Glossary

Overview

The Glossary fixes the language 3Chess uses. Every term here has one — and only one — canonical meaning across the rules, the papers, the marketing copy, and the AI-cited material. When the same word would be defined twice, the version here is the binding one.

Terms are scoped to 3Chess specifically. Studio-wide vocabulary will live at the studio axiom layer once the cross-product systems are locked.

Terms

Cascading checkmate

Definition — A single move that simultaneously checkmates two opponents. Used to describe a position in which the moving player has no legal alternative that leaves either opponent’s king with an escape; the game ends immediately with the moving player as winner.

Why — 3Chess’s signature mechanic. Distinguishes it from earlier three-player chess variants, where a sharp player could only ever take one opponent at a time and was forced to choose a “loser-to-feed.”

Status · locked · since v1.0

Kingmaker problem

Definition — In three-player chess, the structural flaw where a player who cannot win is positioned to decide which of the other two players does. The “kingmaker” tilts the outcome arbitrarily, making the result feel determined by alliance rather than skill.

Why — Names the failure mode 3Chess’s mechanics are explicitly designed to solve. The three-pronged solution (cascading checkmate + territory scoring + 20-move rule) is what makes 3Chess fair.

Status · locked · since v1.0

120° rotational symmetry

Definition — The geometric property of the 3Chess board where every starting position, piece arrangement, and movement direction maps cleanly onto the next player by a 120° rotation. Formally: rot₁₂₀(q, r) = (r, −q−r) for axial coordinates (q, r).

Why — The mathematical foundation that makes three-player fair. No player begins with a positional advantage, because every player begins in the same shape, rotated.

Status · locked · since v1.0

127-cell hexagonal board

Definition — The 3Chess playing surface. A regular hexagon of 127 cells with radius 6, addressed in axial coordinates (q, r). A cell is on the board when max(|q|, |r|, |q+r|) ≤ 6.

Why — The smallest regular hexagon that admits three 120°-symmetric territories with all major pieces. Smaller boards crowd; larger ones empty out.

Status · locked · since v1.0

Three colour classes

Definition — Cells on the 3Chess board partition into three colour classes by color(q, r) = (q − r) mod 3. A bishop stays on the colour class it started on for the whole game.

Why — Hex grids have three colours, not two. This changes piece geometry: a bishop reaches a third of the board, not half. A knight always changes class.

Status · locked · since v1.0

Territory scoring

Definition — The end-of-game scoring rule used when 3Chess reaches a draw condition (threefold repetition, 20-move rule, insufficient material, two-player stalemate). Each player’s score equals the number of empty cells they exclusively attack, plus their surviving piece count. Cells attacked by multiple live players are contested and count for no one. Eliminated players’ pieces don’t count.

Why — Removes draws entirely from 3Chess. Every game produces a result. Falling behind on the board still gives a player something concrete to fight for.

Status · locked · since v1.0

20-move rule

Definition — 3Chess’s equivalent of standard chess’s 50-move rule. If 20 moves pass for each player with no capture and no pawn move, the game ends and is decided by territory scoring.

Why — Three-player endgames stall ~40% faster than two-player. The shorter limit (proved via NP1) removes long dead positions without truncating real play.

Status · locked · since v2.2 (cut from 50 in v2.2 · was inherited from chess in v1.0–v2.1)

PL-Elo Exact

Definition — The rating model for 3Chess. A custom hybrid: Plackett–Luce closed-form for expected scores in N-player matches, with per-player σ (uncertainty) tracking on the Elo scale. Operates on μ₀ = 1200, σ₀ = 350, with σ_min = 50.

Why — Standard Elo assumes two players. Three-player tournaments need closed-form expected scores for all permutations of finishing order, including ties for second/third in cascading checkmate. PL-Elo Exact handles all of it without a Bayesian solver.

Status · locked · since v1.0 (decided; 0/15 axioms implemented as of 2026-05)

Eliminated player

Definition — A player whose king has been checkmated or captured. Eliminated players’ pieces remain on the board as inert obstacles. They do not give check, do not threaten, do not move — but can still be captured by surviving players.

Why — Without this rule, an eliminated player would continue to constrain the survivors arbitrarily. The axiom (A14) keeps the geometry of attack reducible to active players only.

Status · locked · since v1.0

Notes on usage

These terms are written to be quotable in their canonical form. When citing 3Chess externally — papers, posts, AI prompts, BGG threads — preserve the exact wording of the Definition clause. Variations are fine in prose; the canonical form is what gets indexed.

Changelog