At the fifth International Congress of Mathematicians in Cambridge in 1912, German mathematician Ernst Zermelo presented work that would influence mathematical and later computational thinking about strategy games. Its implications for chess separate a reassuring existence result from the difficult question of calculating the answer.
Ernst Zermelo and set theory
Zermelo is best known for foundational contributions to set theory, notably the axiom of choice and Zermelo–Fraenkel axioms, part of mathematics’ standard foundations.
Strategy games offer precise rules, no hidden information and no chance. They make a natural setting for asking whether a theoretically perfect way to play exists. The exact game rules and termination assumptions remain essential.
Zermelo’s theorem: statement and explanation
Consider two-player games of perfect information, without chance, with alternating moves and a finite game tree.
Modern statement in this finite setting: either one player can force a win, or each has a strategy guaranteeing at least a draw. In the latter case, optimal opposition produces a draw. Distinguish this from the exact content of Zermelo’s historical text, discussed below.
For a finite tree, backward induction, as in minimax, explains the result. This teaching presentation is not Zermelo’s original proof: Schwalbe and Walker explicitly note that his 1913 text does not proceed this way. Shannon’s 1950 article appears 37 years later; that interval does not establish direct reuse of the proof unchanged.
Imagine reaching terminal positions, each a White win, Black win or draw. Move one step back. If White is to move, White chooses the most favourable accessible result; Black similarly chooses from Black’s perspective. Continue recursively from terminal nodes to the root, and every node receives a value.
The initial position is therefore determined. White can force a win, or both sides can guarantee at least a draw, or Black can force a win. The statement does not identify which alternative holds in chess.
A proof refined historically
Schwalbe and Walker distinguish the original 1913 text from modern formulations. Zermelo allows infinite move sequences and studies, among other things, lengths needed to force a win. König (1927) and Kalmár (1928/1929) extend the analysis. That history differs from the elementary finite-tree explanation here.
To treat modern chess as finite, include termination rules. Threefold repetition and fifty moves create rights to claim a draw; automatic fivefold repetition and seventy-five moves play a separate role, as the FIDE arbiters’ manual explains.
Why it feels paradoxical
The paradox is practical, not logical. A value exists under the fixed rules, but the existence result does not identify it or provide a practical calculation of it.
Naive exhaustive search considers an enormous number of sequences. Shannon’s order of magnitude rests on assumptions; it is not an exact count of all legal games.
As a hypothetical calculation, examining items at per second takes seconds. That illustrates a cost under stated assumptions, not proof that no other method could establish the game’s value. See the limits of chess calculation for the distinction.
Chess’s unknown truth
The open question is the value of the starting position under perfect play. The theoretical setting is also discussed in game theory applied to chess.
A draw can be conjectured, but neither draws among strong players nor engine evaluations prove it. Those observations concern particular opponents and computing budgets, not every possible reply to a strategy.
A Stockfish evaluation depends on version, depth and time. A small numerical advantage means neither a forced win nor a measured distance to perfect solution. All three initial-position values remain open.
Solved endgames: a window on exact results
Although the starting position remains unsolved, many low-material endgames have been calculated.
Endgame tablebases, developed through work by Ken Thompson, Marc Bourzutschky and others, cover large endgame sets, including seven-piece positions. Coverage and result meaning depend on the table set: castling rights and the fifty-move rule, for example, require particular care.
Ronald de Man’s documentation illustrates why metrics and rules matter with a DTZ50 value of 164 half-moves and a “cursed win”: winning without the fifty-move limit does not necessarily mean winning with it.
DTM and DTZ: different distances
DTM measures distance to mate. DTZ measures distance to a capture or pawn move resetting the counter. Optimising them can select different moves. That creates no contradictory truths: the questions differ, and fifty-move-rule treatment must be specified. Usual Syzygy tables provide WDL and DTZ, not a universal DTM distance.
The human player’s structural limits
Chess has a mathematically defined notion of perfect play while its value from the initial position remains unknown.
A player uses calculation, pattern recognition and positional evaluation. We can observe decisions and compare results with opponents; that does not give a general measurement of distance from an unknown optimal strategy.
Human and engine match performances can be compared. They still do not provide a measured distance to perfect strategy across the game. Positions covered by tablebases, by contrast, allow decisions to be checked under their rules.
Perfect play and the ideal game
Even if a perfect strategy were written down, what would that change about playing?
Imagine, hypothetically, White had a forced win within eighty moves from the initial position. Following that strategy would fix the result regardless of Black’s choices. That would change the theoretical stakes. Human practice could nevertheless continue: knowing a strategy exists differs from applying it in every position.
The unknown solution leaves room for discovery, sporting competition and aesthetic judgement. The idea that solving chess would “kill” it is a philosophical interpretation, not a theorem about how people would respond.
Its present uncertainty contributes to the experience of exploring a game whose possibilities exceed our calculation. That does not require declaring future solution impossible.
Zermelo and solved games
Computing research has solved several games, with the meaning of “solved” tied to the starting position, strategy coverage and rules.
Tic-tac-toe is a draw under perfect play. Connect Four was solved in 1988 as a first-player win. English draughts on 8×8, checkers, was solved in 2007 by Jonathan Schaeffer and colleagues as a draw under perfect play, after eighteen years of work in a game with roughly possible positions; not every position was individually traversed.
Chess and Go remain open from their initial positions. Combinatorial growth hinders current exhaustive methods; it does not prove every future proof impossible. Exact rules must also be specified for each game.
What Zermelo changes for you at the board
Knowing the starting value is unknown does not directly choose your next move. It does change how a claim about a position should be interpreted.
In complex positions, your variations and evaluation are generally partial. Local certainties still exist: a correctly established forced mate or a tablebase-covered position may be solved under precise rules. Distinguish these from uncertainty remaining at the game’s starting point.
Against a stronger opponent, better performance does not guarantee exact solution of every position. Nor does it imply they do not know a particular endgame or forced mate. Compare concrete variations, available time and a move’s justification rather than an assumed closeness to perfection.
Zermelo’s contribution is not a winning strategy from the initial chess position. His text grounds mathematical analysis of game values; modern formulations separate existence under fixed rules from methods of calculation.
After reading: keep one reminder of mathematical humility in a long game: a partial evaluation is not automatically a tablebase result. It can help you decide what still needs checking without demanding impossible certainty everywhere.
Sources
- Zermelo, E. (1913). Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proceedings of the fifth International Congress of Mathematicians, vol. II, 501–504. Translation in the appendix to Schwalbe and Walker.
- Schwalbe, U., & Walker, P. (1999, revised version of the 1997 working paper). Zermelo and the Early History of Game Theory. University of Mannheim / University of Canterbury. Full text consulted: original proof versus modern presentation.
- Shannon, C. E. (1950). Programming a Computer for Playing Chess. Philosophical Magazine, 41, 256–275. Original article.
- De Man, R. (undated). Probetool. GitHub. Syzygy documentation: WDL, DTZ50 and a cursed-win example.
- FIDE (2022). Arbiters’ Manual. Manual and draw rules.
- Allis, V. (1988). A Knowledge-based Approach of Connect-Four. Vrije Universiteit Amsterdam. Thesis.
- Schaeffer, J., et al. (2007). Checkers Is Solved. Science, 317, 1518–1522. Full article.
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