowhere near the proof’s optimal lines in many practical positions.

Chess is heading toward a peculiar middle state: engine play is so strong that it is taken as a proxy for optimal, but it is not formally proven optimal. Top engines disagree on certain positions. New training runs of Leela Chess Zero produce evaluation shifts in lines that had been considered stable. The frontier of the game is being explored by neural networks rather than enumerated by exhaustive search.

What this means for the player is mostly liberating. Chess is, by current evidence, a draw at the highest level — but the path to that draw involves so many decision points that human players will lose and win games for as long as humans play. The game is not just unsolved; it is structurally so much larger than the players who play it that the question of solution does not affect the lived experience of the game.

The wider question

It is sometimes claimed that any game humans play with enough seriousness will eventually be solved. Antichess was solved by computer enumeration. Checkers was. Connect Four was solved in 1988. Reversi has been weakly solved on an 8×8 board. The pattern is that small-state games yield to computation, large-state games do not, and chess sits on the wrong side of that line for the foreseeable future.

Go, which has roughly 10170 distinct positions, sits even further on that side. AlphaGo and its successors did not solve Go; they played it at a level humans cannot match, while leaving the formal question of optimal play untouched. The same pattern holds for chess and, increasingly, for every game humans treat as serious.

The death of opening theory and the impossibility of solving the game are the same observation from two angles. The frontier of the analysable has moved past the frontier of the playable. We can verify near-optimal play in the openings we know. We cannot, even in principle, exhaust the game itself. What is left, for humans, is to play.

References

Cross-links inside Caissly: Antichess covers the solved variant; Atomic and Chess 960 cover other modern variants. Endgame technique entries: Lucena position, Philidor position, Vancura position.

Issue Nº 004 · The Magazine · The Caissly Editorial