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Hexagonal chess

Hexagonal chess is a family of chess variants played on boards composed of hexagons rather than squares, allowing pieces to move in six primary directions and introducing unique strategic elements such as altered pawn chains and bishop color-binding across three board colors.^[1] The most prominent variant, Glinski's hexagonal chess, was invented by Polish mathematician Władysław Gliński in 1936 and features a 91-hexagon board with three alternating colors, where each player starts with 18 pieces including three bishops and nine pawns (one extra bishop and one extra pawn compared to orthodox chess).^[1]^[2] In Glinski's version, pieces retain familiar roles but adapt to the hexagonal grid: the king moves one step in any of six directions, rooks slide along three orthogonal axes, bishops along three diagonal axes (each confined to one color), knights leap in an L-shape adjusted for hexes, and the queen combines rook and bishop moves, while pawns advance orthogonally toward the opponent's baseline but capture diagonally forward with an initial two-step option and en passant captures possible.^[1] Special rules include no castling due to the board's geometry, pawn promotion on the opponent's two opposite edges to a queen, rook, bishop, or knight, and stalemate awarding three-quarters of a point to the winner and one-quarter to the stalemated player rather than a draw.^[1]^[2] First commercialized in Britain in 1949, Glinski's hexagonal chess gained significant popularity in Eastern Europe, particularly Poland, Czechoslovakia, Hungary, and Russia, with estimates of up to 500,000 players by the late 20th century and media coverage in the 1970s.^[1]^[2] The International Hexagonal Chess Federation (IHCF), reorganized in 1996 and based in Hungary, organized world championships, including the inaugural event in Beijing in 1990, though the game's organized play declined after Gliński's death in 1990 before a partial revival.^[1] Other notable hexagonal variants include McCooey's Hexagonal Chess on a smaller 66-hex board and three-player adaptations, but Glinski's remains the benchmark for its balance of familiarity and innovation in tactics and opening theory.^[1]

Fundamentals

Board Geometry and Setup

Hexagonal chess variants are played on boards composed of hexagonal cells, which differ fundamentally from the square grids of traditional chess. In a hexagonal grid, each cell has six adjacent neighbors, allowing movement in six primary directions: three orthogonal (sharing a full edge) and three diagonal (sharing a vertex). This contrasts with the square grid's four orthogonal directions and four diagonal directions, creating more interconnected paths and altering strategic dynamics such as piece control and mobility.^[1] Common board sizes vary across variants but are typically shaped as regular or irregular hexagons, rhombi, or other forms to accommodate two or more players. For instance, Gliński's Hexagonal Chess uses a symmetric regular hexagon with 91 cells, arranged in 11 horizontal rows with lengths 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6 hexes, forming a regular hexagonal shape. Shafran's Hexagonal Chess employs a narrower irregular hexagon of 70 cells, derived from a size-6 hexagon with peripheral cells removed. De Vasa's Hexagonal Chess features a rhombus-shaped board of 81 cells, while Brusky's Hexagonal Chess utilizes an irregular hexagon of 84 cells. Smaller boards, such as the 37-cell hexagram in Starchess (a four-player variant), provide more compact play areas.^[1]^[3]^[4]^[4]^[5] The cells are colored in a repeating cycle of three colors—often light, dark, and medium—to reflect the grid's tripartite structure, unlike the bichromatic pattern of square chessboards. This three-color scheme ensures that bishops remain confined to cells of one color, with each player typically starting with three bishops, one per color, to balance coverage across the board.^[1] In standard two-player setups, such as Gliński's, each side deploys 18 pieces along three edges of the board: a king, a queen, two rooks, three bishops, two knights, and nine pawns. The pieces are arranged in a V-shaped formation across the starting ranks, with the king and queen positioned centrally to defend the bishops, and pawns forming a forward line of nine. The board is oriented vertically, with opposing players facing each other across parallel edges, facilitating pawn advances toward the opponent's promotion zone. Multi-player variants, like Starchess, may use radial orientations around a central area. In some setups, such as McCooey's variant, the pawn promotion zones are influenced by the board's edges to mirror traditional chess endgames.^[1]

Piece Movements and Captures

In hexagonal chess, piece movements are adapted to the six-sided geometry of the board, where directions are defined by orthogonal (along the hex edges, six possible) and diagonal (along the hex corners, six possible) paths. This results in greater overall mobility compared to traditional square chess, as pieces can access more lines of travel without the constraints of a four-directional grid. For instance, non-pawn pieces generally exhibit expanded reach, with knights potentially controlling up to 12 squares from a central position due to the L-shaped jump tailored to the 60-degree angles.^[1] The rook moves any number of unoccupied hexes in one of the six orthogonal directions, akin to sliding along straight lines that connect hex edges. This allows rooks to traverse rows, columns, or angled paths unimpeded until blocked by another piece.^[1] The bishop moves any number of unoccupied hexes in one of the six diagonal directions, following lines that connect the corners of adjacent hexes. In the three-color hexagonal board system, each bishop remains confined to hexes of a single color, eliminating the strict opposite-color binding seen in square chess where bishops operate on complementary black and white squares. Thus, with three bishops per side, one for each color, they collectively cover the entire board without the limitations of a bichromatic setup.^[1] The queen combines the rook's and bishop's movements, allowing it to slide any number of unoccupied hexes in any of the 12 possible directions (six orthogonal and six diagonal). This makes the queen the most versatile piece, capable of rapid traversal across the board's multifaceted geometry.^[1] The king moves one hex in any of the 12 directions—either orthogonally or diagonally—providing it with enhanced evasion options compared to its eight possible moves in square chess.^[1]^[6] The knight executes an L-shaped jump: two steps orthogonally in one direction followed by one step at a 60-degree angle (diagonally outward), or equivalently one orthogonal step followed by one diagonal outward step, leaping over intervening pieces without capturing them. From a central hex, this reaches up to 12 possible squares, significantly increasing its tactical influence over the square chess knight's maximum of eight.^[1]^[7] Captures occur universally by any piece landing on the occupied hex of an opponent's piece, replacing it and removing the captured piece from the board. Unlike square chess, castling maneuvers are omitted due to the board's geometry, while en passant captures are possible in variants like Gliński's.^[1]

Core Rules and Winning Conditions

Hexagonal chess variants are played by two players, designated White and Black, with White always moving first and players alternating turns thereafter.^[8] The objective is to checkmate the opponent's king, placing it under attack with no legal move to escape, or to win by resignation.^[6] Check occurs when the king is under direct attack by an opponent's piece, requiring the player to resolve the threat on their turn.^[8] Pawns promote upon reaching any of the 11 hexes defining the opponent's opposite borders, a configuration arising from the hexagonal board's geometry.^[8] Promotion options include queen, rook, bishop, or knight, chosen by the promoting player.^[6] In many variants, pawns lack an initial two-step advance option, though this rule varies; for instance, some allow a double step forward on the first move.^[7] Stalemate, where the player to move has no legal moves but their king is not in check, is typically treated as a draw in hexagonal chess, though variant-specific scoring exists—such as awarding ¾ point to the delivering player and ¼ to the stalemated player in Gliński's variant.^[9] Other draw conditions include threefold repetition of the same position with the same player to move, and a 50-move rule prohibiting claims after 50 consecutive moves without captures or pawn advances, adapted to account for longer paths on hexagonal boards.^[10] Moves in hexagonal chess are recorded using adapted algebraic notation, often employing letters a through i (skipping j to avoid confusion with 1) for files and numbers for ranks, aligned with the board's axial orientation.^[11] More advanced systems, such as axial (q-r coordinates) or cubic (x-y-z where x+y+z=0) coordinates, facilitate precise tracking of hexagonal positions in analysis or programming.^[12]

Historical Development

Early Concepts and 19th-Century Origins

The concept of chess on a hexagonal board emerged in the mid-19th century as part of broader explorations in recreational mathematics and board game design, where geometric variations were experimented with to enhance strategic depth. The earliest documented attempt is attributed to Thomas Hanmer Croughton, who developed a rudimentary hexagonal chess variant around 1853, though details of its rules remain sparse and it saw no commercial release.^[13] This was followed by the first published and commercialized version, Hexagonia, released in 1864 by the London-based game manufacturer John Jaques & Son. Hexagonia utilized a hexagonal board consisting of 127 cells arranged in a larger hexagon, with each player controlling a single king and a reduced set of pieces adapted from traditional chess, emphasizing simple movement patterns to fit the grid's six-directional geometry. The game aimed to introduce novelty through the board's radial symmetry but retained core chess-like objectives of capturing the opponent's king.^[14]^[4] These early efforts were heavily influenced by standard square-grid chess, with designers adapting piece movements—such as orthogonal slides for rooks and leaping patterns for knights—to the hexagonal lattice, which offered three primary directions instead of two, thereby increasing mobility and tactical complexity without fundamentally altering the game's confrontational nature. However, such adaptations often resulted in unbalanced play due to the grid's inherent asymmetries for two-player setups.^[15] Pioneering ideas for multi-player hexagonal chess appeared in the early 20th century, notably with Sigmund Wellisch's Three-Handed Hexagonal Chess, published in 1912 in the Austrian chess journal Wiener Schachzeitung. Wellisch's variant employed a 91-cell hexagonal board with radial symmetry for three players, each starting with an augmented army including an extra knight, and rules that allowed pawns to advance in two forward directions, with piece promotion to previously lost pieces and takeover of eliminated players' remaining pieces by the capturer, highlighting the hexagon's potential for multi-player engagement without formal alliances. This design addressed limitations of linear square boards.^[16]^[17] Despite these innovations, 19th- and early 20th-century hexagonal chess suffered from a lack of standardization, with primitive rules varying widely across unpublished manuscripts and limited commercial products, preventing widespread adoption and leading to obscurity until later refinements.^[13]

20th-Century Innovations and Key Inventors

The 20th-century advancements in hexagonal chess were marked by several key inventions that refined board designs, piece arrangements, and rules to better adapt traditional chess mechanics to hexagonal grids. In 1936, Polish mathematician Władysław Gliński developed what became the most influential variant, featuring a symmetric 91-cell hexagonal board that balanced piece mobility across three cell colors and incorporated three bishops per side to account for the geometry.^[6] Gliński's design was first launched commercially in Britain in 1949, sparking initial interest among enthusiasts.^[4] Independently, Soviet geologist Isaak Grigorevich Shafran invented his variant in 1939, utilizing a narrower 70-cell board and notably including castling rules adapted for hexagonal movement, which added strategic depth absent in some contemporaries.^[3] Shafran's rules were formally registered in 1956 and demonstrated at the 1960 Worldwide Chess Exhibition in Leipzig.^[4] The 1950s saw further innovation with Helge E. de Vasa's 1953 proposal, published in Joseph Boyer's 1954 book Nouveaux Jeux d'Echecs Non-orthodoxes, which introduced a rhombus-shaped 81-cell board with a linear starting array to streamline setup and promote fluid early-game play.^[17] By the mid-1960s, Yakov Brusky created his variant in 1966, employing an irregular 84-cell hexagon that adjusted pawn counts to ten per side while retaining Gliński's core movement principles, aiming for a more compact yet dynamic contest.^[18] Toward the decade's end, in 1978–1979, American mathematicians Dave McCooey and Richard Honeycutt devised another 91-cell symmetric board variant, emphasizing balanced pawn promotion and bishop coverage similar to Gliński's but with refined en passant rules.^[19] Hexagonal chess reached its zenith of organized play between 1976 and 1982, fueled by growing European interest. The inaugural Hexagonal Chess Congress convened in London in June 1976 at the Bloomsbury Centre Hotel, establishing the British Hexagonal Chess Federation and crowning David Springgay as the first British champion.^[4] This event led to annual British championships, a first European Championship in 1980, and national tournaments across countries like Poland, Hungary, and Czechoslovakia.^[13] International matches, such as the 1978 Poland versus Great Britain team contest broadcast on BBC and Australian television, highlighted the game's appeal.^[4] This period also saw the first World Hexagonal Chess Championship in Beijing in 1990-1991, shared by Hungary's László Rudolf and Poland's Marek Mackowiak.^[13] Gliński's variant drove much of this surge, attracting over 500,000 players in Poland alone, with more than 130,000 books sold there by the early 1980s.^[4]^[20] Despite this peak, hexagonal chess waned in the late 20th century due to the proliferation of competing variants without a unified standard, which fragmented communities and hindered widespread adoption. Additionally, the dominance of traditional square-grid chess, supported by established federations and media, overshadowed these innovations. After Gliński's death in 1990, international coordination lapsed, though the International Hexagonal Chess Federation (IHCF) was established in 1996 in Hungary, leading to a partial revival; however, organized play remained limited compared to its earlier peak.^[13]

Two-Player Variants

Gliński's Hexagonal Chess

Gliński's Hexagonal Chess is a prominent two-player variant of hexagonal chess, invented by Polish engineer Władysław Gliński in 1936 and first commercially launched in Britain in 1949 following his relocation after World War II.^[21]^[22] The game is played on a symmetric hexagonal board consisting of 91 cells arranged in 11 rows, with the cells colored in three alternating colors to accommodate the geometry.^[8] Each player deploys an army of 18 pieces: one king, one queen, two rooks, two knights, three bishops (one confined to each color), and nine pawns, mirroring the expanded setup necessitated by the board's size and shape.^[8] This variant became the benchmark for hexagonal chess due to its balanced design and widespread adoption, particularly in Eastern Europe. The board's hexagonal geometry alters piece movements compared to square chess, with directions expanding to six primary axes. Pawns advance forward orthogonally (along the three straight-line directions toward the opponent's side) one cell at a time, but may move two cells on their initial advance from the starting row; captures occur at 60-degree angles to the forward path, allowing diagonal-like attacks to adjacent cells.^[15] En passant captures are permitted under analogous conditions to square chess, enabling a pawn to capture an opponent's pawn that has just advanced two cells, skipping an attackable position.^[15] Bishops gain enhanced power in this setup, as each of the three per side operates exclusively on one of the board's three colors, providing comprehensive coverage without the colorbinding limitations of square chess bishops; knights, meanwhile, exhibit superior mobility due to their L-shaped path combining orthogonal and angled steps.^[15] Unlike traditional chess, there is no castling, emphasizing direct king safety through piece coordination.^[6] Unique scoring rules distinguish tournament play: stalemate is not a draw but awards ¾ of a point to the player delivering it and ¼ to the stalemated player, incentivizing aggressive endgames over defensive stagnation.^[9] The game concludes with checkmate or resignation, with pawns promoting to any non-pawn piece (except king) upon reaching either of the opponent's two rear edges. At its peak, Gliński's variant attracted over 500,000 players, predominantly in Eastern Europe, including strong followings in Poland, Czechoslovakia, Hungary, and Russia, where it dominated club and competitive scenes during the 1970s and 1980s, featuring national championships in Poland.^[22]^[23] This era marked its historical impact as the most influential hexagonal chess form, fostering studies on piece dynamics and endgames adapted to the board's expanded tactical possibilities.

Shafran's Hexagonal Chess

Shafran's Hexagonal Chess is a two-player variant played on a compact board consisting of 70 hexagonal cells arranged in a rhombus-shaped layout with 9 files (labeled a to i) and varying ranks from 6 to 10 cells in length.^[3] The game was invented in 1939 by Soviet geologist Isaak Grigorevich Shafran and officially registered in the USSR in 1956, with a demonstration at the Worldwide Chess Exhibition in Leipzig in 1960.^[4] Each player deploys a standard set of pieces adapted for the hexagonal grid: one king, one queen, two rooks, two knights, three bishops (to account for the three board colors), and six pawns, positioned on the first two ranks with the back rank featuring the major pieces from right to left as rook, knight, bishop, queen, king, bishop, knight, rook, bishop.^[24]^[25] Piece movements follow hexagonal adaptations similar to other variants, with rooks moving any number of cells orthogonally in six directions, bishops along the six diagonal directions, queens combining both, knights in an L-shape of two orthogonal steps followed by one perpendicular, and kings one cell in any direction.^[26] Pawns advance toward the opponent's baseline, capturing diagonally forward in bishop directions; on their initial move, central pawns (d-, e-, and f-files) may advance up to three cells, flank pawns (b-, c-, g-, and h-files) up to two, and edge pawns (a- and i-files) one, with en passant possible against multi-step advances and promotion occurring on any of the opponent's back-rank cells to queen, rook, bishop, or knight.^[26] The core winning condition is checkmate, with stalemate resulting in a draw.^[3] A distinctive feature is the inclusion of castling, permitted under standard conditions (neither piece moved, no pieces between, king not in check), available in short and long forms on both queenside and bishopside flanks. In short castling, the king shifts two cells toward the rook while the rook jumps three cells to the king's opposite side; in long castling, the king moves three cells and the rook two.^[26] This mechanic, absent in many hexagonal variants, enhances king safety by allowing rapid sheltering amid the board's constrained geometry. The tighter 70-cell board compared to larger hexagonal setups promotes quicker development and engagements, often leading to faster-paced games where central control and early castling prove crucial for defensive stability.^[3]

De Vasa's Hexagonal Chess

De Vasa's hexagonal chess is a two-player variant invented in 1953 by Helge E. de Vasa.^[27] The game is played on a rhombus-shaped board consisting of 81 hexagonal cells, arranged in a diamond configuration with 9 cells per side.^[4] The initial setup features the standard chess pieces—two rooks, two knights, two bishops, a queen, and a king—placed on the back rank for each player, with 8 pawns positioned on the third rank, totaling 16 pieces per side.^[28] Pawn movement in De Vasa's variant emphasizes straightforward advancement, with pawns advancing one cell at a time to an adjacent forward position, either left or right in the direction toward the opponent's side, but may move two cells on their initial move; captures occur diagonally forward only, aligning with the bishop's diagonal paths on the hexagonal grid, which promotes aggressive forward play by limiting defensive maneuvering for pawns.^[28] This design fosters a gameplay style centered on rapid central control and pawn exchanges, as the double-step option accelerates pawn confrontations compared to variants without flexible initial advances.^[17] Castling is permitted under standard conditions, with the king and rook moving two cells for kingside castling or three cells for queenside, adapting the orthogonal mechanics to the hexagonal layout while ensuring the pieces remain unmoved prior to the maneuver.^[4] Non-pawn pieces follow movements analogous to their orthogonal counterparts translated to the hex grid, such as rooks along straight lines and bishops along diagonals, with the two bishops per side balanced across the board's color classes to allow equitable access to all playable cells.^[29] Promotion occurs when a pawn reaches the opponent's back rank, typically to a queen or other piece, maintaining the variant's focus on dynamic endgames driven by pawn structure.^[27]

Brusky's Hexagonal Chess

Brusky's hexagonal chess is a two-player variant invented by Yakov Brusky in 1966.^[30] It is played on an irregular hexagonal board consisting of 84 cells arranged in a pattern with sides of lengths 9, 5, and 4, featuring horizontal ranks similar to those in Gliński's hexagonal chess.^[4] The initial setup places 10 pawns per side in the second rank behind the pieces, one more than in Gliński's variant, using a linear arrangement inspired by Helge E. de Vasa's first-rank array.^[4] This design emerged during the 1960s wave of hexagonal chess innovations that built upon earlier 20th-century concepts.^[30] Piece movements follow the conventions of Gliński's hexagonal chess, with the rook moving orthogonally, the bishop diagonally, the queen combining both, and the knight in its characteristic L-shape adapted to the hexagonal grid.^[30] Pawns advance one cell forward along rook-lines (with an initial two-step option) or capture one step obliquely along bishop-lines; unmoved pawns may also capture straight ahead, and en passant captures are permitted.^[4] A key unique rule is the pawn blocking mechanism: if an opponent's pawn obstructs one of a pawn's two possible forward paths, the blocked pawn cannot move at all, even along the unblocked path, emphasizing adjacent file interactions in the hexagonal setup.^[30] Castling is allowed under standard conditions, with the king moving two cells kingside or three cells queenside toward the respective rook.^[4] Promotion occurs when pawns reach the opponent's back rank, typically to queen, rook, bishop, or knight. Strategically, the additional pawn per side fosters denser pawn chains and stronger defensive walls, as the extra units enhance blocking potential across the board's irregular geometry.^[4] This setup promotes robust early-game pawn structures that can lock central files, shifting focus to flank maneuvers and knight activity. In the endgame, the variant encourages dynamic pawn promotion races, where breakthroughs in pawn lines can rapidly alter material balance due to the interconnected hexagonal paths.^[30]

McCooey's Hexagonal Chess

McCooey's Hexagonal Chess is a two-player variant played on a board composed of 91 hexagonal cells arranged in a larger hexagon with six cells along each side, similar to the board used in Gliński's variant.^[31] Developed by Dave McCooey and Richard Honeycutt between 1978 and 1979, it aims to adapt orthodox chess rules to a hexagonal grid while preserving strategic depth and balance.^[31]^[32] The game employs standard chess pieces—king, queen, two rooks, two knights, three bishops, and seven pawns per side—arranged with side-to-side symmetry and no unoccupied spaces immediately behind the pawns, ensuring all pieces start 33 cells from the opponent's forces.^[31] Pawn movement in McCooey's variant introduces refinements for hexagonal geometry: pawns advance one cell straight forward toward the opponent's baseline or, on their first move, two cells except for the central pawn, which is restricted to a single step to maintain balance.^[31] Captures occur diagonally forward along the three possible bishop-like directions, functioning as short-range bishop moves limited to adjacent cells, which adds tactical complexity compared to orthogonal captures.^[31] En passant is permitted under conditions analogous to orthodox chess following a two-cell advance, and pawns promote upon reaching any of the 11 cells on the opponent's two opposite edges to a queen, rook, bishop, or knight.^[31] Other pieces move as in standard chess adapted to hexagons: rooks along straight lines, bishops along diagonals (with three color classes on the tricolored board), knights in an L-shape jumping over pieces, and the queen combining rook and bishop paths; no castling is allowed.^[31] A key draw condition is stalemate, which results in a half-point draw for both players, aligning with orthodox chess conventions unlike some other hexagonal variants.^[31] Relative to Gliński's Hexagonal Chess, McCooey's uses fewer pawns (seven instead of nine) to promote a more open board and fluid gameplay, while the diagonal pawn captures introduce greater mobility and strategic nuance.^[31] These adjustments, detailed in the variant's ruleset on ChessVariants.com, emphasize equitable promotion distances for all pawns at seven cells from the baseline.^[31]

Starchess

Starchess is a compact variant of hexagonal chess invented by Hungarian chess pedagogue László Polgár around 2004.^[33] The game is played on a star-shaped board composed of 37 hexagonal cells, forming a hexagram that emphasizes central control and rapid piece interactions.^[5] This design, patented by Polgár, uses uniform hexagonal cells without color differentiation, promoting a focus on geometric movement patterns inherent to hexagonal grids.^[34] The rules adapt standard chess mechanics to the hexagonal environment while introducing unique elements for variability and speed. Each player begins with one king, one queen, one rook, one bishop, one knight, and five pawns, but unlike traditional chess, there is no fixed starting position for the non-pawn pieces. Instead, players alternate turns placing these pieces on their back rank behind the pawns, resulting in up to 14,400 possible initial configurations and encouraging strategic experimentation from the outset.^[35] Piece movements follow hexagonal adaptations: the rook moves along vertical lines, the bishop diagonally, the queen in any direction, the knight in its characteristic L-shaped jump adjusted for six directions, and the king to adjacent cells. Pawns advance forward (with an optional initial double step), capture diagonally, and promote upon reaching any cell on the opponent's edge to a queen, rook, bishop, or knight; notably, there is no castling or en passant to streamline play.^[35] The objective remains checkmate, with draws possible via stalemate, repetition, the 50-move rule, or agreement.^[35] Due to its small board size, Starchess facilitates faster games that prioritize tactical skirmishes over prolonged openings or complex endgames, often resolving in under 30 moves.^[36] This emphasis on immediate combat appeals to variant enthusiasts seeking dynamic, educational play, as intended by Polgár for developing tactical acumen.^[33] The game has been promoted through Polgár's official website and international tournaments, including annual world championships and events at festivals like the Global Chess Festival, fostering a niche but dedicated community.^[37]^[38]

Multi-Player Variants

Three-Player Variants

Three-player variants of hexagonal chess extend the game's geometry to support radial play among three participants, typically positioning each player's forces in a corner of the board to converge toward a shared central area. These adaptations emphasize rotational symmetry for fairness, with turns proceeding counterclockwise and rules adjusted to handle multiple opponents, such as modified promotion and elimination mechanics. The pioneering example is Siegmund Wellisch's Three-Handed Hexagonal Chess, introduced in 1912. Played on a 91-cell hexagonal board, each player's setup occupies one of three corners, featuring a back rank of knight-rook-queen-king-rook-knight, a middle rank of pawn-pawn-pawn-knight-pawn-pawn-pawn, and two additional pawns on the forward rank. To suit the board's three-color scheme, each side includes an extra knight but omits bishops.^[16] Piece movements follow hexagonal logic: the king steps to any adjacent cell (up to six directions), the rook slides along straight lines, the knight jumps to the nearest cell of the same color in any direction, and the queen combines rook and knight paths. Pawns advance one cell orthogonally forward in either of two directions toward the board's center, capturing orthogonally forward in those directions (same as non-capturing move), without an initial double step or en passant. Castling is possible by the king exchanging places with a rook.^[39]^[16] Players alternate turns in counterclockwise order, with the goal of capturing both opponents' kings through checkmate. Upon capturing a king, the victor assumes control of the defeated player's remaining pieces, including pawns which retain their original forward direction to prevent immediate reversal advantages. Pawns promote upon reaching any end row—effectively an opponent's back rank—to a type of piece previously lost from the promoting player's army. This elimination and takeover mechanic supports continued play until one player remains, addressing the logistical challenges of three-way conflict without formal alliances.^[16] The variant's radial design promotes central control as a key strategic element, though it poses balance challenges, as the player dominating the center may gain disproportionate power; rotational symmetry helps equalize starting positions. Modern three-player adaptations often scale to larger boards of 127 cells or more, providing 126–150 cells total to reduce congestion while preserving corner setups, counterclockwise sequencing, and flexible promotion rules. These expansions enhance maneuverability without castling, focusing on alliance-like dynamics through temporary piece inheritance in eliminations.^[15]

Four-or-More Player Variants

Variants of hexagonal chess for four or more players typically employ expanded hexagonal boards to accommodate multiple participants, often divided into sectors or arranged radially around a central area. These boards are larger than those used in two-player games, such as a 9-hexagon-per-side board comprising 217 cells for six players, allowing space for individual armies without immediate overcrowding.^[40] The setup emphasizes strategic depth through alliances and betrayals, with pieces starting in designated corners or sectors to prevent early clashes.^[41] A prominent example is Chexs, designed by Stephen P. Kennedy, which supports 2 to 6 players on a modular hexagonal board—using a 6-sided version (91 cells) for fewer players and a 9-sided one (217 cells) for four or more.^[41] In this variant, each player commands a standard chess army of 16 pieces, including one king, queen, two rooks, two knights, two bishops, and eight pawns, placed in opposing corners.^[41] Turns proceed sequentially around the board, with players moving one piece per turn; checkmate eliminates a player, transferring control of their pieces to the victor, enabling dynamic team formations or free-for-all competition.^[41] Victory occurs when one player controls all piece sets, often through diplomacy where temporary alliances can form via piece control, though such pacts are non-binding and prone to betrayal.^[41] Pawns promote in shared zones on the opposite side of the board, adding tension to central confrontations, and piece movements adapt standard chess rules to the hexagonal grid, with knights limited to specific leaps and bishops confined to their color.^[41] Another key variant is Echexs, developed by Jean-Louis Cazaux in 1998, tailored specifically for six players on a 217-cell hexagonal board divided into six radial sectors.^[40] Each player deploys 16 pieces in a corner sector, with the king positioned to the right of the queen, and armies facing outward in a one-against-all format.^[40] Play proceeds clockwise, with each participant moving one piece per turn according to Glinski's hexagonal rules: rooks and queens move any distance along straight lines, bishops stay on their color, knights jump in L-shapes adapted to six directions, and pawns advance forward with diagonal captures, including initial double steps and en passant.^[40] Checkmate removes a player from the game, granting their pieces to the mating player, who then controls them without taking an extra turn; the game ends with the last standing player as winner, or by mutual agreement among survivors.^[40] Promotion occurs after crossing 17 cells to the opposite sector, typically to queen or other major pieces, fostering aggressive mid-game expansions.^[40] This design builds on earlier multi-player concepts from Kennedy's Chexs and Dave McCooey's hexagonal adaptations, prioritizing balance in radial symmetry.^[40] These four-or-more-player variants emerged in the late 20th century, with Chexs originating in the 1990s and Echexs formalizing six-player play in 1998, though they remain less standardized than two-player forms due to the challenges of coordinating multiple armies and variable alliances.^[41]^[40] Unlike simpler multi-player setups, they incorporate increased piece counts per player—up to 16 standard units plus captured additions—and shared central zones for promotion, heightening diplomatic elements where players may form temporary teams to eliminate threats before turning on each other.^[41] Such mechanics distinguish them from two-player games by emphasizing survival and opportunistic control over direct confrontation.^[40]

Special Features and Studies

Pawn Movement Variations

In hexagonal chess variants, pawns generally advance orthogonally along one or more forward directions and capture along adjacent 60-degree diagonals, adapting the standard chess pawn mechanics to the board's geometry.^[42] This base allows for controlled progression toward promotion while enabling lateral threats, though specifics vary to balance board shape and piece power.^[43] In Gliński's Hexagonal Chess, pawns move passively straight forward orthogonally one step, with an optional initial two-step advance along the same path.^[42] They capture to the 60-degree right or left of forward, and en passant captures are permitted when an opponent advances two steps past an adjacent pawn.^[42] Promotion occurs upon reaching any of the eleven hexes on the opponent's border row.^[42] Shafran's Hexagonal Chess expands initial mobility, permitting a pawn's first move to advance up to two steps along any of three possible forward directions, reaching one of six potential hexes.^[43] Subsequent non-capturing moves are limited to one step along those directions, while captures occur diagonally forward to two possible squares.^[43] En passant is allowed, with the capturing pawn moving behind the advanced pawn to its original square.^[43] Pawns promote upon reaching squares originally occupied by the opponent's major pieces or the final rank of their file.^[43] De Vasa's Hexagonal Chess modifies the base by providing two forward orthogonal movement directions for pawns, with an initial two-step option, and restricts captures to forward diagonals.^[30] This linear startup and dual-forward paths aim to preserve orthodox pawn dynamics on the rhombus-shaped board, with no en passant rule specified.^[44] Promotion follows standard variant conventions upon reaching the opponent's rear ranks.^[30] Brusky's Hexagonal Chess employs ten pawns per side to fill the board's files, with each advancing one or two steps forward along its file to unoccupied hexes, though no initial double-step from the starting position to avoid en passant complications.^[45] Captures target pieces on the two diagonally forward hexes, akin to a bishop's shortest move.^[45] No en passant exists, and promotion is optional on the opponent's back three ranks or mandatory on the final rank, limited to recaptured lost pieces.^[45] The extra pawns introduce file-blocking potential, where central files may congest early.^[30] McCooey's Hexagonal Chess alters captures to bishop-like advances along diagonals, with non-capturing moves limited to one step along either forward file, or optionally two steps initially if starting from the home ranks.^[46] Edge pawns lack the two-step option to prevent board asymmetry.^[46] En passant is permitted, and promotion is optional on the second or third rear ranks (to captured pieces only) but required on the final rank.^[46] In multi-player variants like Trihex, pawns retain orthogonal forward movement and diagonal captures with en passant, but promotion occurs along radial paths defined by the two farthest board sides from a player's starting corner, accommodating the expanded geometry.^[47] This allows promotion to enhanced pieces such as cardinals or marshalls, reflecting the need for versatile endpoints in three-player setups.^[47]

Endgame Analysis and Studies

Endgame analysis in hexagonal chess reveals significant deviations from square chess principles due to the board's geometry, which grants pieces greater mobility and alters fundamental concepts like opposition and promotion dynamics. In variants such as McCooey's Hexagonal Chess, the king's ability to move to any of six adjacent hexes complicates traditional opposition, requiring players to consider multiple approach vectors rather than the linear duels of square chess; this often results in more intricate maneuvering to gain tempo or restrict the enemy king. Pawn promotion races are uniquely challenging on multiple fronts, as pawns in Brusky's Hexagonal Chess advance along files, allowing chains to form linearly and enabling aggressive breakthroughs where a pawn captures diagonally to shatter an opponent's structure and accelerate promotion.^[48]^[49] Computer-generated endgame databases, particularly for McCooey's variant on a 91-hex board, highlight strategic differences including higher draw rates from enhanced piece mobility, which facilitates fortress positions and perpetual checks more readily than in orthodox chess. For instance, in king + Amazon vs. king + Amazon endgames, fortress draws occur in 21.18% of positions, underscoring how the hexagonal layout allows defensive setups that are harder to breach. Queen vs. lone king endgames resolve more swiftly, with the queen's extended range across the interconnected board enabling checkmate in far fewer moves compared to square chess equivalents, with the longest win in 7 moves (14 half-moves) in optimal play. In contrast, some combinations like king + bishop vs. king are pure draws 100% of the time due to the bishop's colorbound nature on the three-color hex board, preventing any forcing sequence.^[50] Studies in these variants emphasize pawn interactions and piece synergies unique to hex geometry. In Brusky's Hexagonal Chess, pawn chain breakthroughs involve leveraging the file advances to create passed pawns on adjacent files, where a coordinated advance can overrun a defender stretched across the wider board. McCooey's analyses detail bishop-pawn interactions, where the three bishops (one per color) can support pawn advances more effectively than in square chess, but also risk overextension if the pawn structure fragments under hex-specific captures. Composed problems often showcase hex-exclusive zugzwang scenarios; for example, in a knight-lance endgame from McCooey's database, white achieves checkmate in 164 half-moves from the position with white king on a6, knight on e8, lance on d8, and black king on e6, forcing black into a zugzwang where any move exposes the king to capture due to the knight's oblique leaps across hex lines. These studies confirm elevated draw tendencies overall, with about 10-20% of two-piece endgames resulting in non-win outcomes across analyzed tables.^[50]

Other Hexagonal Chess Variants

Hexagonia, published in 1864 by John Jaques & Son, represents the earliest commercial hexagonal chess variant, featuring a 127-cell hexagonal board with pieces adapted from standard chess including kings, cannons, and infantry that move along the six primary directions.^[51]^[13]^[14] Among modern niche variants, C'escacs, developed in 2007, expands on Glinski's hexagonal chess with a larger 169-hexagon board and introduces unique pieces such as a dragon (combining queen and knight movements), pegasi (queen and bishop compounds), and almogavars (enhanced pawns), while pawns advance orthogonally in three forward directions with initial double- or triple-step options.^[52]^[53]^[4] Similarly, 88 Hexagons Chess, introduced in 2020, uses an 88-hexagon board where pawns can advance up to three hexes initially and castling is permitted, aiming for a more compact two-player experience.^[54] Unique adaptations include single-player hexagonal chess puzzles, which challenge solvers to achieve checkmate or specific configurations against an imagined opponent, as exemplified by tactical problems involving forks and decoys on hexagonal grids.^[55] Computer-only variants, supported by dedicated engines like Hexodus, enable automated play and analysis exclusively through software, facilitating experimentation without physical boards.^[56] In the 2020s, app implementations have increased accessibility, such as HexChess One on Google Play, which supports ranked online matches in Glinski's variant, and Hexagonal - Chess Variants, offering multiple hexagonal rulesets for mobile play.^[57]^[58] Community discussions in 2024 have explored extensions like 6-player rotational variants on expanded hexagonal boards, though these remain informal.^[59] Organized hexagonal chess tournaments have been limited since the 1980s, with the First European Hexagonal Chess Championship in 1980 marking a peak in competitive interest, after which activity shifted toward online and casual formats.^[60] Recent research highlights potential for AI analysis, including Monte Carlo Tree Search combined with reinforcement learning to develop efficient hexagonal chess programs capable of strong play.^[61]^[62]

References

[1]
Glinski's Hexagonal Chess
### Summary of Glinski's Hexagonal Chess
[2]
[PDF] Checkers variants • A new perspective on Octagons
First Theories of Hexagonal Chess. Wladyslaw Glinski, Hexagonal Chess Publications, London, 1974. I have a soft spot for Glinski's Hexagonal Chess. It was one ...
[3]
Ludii Portal
### Summary of Shafran Chess
[4]
Hexagonal chess - chess variant
Gliński's hexagonal chess, invented by Władysław Gliński in 1936 and first launched in Britain in 1949, is probably the most widely played of the hexagonal ...Missing: Glinski's | Show results with:Glinski's
[5]
Starchess - Ludii Portal
Starchess is played on a star-shaped hexagonal board that can be thought of as a hexagon-shaped board with each side having length 6, but with four spaces ...Missing: cells | Show results with:cells
[6]
https://ludii.games/details.php?keyword=Glinski%20Chess
[7]
What are the rules of Glinski's Hexagonal Chess?
Nov 12, 2022 · What are the rules of Glinski's Hexagonal Chess? Ask Question. Asked 2 years, 11 months ago. Modified 2 years, 11 months ago. Viewed 681 times.Opening Theory in Hexagonal Chess? - Chess Stack ExchangeHow should the minor pieces move in hexagonal chess?More results from chess.stackexchange.com
[8]
Gliński's Chess - Green Chess
Other rules. Otherwise the rules of chess apply, with the following difference ... External links. Wikipedia: Glinski's hexagonal chess · ChessVariants.com ...
[9]
Board and Pieces - Hexagonal Chess - Google Sites
Gliński's Hexagonal Chess, which is the primary version discussed here, is played on a hexagonal board composed of 91 cells of three different colors, as shown ...
[10]
Rules - Hex Chess
The player who reaches the center of the board with their king (the f6 hexagon, marked on the board) wins the game. You still have to play legally according to ...Missing: conditions | Show results with:conditions
[11]
Why does a hexagonal chess board skip the letter J?
Jan 3, 2024 · For every Glinski's Hexagonal Chess game I see online the algebraic notation skips J for IK L. Why? What's wrong with J?
[12]
[PDF] hexagonal chess
May 19, 1975 · This means that any complete row of cells running in any direction on the Hexagonal Board represents the move of either a Rook or a Bishop.
[13]
Issue24 - THE NEW ABSTRACT GAMES
Subsequently, John Jaques of London, mentioned above, published Hexagonia in 1864, which was the first commercial version of chess on hexagons.
[14]
Hexes Chess - Chess on Hexagons
The hexagonal idea has been around since the mid-nineteenth century. John Jaques and Son developed the first hexagonal chess variant in 1864 calling it ...
[15]
A Discussion of Hexagonal Chess
Although the layout of the board is very similar to that of Gliński's variant, to be described next, he chose a different arrangement of colors for the squares.
[16]
[PDF] Chapter 37, Games for three - JSB
37.2 Rotationally symmetric boards based on hexagons. Three-Handed Hexagonal Chess [Wellisch]. (Sigmund Wellisch, 1912). Wellisch believed, with reason, that ...Missing: Wellisch's | Show results with:Wellisch's
[17]
Hexagonal Chess II
A version of Hexagonal Chess was devised in 1953 by Helge E. De Vasa which used these moves. Its board and initial array are shown below.<|control11|><|separator|>
[18]
https://ludii.games/details.php?keyword=Brusky%20Chess
[19]
https://ludii.games/details.php?keyword=McCooey%20Chess
[20]
Hexagonal Chess - Kruzno
This game is based on the rules developed by Wladyslaw Glinksi in 1936. Glinski's Chess is one of the most popular chess variants with more than a half million ...Missing: Piotr invention 1949
[21]
https://www.boardgamegeek.com/boardgame/19432/hexagonal-chess
[22]
About Hexes Chess
rules · 3-Pawn rules · 4-Pawn rules · 6-Pawn rules · 9-Pawn rules · play · chess boards ... Right: Glinski's hexagonal chess, launched in 1949, became popular in ...
[23]
C'escacs: Rules
There are twenty four total possible moves, but only the twelve simple knight moves allow a capture. The color of the destination hex always changes with ...
[24]
Hexagonal Chess Board (70 Hex Cells) - The Game Crafter
In stockOct 2, 2021 · A Rook can move any number of spaces in a straight line from any adjacent space. The bishop can move diagonally along the apex lines any number ...Missing: geometry | Show results with:geometry
[25]
Hexagonal chess | BoardGameGeek
Glinski's hexagonal chess popularized from 1936 the hexagonal chess pieces ... Pieces moves have its own rules that difer from the common hexagonal chess piece ...
[26]
Shafran's Chess - Green Chess
Hexagonal chess variant played on an irregular hexagon.Missing: geometry | Show results with:geometry
[27]
Ludii Portal
### Summary of De Vasa's Hexagonal Chess
[28]
Chess Variants
Jan 13, 2018 · In these variants, the same pieces and rules as in chess are used, but the board is different; It can be smaller or larger, the shape of either ...
[29]
De Vasa's Chess - Green Chess
Chess variant played on a diamond-shaped board with hexagonal fields. Movement of the pieces More details about non-pawn piece movement.Missing: size | Show results with:size
[30]
[PDF] Variant Chess - Mayhematics
In this issue: Hexagonal Chess,. Solutions, Semi-Pieces, Variants. Volume 1 complete (issues 1-8,. Modern Courier Chess, Escalation,.
[31]
[PDF] THE CLASSIFIED ENCYCLOPEDIA OF CHESS VARIANTS - JSB
Reflect that chess is but an imperfect variant of a game that was itself a variant of a germinal game whose origins lie somewhere in the darkness of time. Page ...<|separator|>
[32]
Hexagonal Chess - McCooey's variant
### Summary of McCooey's Hexagonal Chess
[33]
McCooey's Chess - Green Chess
One of the most known hexagonal chess variants, played on a regular hexagonal board. Each side has three bishops and seven pawns.Missing: John | Show results with:John
[34]
Superstar Chess | Board Game - BoardGameGeek
Rating 6.9/10 (5) Superstar Chess was invented by László Polgár, Hungarian chess master coach. It is a hexagonal chess variant, created mainly for educational purposes.
[35]
US20070063436A1 - Logical board game and game of chance on a ...
The cells on the Polgár Szupersztár® six-pointed star-shaped board are hexagonal. The cells are of a single colour, and there are two varieties of the board.
[36]
[PDF] Polgar Superstar Chess (Starchess)
Invented by László Polgár, father of the famous Polgár sisters, Starchess is played on a star-shaped board with rules which are very similar to those from ...
[37]
Starchess, a game by Laszlo Polgar - Chess Forums
Jan 30, 2023 · Laszlo Polgar holds yearly world championship matches and has written dozens, if not hundreds, of starchess books on tactics, endgames and ...The *other* Laszlo Polgar booksGM/IM/FM/NM games from László Polgár's 5334 bookMore results from www.chess.com
[38]
https://globalchessfestival.com/index.php/en/about-us/programs-2024?view=article&id=242:polgar-star-chess&catid=13
[39]
POLGAR STARCHESS - Global Chess Festival
Sep 17, 2025 · Invented by Laszlo Polgar this game is fast, dynamic and highly enjoyable. Amongst the large variety of board games, Polgar StarChess is one of the fastest and ...
[40]
http://www.chessvariants.com/hexagonal.dir/echexs.html
[41]
http://www.chessvariants.com/hexagonal.dir/chexs.html
[42]
http://www.chessvariants.com/hexagonal.dir/hexagonal.html
[43]
http://www.chessvariants.com/hexagonal.dir/hexch.html
[44]
https://gambiter.com/chess/variants/List_chess_variants.html
[45]
List of chess variants
De Vasa's hexagonal chess: Chess on a rhombus-shaped board of 81 hex cells. Same as Gliński's hex chess, but linear startup, two forward move directions for ...<|control11|><|separator|>
[46]
http://www.chessvariants.com/hexagonal.dir/a9hexachess.html
[47]
http://www.chessvariants.com/hexagonal.dir/trihex.html
[48]
https://greenchess.net/rules.php?v=brusky
[49]
Brusky's Chess - Green Chess
Hexagonal chess variant played on an irregular hexagon. Movement of the pieces More details about non-pawn piece movement.
[50]
http://www.chessvariants.com/hexagonal.dir/hexendgame.html
[51]
https://boardgamegeek.com/boardgame/153683/hexagonia
[52]
Hexagonia | Board Game | BoardGameGeek
Hexagonia is played on a hex-shaped board comprised of 127 smaller hexagons, seven on each side. The hexes are alternately colored red, white, and blue.
[53]
C'escacs hexagonal chess: C'escacs
Glinski's grand chess variant. · Setup · Rules · Hexagonal moves · Pieces and piece moves · Pawn moves · Castle · Variant: C'escacs-L.
[54]
C'escacs hexagonal chess | Board Game - BoardGameGeek
Rating 7.5/10 (2) Pawns move orthogonally on any of the three forward directions, and capture diagonally. Pawns have a 2 or 3 step initial move option, so en passant captures ...
[55]
88 Hexagons Chess - new hexagonal chess variant - Chess Forums
Sep 2, 2020 · On hexagonal boards, the "diagonals" would be those lines that maintain the same color as the beginning square which the piece moves from.Missing: Starchess | Show results with:Starchess
[56]
Hexagonal Chess Puzzle - Black to play and win
May 8, 2023 · Black can decoy the white queen to g7 by playing the fork Bi6-g7+. After the queen takes the bishop Ne5+ uncovers an attack on the white queen from the black ...
[57]
Where can I get a free engine to play Hexagonal Chess?
Dec 15, 2022 · This page on the chess-variants website has links to various hexagonal chess playing engines. I particularly like Hexodus which you can download from this link.Recent research, or analysis sites, about some chess variants?What are the rules of Glinski's Hexagonal Chess?More results from chess.stackexchange.com
[58]
HexChess One - Apps on Google Play
Apr 23, 2024 · HexChess One is a multi-platform client for hexagonal chess that allows you to compete against other players in ranked matches to climb the leaderboard.
[59]
Hexagonal - Chess Variants - Apps on Google Play
Rating 2.9 (62) · Free · AndroidHexagonal chess refers to a group of chess variants played on boards composed of hexagon cells. The best known is Gliński's variant.Missing: 2020s | Show results with:2020s
[60]
New chess variant: 6 player chess - Chess Forums
Apr 15, 2025 · How it would work is a normal chess game, but instead 1 person makes 5 moves, and then the next person makes 5 moves, etc. on both sides.Hexagonal Chess - Chess Forums - Page 288 Hexagons Chess - new hexagonal chess variant - Chess ForumsMore results from www.chess.comMissing: hexagonal | Show results with:hexagonal
[61]
Aug. 08, 1980 - First European Hexagonal Chess Championship ...
Aug 8, 1980 · Players from Britainm Poland m Hungary and Austria will be taking part at the Polish Cultural Institute in London.Missing: 1970s | Show results with:1970s
[62]
A Study of the Hexagonal Chess Game Program Based on the ...
Aug 20, 2024 · This article deeply discusses how to use the Monte Carlo Tree Search (MCTS) algorithm combined with reinforcement learning technology to develop an efficient ...
[63]
Research on Hexchess Game System based Artificial Intelligence
Sep 11, 2022 · Hex chess became popular because of its simple rules, but it also brought complex algorithms. Although the simple Monte Carlo tree search can be ...<|control11|><|separator|>