No reachable chess position with more than 218 moves

Clarifying the problem (“218 moves”)

• Many readers initially misread the title as about game length or distance-to-reach.
• The article is about branching factor: a position where it’s one side’s turn and they have at most 218 legal moves available.
• Several comments suggest clearer phrasings like “no position with more than 218 legal/possible next moves” or “moves to choose from.”
• Once clarified, confusion about the initial position’s move count (20 from the standard start) disappears.

Chess rules, reachability, and longest games

• “Reachable” means attainable from the standard starting position by legal play.
• There’s side discussion on rules that bound game length:
  • 50‑move and 75‑move rules (pawn move or capture resets counter).
  • 3‑fold vs 5‑fold repetition (optional vs automatic draw).
• Links are shared claiming an upper bound of ~8848.5 moves under modern rules, and others derive rough bounds by counting pawn moves and captures.
• Some debate exists over whether certain rules are “optional” and thus must be considered in theoretical upper bounds.

MILP / integer programming modeling

• Commenters note that the author effectively used mixed‑integer linear programming tricks (row generation / lazy constraints / branch‑and‑bound).
• A solver expert points out these ideas map well to standard MILP features like lazy constraints.
• The author explains that simplifying chess rules (e.g., temporarily ignoring check on the white king) and tightening the LP relaxation were crucial for speed.
• The relaxed model gave an upper bound of ~271.67; after model improvements the solver proved optimality at 218.

Human constructions vs computer proof

• Historical composers produced 218‑move positions by intuition and iterative refinement.
• Commenters outline plausible human heuristics: maximize centrally placed queens, minimize black material while avoiding illegal check or stalemate, push black king into the corner, keep extra pieces on board edges.
• The article’s contribution is not a new 218‑move position but a proof that 218 is maximal among all reachable positions.

Legality, reachability, and many queens

• A distinction is made between:
  • “Legal but non‑reachable” positions in problem‑composer jargon (no immediate rule violations, but not derivable from the normal start).
  • FIDE’s stricter definition where such positions are simply “illegal.”
• This surfaces in diagrams with >9 queens: logically consistent under local rules, but unreachable given normal starting material.

Encoding moves and positions

• One motivation discussed: confirming that 8‑bit storage (≤255) always suffices to index legal moves from any position.
• Several commenters play with bit‑level encodings of board states vs theoretical minimum (~149–153 bits based on position counts).
• There’s debate over fixed‑length encodings, sparse encodings, and practical engine design (arrays of piece objects vs compressed indexes).
• It’s noted that storing full game state also needs castling/en‑passant info and, if modeling repetition explicitly, some form of history.

Other games and extensions

• A Go researcher notes that Go has a trivial maximal branching factor (361 from the empty board) and characterizes hardest‑to‑reach positions there, without brute forcing all legal positions.
• People suggest follow‑ups: repeating the analysis for Chess960 or exploring bounds on “hardest‑to‑reach” chess positions.

Reception, clarity, and lichess

• Multiple commenters praise the article and lichess generally (free, open source, strong variant support).
• Some readers found the LP/relaxation part under‑explained or “and then a miracle occurred,” prompting clarifying replies from the author.
• The author acknowledges the original title was suboptimal and has since updated it, inviting further questions about similar proof techniques.