The math of infinite chess | Joel David Hamkins and Lex Fridman

Quick Overview

Infinite chess is played on an infinitely extended chessboard where pieces move according to standard rules but without boundaries, and the key mathematical feature is the existence of positions where White has a guaranteed win, but Black controls the length of the game, meaning the position is not mate in N for any finite N, yet White wins infinitely many moves, a concept related to transfinite ordinals like omega.

Key Points: Infinite chess takes place on an integer board infinite in all four directions, maintaining the chessboard pattern, and unlike standard chess, it starts from complex, pre-existing positions rather than a standard setup. A crucial distinction in infinite chess is the existence of positions where White has a winning strategy that requires an infinite number of moves, meaning it is "not mate in n for any n," but Black controls the duration of the game, which is doomed for Black. The rules involve standard piece movement without intervening pieces, and the three-fold repetition rule is replaced by the actual rule: infinite play is a draw, meaning the only way to win is by checkmate at a finite stage. Because there is no edge on the infinite board, pawn promotion does not occur in infinite chess. The game value of such positions is described using ordinals; a position with game value omega means White wins, and Black, playing as the second player, essentially counts down from omega by choosing a finite number initially, and then subtracting one move for every subsequent turn. Joel Hamkins developed the concept partly through a Math Overflow question and collaborated with US National Master chess player Corey Evans to construct and prove the validity of these complex positions, correcting initial flaws in piece placement. Initial work established positions with game values up to omega cubed, and subsequent work with Norman Promotter reached omega to the fourth, leading to the current understanding that every countable ordinal arises as the game value of some infinite chess position.