Glossary

3Chess Glossary

Definitions of the terms 3Chess uses. The page is the canonical source — every page on this site that mentions these terms links back here.

3Chess is a three-player hex chess game built from a formal axiom system. Some of the terminology is unique to it (cascading checkmate, territory scoring, the 20-move rule as applied here, the PL-Elo Exact rating); some is borrowed from chess and game theory (paranoid search, axial coordinates); some is geometric (120° symmetry, the three colour classes).

Game mechanics

Cascading checkmate #

A single move in 3Chess that delivers checkmate to two opponents at once. Made possible by the geometry of the 127-cell hex board and the three-territory layout: pressure on the shared structures that both opposing kings depend on can collapse both at the same time.

This mechanic is load-bearing for the design. It means a strong player in a winning position can simply win the whole game on the board, rather than being forced to pick which surviving opponent to hand it to. The path to victory runs through your own play, not through someone else's choice — the central fix to the kingmaker problem.

See also: The kingmaker problem essay · /rules · check, checkmate & stalemate

Kingmaker problem #

The structural flaw of three-player games where a player who cannot win still decides who does — by choosing whom to attack. Strike the leader and the trailing rival wins; strike the rival and the leader wins; do neither and you fall further behind. The result is decided by the losing player's grudges or mood rather than by skill.

The problem is not exclusive to chess but is most galling there, because chess players expect the result to be earned. 3Chess addresses it through three mechanisms: cascading checkmate (a strong player can win outright), territory scoring (a trailing player has a real goal that isn't "pick a winner"), and the 20-move rule (kills the long dead endgames where kingmaker dynamics are worst).

See also: The kingmaker problem essay

Territory scoring #

The 3Chess draw-resolution mechanism. When a game reaches a draw condition — threefold repetition, the 20-move rule, two-player stalemate, or insufficient material — it is decided by territory: the empty cells a player alone attacks, plus their surviving piece count. Highest score wins.

The point isn't to add a side goal during play; it's to replace a tie outcome with a skill-expressing result, so a player who is "behind" still has somewhere meaningful to fight for. Pawns count for territory purposes because they always threaten their two capture squares — even when those squares are empty.

See also: /rules · territory score

20-move rule #

3Chess's equivalent of chess's 50-move rule, tightened. The game ends and is decided by territory if 20 moves pass for each player with no capture and no pawn move.

Cut from 50 to 20 because three-player endgames stall faster: there is no incentive to stall when stalling just freezes a result you may not like. The shorter limit removes long dead positions without cutting off real play. Combined with threefold repetition and the two-player stalemate, every draw condition in 3Chess routes to territory rather than to a tie.

See also: /rules · draw → territory

Neutral obstacle (eliminated player) #

When a player is checkmated in 3Chess they are eliminated, but their pieces stay on the board as captures-only obstacles. They no longer move; they no longer attack; pieces of an eliminated player do not deliver check. But they still occupy cells, block lines, and can be captured for material or territory.

The mechanic preserves a sensible board state after one player goes out (the game doesn't collapse into a thin 2-player endgame on an empty board) and contributes to territory calculations.

See also: /rules · capture & elimination

Pawn facing #

Each player's pawns move toward the opposite edge of the hex board, so the three armies have three different forward direction sets. In axial coordinates: White pawns move {SW, SE}; Black pawns move {E, NE}; Red pawns move {NW, W}. Captures are the outer flanking diagonals of these forward sets.

Pawns also threaten their two capture squares even when those squares are empty — a fact that matters for territory calculations.

See also: /rules · the pawn

Board geometry

127-cell hex board #

The 3Chess playing surface. A hexagonal board of 127 cells, laid out so three armies of 15 pieces each sit on three 120°-rotationally-symmetric territories. Each cell is addressed in axial coordinates (q, r); a cell is on the board when max(|q|, |r|, |q+r|) ≤ 6.

Cell count is the natural consequence of the radius-6 hexagonal region (3·6² + 3·6 + 1 = 127). The 45 pieces (15 per player) cover ~35.4% of the board at the start.

See also: /rules · board & setup

120° rotational symmetry #

The geometric property that makes 3Chess fair across all three seats. The board, the three territories, and the starting positions are identical under a 120° rotation around the centre cell. The rotation rot₁₂₀(q, r) = (r, -q-r) maps each player's setup to the next.

Consequence: whatever holds for one army holds for all three. There is no "seat advantage" in the starting position — a fact that has to be true for a three-player game to feel fair, and that a square board cannot deliver.

See also: /three-player-chess · why a hexagon

Axial coordinates #

The (q, r) coordinate system used to address cells on the hex board. Each hex cell has two free axes; the third coordinate is implicit as s = -q - r. Piece movement can then be expressed as small vector offsets rather than as a tangle of special cases.

The system is standard for hex grids in games and graphics (see Red Blob Games' hex-grid reference). 3Chess uses it internally for board state, the AI search, and the move-pattern diagrams on the rules page.

Three colour classes #

On a hex board, cells partition into three colour classes given by the formula color(q, r) = (q − r) mod 3. The three classes replace the two-colour light/dark of square chess.

Consequences for piece movement: a bishop is locked to one of the three classes for the entire game (parallel to the "dark-squared bishop" idea in square chess, but with three classes). A knight always changes class on every move. The three-class structure shapes long-term piece reachability and adds depth to opening theory in 3Chess that has no analog in regular chess.

AI and rating

PL-Elo Exact (N = 3) #

The 3Chess rating system. A custom hybrid of the Plackett–Luce model (closed-form expected scores for 3-player placements) with σ tracking on Elo's scale: μ₀ = 1200, σ₀ = 350, σ_min = 50, β = 200, K_max = 48, K_min = 16. Ratings update only on placement (not on mid-game metrics), per Snellman's Rule.

A cascading checkmate counts as a tie for 2nd/3rd at 0.25 each. The displayed rating is R_display = max(0, μ − 2σ); a new player starts at 500. Provisional status (σ > 200 or games < 15) signals ratings are still being calibrated. Inactivity drifts σ upward (back toward provisional) so a long-absent player is treated as uncertain again on return.

Axiom system (3Chess) #

The formal set of rules from which the 3Chess game emerges. Twenty-four axioms (A1–A24) cover the board, the pieces, moves, capture, check and checkmate, elimination, territory scoring, and the draw-resolution conditions. Fifteen theorems (T1–T15) are proved from those axioms.

The axioms aren't documentation; they are the source-of-truth. Every later design decision — AI evaluation, the rating system, balance tuning, online protocol — has to be consistent with the axioms or the axioms are revised. The system has gone through two major revisions (v1 and v2.2); each revision is a deliberate event, not a patch.

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