Three-dimensional chess (or 3D chess) refers to a family of chess variants that replaces the two-dimensional board with a three-dimensional array of cells, usually stacked chessboards. There are numerous 3D chess board arrays depending on the design of the specific game. Unlike standard chess, where pieces move solely on a single plane, 3D chess requires players to navigate across multiple levels, introducing a vertical z-axis to the game.
The expression "three-dimensional chess" is sometimes used as a colloquial metaphor to describe complex, dynamic systems with many competing entities and interests, including politics, diplomacy and warfare. To describe an individual as "playing three-dimensional chess" implies a higher-order understanding and mastery of the system beyond the comprehension of their peers or ordinary observers, who are implied to be playing "regular chess".e.g.
The basic rules of most 3D chess variants generally preserve the movement logic of traditional pieces while extending their range into the third dimension. For example, in many variants, a rook can move vertically through the levels as if traveling along a column. Capturing and checkmate remain the primary objectives, but the number of available squares and the complexity of the "lines of sight" make the 3D game significantly more difficult than two-dimensional versions. Some variants also introduce new pieces that take advantage of the three-dimensional play space.
Three-dimensional variants have existed since at least the late 19th century. One of the oldest and most enduring variants is Raumschach (German for "Space chess"), invented in 1907 by Ferdinand Maack, which uses five stacked 5×5 boards (though there were different board spaces designed). Maack's game is considered by chess variant enthusiasts to be the classic form of 3D chess. The game was played in clubs in Hamburg and London during the early 20th century. Numerous other variants of 3D chess have been invented since. David Pritchard's The Classified Encyclopedia of Chess Variants discusses some fifty 3D chess variations as well as a handful of higher-dimensional variants in Chapter 25. Chapter 11 also covers variants using multiple boards normally set side by side, which can also be considered to add an extra dimension to chess.
The modern cultural recognition of 3D chess is linked to the Star Trek franchise, which featured a "Tri-Dimensional Chess" set as a prop.Smith, L. Lynn. "The History of 3D Chess Part Two: Kogbetliantz and Weaver", Abstract Games — Issue 11 Autumn 2002 The show provided no concrete rules, but fans and designers later codified various rule sets to make it playable. Subsequently, various Science fiction media have also featured some form of 3D chess as a prop or plot device, usually to indicate the intelligence of the players. The rising popularity of the concept also led to various commercial products using 8×8×3 layout (three stacked boards), including a 3D Chess game published in 1967 by Dimensional Games, Inc, as well as Strato Chess (1973) by Dynamic Games and Space Chess (1970) by Pacific Game Co. According to Pritchard, the variants that use a 8×8×3 layout or boards smaller than 8×8 are "less demanding on spatial vision, and hence more practical".
Early history
Perhaps the first extension of chess into a three-dimensional space was accomplished by Alexandre-Théophile Vandermonde in 1771. Vandermonde is known for his unique chess problem, which was extension of the knight's tour into a 4×4×4 playing field.Smith, L. Lynn. "The History of 3D Chess Part One: Raumschach and the Unicorn", in Abstract Games Magazine Issue 10 Summer 2002
There are various reports of Baltic-German chess master Lionel Kieseritzky (1806–1853) demonstrating a game called Kubikschach (Cube chess) to the German master Adolf Anderssen at a London tournament. This game was also reported on in the Deutsche Schachzeitung. However, the exact rules and playing setup for Kubikschach are unknown. It is also unknown who exactly designed this game, though it is often assumed it was designed by Kieseritzky himself and various publications list him as the inventor.
Kubikschach influenced Ferdinand Maack in his development of Raumschach, and he discusses Kubikschach in his first Raumschach periodical. However, we know almost nothing about the rules and configuration of the board for this game.
Raumschach
The German mathematician Ferdinand Maack (1861–1930) developed Raumschach (German for "Space chess", also called Schachraumspiel) in the early 1900s, patenting it with the German Imperial Patent Office in 1907.Binnewirtz, Ralf. Ein Streifzug durch die Raumschach-Historie – von Vandermondes Raumr¨ossel zum spieltauglichen Stereoschach (28.10.2023) Maack contended that for chess to be more like modern war, attack should be possible from above and below, writing: "if the analogy to war is to be maintained, the third dimension is required. Modern strategy, with its steerable airships and submarines, uses the whole of space. Thus, in chess, attacks from above and below must be made possible" (Anleitung zum Raumschach).
Maack's original formulation presented in 1907 was for an 8×8×8 board, as with Cubic chess. He labeled each plane with Greek letters (with Alpha as the lowest board and Theta as the highest). However, the long and difficult games that arose from the 8×8×8 space led Maack to experiment with numerous other smaller play spaces over the course of the following years. Maack eventually settled on a 5×5×5 playing field as the best, though he also taught a version played on a 4×4×4 space. Other obvious differences between standard chess and Raumschach include two additional pawns per player, and a special piece (two per player) called the unicorn. The unicorn has a special triagonal move which can shift one space in rank, file and level. In Spielregeln zum Raumschach (1913), Maack explains the rules for two main versions: a "five level game" (5×5×5) and a "four level game" (played on a 4×4×4 board).
Maack promoted the game with demonstrations, articles, magazines and several books, such as Das Schachraumspiel: Dreidimensionales Schachspiel (1907), Spielregeln zum Raumschach (Rules of Space Chess) and Raumschach: Einführung in die Spielpraxis (1919). An article first appeared in a 1907 edition of the Frankfurter Zeitung, though the reception was initially mixed. Spielregeln zum Raumschach also states that sets of the game were available at German toy stores. Maack gave the first public presentation of the game in the 1907 International Chess Tournament in Carlsbad. Maack also edited and published a journal on the game called Mitteilungen über Raumschach und wissenschaftliche Schachforschung (Reports on Space Chess and Systematic Chess Research). Maack later founded the Hamburg Raumschach Club (possibly 1909), which remained active until WW2. Members included chess problemists like Hans Klüver and Willibald Roese. Raumschach achieved some limited popularity in England and Germany in the early 20th century, and is considered to be the classic form of 3D chess.
Raumschach was studied in detail by Thomas Rayner Dawson (1889–1951), an English chess problemist. He wrote a series of articles in The Chess Amateur on the game (1926–27), and composed problems for it. Dawson also wrote a manuscript on Raumschach, which was picked up by Hans Gruber and Kjell Widlert, who published it in two parts under the title Raumschachfunken (Space Chess Sparks) in 1993 and 1995. Alexey Troitsky is also known to have studied the game and its endgame problems.
While Raumschach is not a widespread game, it has been implemented in several modern computer programs, including the Windows software Zillions of Games. Numerous variants for Raumschach have also been designed by fans and some are also accessible in the Zillions program.
Board and starting position
The Raumschach 3D board can be thought of as a cube sliced into five equal spaces across each of its three major coordinal planes. This sectioning yields a 5×5×5 (125 cube) game-space. The cubes (usually represented by squares and often called cells) alternate in color in all three dimensions.
The horizontal levels were originally denoted by Greek letters (Alpha α, Beta β, Gamma γ, Delta δ and Epsilon ε) in Maack's publications. Some English writings use capital letters A through E instead. Ranks and files of a level are denoted using standard chess algebraic notation.
As for the starting position, the armies are arrayed on four levels, two for each color. White starts on the α and β levels and Black starts on δ and ε.
Standard rules
While different rules have been used, the standard rules that the Hamburg Raumschach club settled on after the first world war are found in Maack, F. Raumschach: Einführung in die Spielpraxis (1919). They are the following:
White moves first. The game objective, as in standard chess, is checkmate. Rooks, bishops and knights move as they do in chess in any given plane. The three dimensional movements are as follows:Maack, F. Spielregeln zum Raumschach (Rules of Space Chess) 1913
Variant move sets Regarding the pawn, the key principle is that they move one step like a rook or capture one step like the bishop. However, Maack and the Hamburg space chess club played with four different move sets for the pawns. There seems to have been much discussion and testing around the movement of the pawn, as Maack writes in Raumschach: Einführung in die Spielpraxis that "from the very beginning, the greatest difficulties and the liveliest controversies among space-chess players have arisen in answering the two questions: How are the pawns to move? and how are the pieces to be set up at the beginning of the game?"Maack, F. (1919) Raumschach: Einführung in die Spielpraxis (Space chess: Introduction to the Game's Praxis), pp 40–41.
The four options outlined by Maack in Einführung in die Spielpraxis are as follows:
: A pawn moves one step like a rook or captures one step like the bishop, but only forwards towards the promotion rank and not backwards, although it can move from bottom to top and vice versa. "Backwards" is defined here as the home rank of its king. In other words, the pawn cannot move towards the rank of his own king when moving on the y axis (which is also the rank where the opponent's pawn can promote), but it can move or capture one step up or down on the z axis as it likes. It also does not capture sideways when moving up or down, only towards the front edge. In Raumschach: Einführung in die Spielpraxis Maack argues that this option which has the pawn "move in three directions and capture in four directions" is the best option, and that this has been play-tested for years by the Hamburg club. Maack also says that this movement is "recommended for beginners". : A more restricted form of move A in which pawns only move and capture toward the promotion rank (rank E5 for White, rank A1 for Black). This includes moving one step directly upward (for White) or downward (for Black), and capturing one step diagonally upward (White) or diagonally downward (Black), through a front or side cube edge. This is the most restrictive move, since it removes the pawn's freedom to move as it likes on the z axis, forcing it to always move towards its promotion rank on the z and y axes. : This move is similar to B, but also allows capturing sideways upward. Maack says this is the "new method of play" that was adopted by the Hamburg club after years of play and was "vigorously advocated by perceptive and unprejudiced strong players and composers well known in the official chess world", including Hans Klüver and Wilhelm Roese.