Instructions for Raumschach
(Three-Dimensional Chess)
Chess is composed of three factors: space (the playing field), time (the sequence of moves), and movement or force (the manner in which pieces move). To these are added certain rules and laws of play. Over the centuries, the terrain, pieces, and rules have undergone many changes and improvements. Yet all such "variants" have essentially remained within chess as a board game.
Depending on whether the chess space is one-, two-, or three-dimensional, one may distinguish line, surface, and space chess games. The Chinese played on lines, though their intersection already creates a two-dimensional playing surface. The ordinary chess game in common use is a surface or board game. To these is now added the three-dimensional chess space game. This space game surpasses the board game by as much as a cube surpasses a square.
The extension of the playing terrain into the third spatial dimension is not an arbitrary act, nor an artificial complication of chess — it is a logically and mathematically necessary postulate, a natural completion of the game.
First of all, chess is a game of movement par excellence. One must therefore give the pieces maximum freedom and possibility of movement, thus placing the entirety of space at their disposal. Secondly, chess is more than a game: it is the exemplification of an ancient spatial science — one that can only be understood in three dimensions. And thirdly, chess, if the analogy with warfare is to be maintained, demands the third dimension. For modern strategy, with its steerable airships and submarines, also exploits the full volume of space. Attacks from above and below must therefore also be possible in chess.
Finally, it should be noted that the three-dimensional chess space game is probably the primary "ur-chess," of which ordinary two-dimensional board chess is merely a secondary derivation — a projection onto a plane. All the laws of the board game are artificial surrogates for the natural laws of the space game. This establishes a "causal chess science." However, this short guide — serving only practical purposes — cannot engage with theoretical questions at any length. The hypothesis of a cubic ur-chess has, in any case, nothing to do with the game itself.
Whereas ordinary chess is played on a board divided into 8×8 = 64 square cells, space chess uses a cube divided into 8×8×8 = 512 cubic cells. Since playing within small cubes is not practically feasible, the 8×64 theoretical cubes are replaced by 8 chessboards arranged one above another in layers, with sufficient space between them for placing, moving, and surveying the pieces. The colours of the cells alternate from bottom to top as well (see diagram). The cells are designated, as in ordinary chess, with Latin letters and numbers; additionally, the 8 levels are labelled from bottom to top with the Greek letters α, β, γ, δ, ε, ζ, η, θ.
The practical construction of the chess space from 8 levels should not lead to the mistaken belief that 8 separate chessboards have merely been stacked — some of which might even be omitted. What we have here is, rather, an organic unity of 8³ cells, corresponding to the unity of 8² cells on the chessboard.
Space chess is played with the same pieces as ordinary board chess. However, since in the latter the pawns form a protective barrier for the officers against the enemy advancing from the front (i.e., along the second dimension), in space chess a second row of pawns must be placed above the officers as protection against attacks from above — that is, from the third dimension.
Each of the two players therefore has:
A total of 2 × 24 = 48 pieces are required. If no second chess set is available for the top pawns, they may be replaced by halma stones or other markers.
The lowest level, α, serves as the operational base. Here the officers and pawns are arranged exactly as in ordinary chess:
(See also the section on Game Variations.)
The manner in which each piece moves follows from the mathematical (stereometric) properties of the cubic cell in which it theoretically stands. Each cell-cube has three stereometric elements: 1) 6 faces, 2) 12 edges, and 3) 8 corners. The beginner should take care to keep edges and corners clearly distinct — they are easily confused.
For orientation and practice, leave any central cell (e.g. ε d3) in the three possible ways. For instance, ε d3–ε d2 or ε d3–δ d3 etc. are "face moves". Moves such as ε d3–ε e2 or –δ e3 or –ζ d2 etc. are "edge moves". Moves such as ε d3–δ c2 or –δ e4 or –δ e2 etc. are "corner moves". Edge moves connect cells of the same colour; face and corner moves connect cells of different colours.
Each of the three mathematical elements of the cube corresponds to the movement of one type of officer — a "long-range" piece:
The Rook moves through the faces. It may therefore leave its cell in up to 6 directions. Rook moves represent the 3 mutually perpendicular axes (coordinates) of space.
The Bishop moves through the edges. It may leave its cell in up to 12 directions. It remains "colour-bound."
The Knight moves through the corners. It may leave its cell in up to 8 directions. From this follows an important law for cubic chess: the Knight does not jump in space, but slides — just like the Rook and Bishop — as far as it wishes or is able, until it meets resistance, an occupied cell, or the boundary of the chess space.
From the corner moves of the Knight follows its peculiar passive behaviour on its current level: since a plane has no three-dimensional "corners," the Knight is condemned to inactivity on its own level. (See, however, "Game Variations" below.)
The Queen combines face, edge, and corner moves — those of the Rook, Bishop, and Knight — for any number of steps. She therefore commands 26 directions.
The King likewise combines all three movement types, but only for one step. The King thus controls, including his own cell, a terrain of 3×3×3 = 27 cubic cells.
Pawns move through the faces like the Rook — one step — and capture through the edges like the Bishop — one step. They may move and capture in all directions (see "Game Variations"). Pawns promote on the 8th or 1st rank of all 8 levels into any officer of the player’s choice.
As can be seen, the relative value of the pieces in space has shifted considerably compared to their value on the board. This is due to the mathematical properties of the cubic cell: the Rook, for example, has only 6 exits, while the Bishop has 12.
Space chess is played generally in the same way as board chess. However, one must be careful not to transfer theoretical knowledge from the board to space! Cubic chess has its own entirely distinct theory, which is still awaiting development.
Since all officers (except the Rook) are mobile from the very first move — they can reach higher levels immediately — the initial double-step of pawns (and thus en passant capture) is forbidden; as is castling. The King can easily move upward to safety.
The most common openings will be "officer games," in contrast to the initial pawn games typical of board chess. In board chess, the officers must first be freed by pawn moves; in space chess, they are mobile from the outset.
Rome, as the saying goes, was not built in a day. Ordinary chess also was not finished all at once, but required centuries to reach its current form. The same applies to space chess — it is still in the early stages of its development. Settling and fixing its methodology and strategy will take much more time and effort. Above all, a universally binding agreement among chess players will be required. Until then, players must carefully agree on the rules to be followed each time.
In addition to the method described above, the following variations may be adopted:
Nobody will likely want more than 8×8×8 cubic cells — say 9³ or 10³ = 1000. However, one might consider fewer cells, e.g. 7³. In that case the Queen would be omitted. Seven levels would offer the advantage of a central level (the 4th) and a central cell δ d4.
Instead of the lowest level, one might choose a middle level (e.g. δ) as the operational base — this would allow attacks from below from the outset.
One might also consider placing the White officers on the lowest level (α1) and the Black officers diagonally opposite on the highest level (θ). However, in that arrangement both queens and all four bishops would be en prise from the start. This could be mitigated with additional pawn rows, but the diagonal setup finds no justification in pawn behaviour.
If a middle level is the operational base, or if one army stands higher and the other lower, officers must also be protected from attacks from below. In that case, 8 "bottom pawns" are added directly beneath the officers, in addition to the 8 front and 8 top pawns.
Instead of one row of top and bottom pawns, one might use two rows each, to protect officers not only from direct face attacks but also from oblique edge and corner attacks. Each player would then have three or even five pawn rows in addition to the 8 officers. This improves the ratio of cells to pieces, making it easier to reinforce the officer corps through pawn promotion — which is desirable in the endgame.
1) Knight. The logically and mathematically consistent movement for the Knight is the straight corner move. The familiar angular "jump" on the board is probably only an artificial surrogate for the corner move in space: for example, the Knight move α b1–β c2 is identical to α b1–β b1 plus β b1–β c2, i.e., a direct corner move equals a face move plus an edge move.
The corner moves result in the Knight’s passivity on its current level. To avoid this, one might allow the Knight the angular (jumping) moves on its current level. One might go even further and transfer the board-style knight jumps into space — so the Knight could move, for example, from α b1 to β b3 or to γ b2.
Placing the Knight in the centre of the space (e.g. δ d4) and connecting all the cells it commands through "jumps" yields the stereometric shape of a truncated octahedron. The Knight is thus the type of dynamic convergence. The Bishop, by contrast, represents dynamic divergence.
In summary, one may either: (1) allow the Knight only corner moves; (2) allow only angular jumps in space; (3) allow both; or (4) allow corner moves plus angular jumps on its current level only. For practical reasons, option (4) seems best: without any movement on its level, the Knight cannot approach certain enemy pieces or be well manoeuvred.
2) Queen and King. The Knight’s planar jumps do not transfer to the Queen and King — only the corner moves apply to them. (It has also been felt in board chess that a piece combining the Knight’s function with others would be useful.)
3) Pawns. Since space chess is presumably the ur-chess, and board chess only a simplified derivation — a "popular edition" — the peculiarities of the board should not be retroactively imposed on the space game. This applies particularly to the "jumping" movements (the Knight, the King at castling), but also to the pawn’s double step and probably to the pawn’s one-sided movement and capture.
On the board, the pawn may only move and capture forward. A direct analogy would restrict the pawn in space to moving only forward and upward, not backward or downward. One may play this way, but it is more correct to allow the pawn to move and capture in all directions — with one exception: from its original square, the pawn may only move forward and upward (otherwise the officer row would be unreachable). Thus, for example, δ d1 is not protected by pawn β e1, but is protected by pawn β d2.
Pawn promotion most practically takes place on all levels, although this too may belong to board-game conventions. Opposing officers stand on only one of the eight levels. But ranks 1 and 8 exist on all levels.
For orientation in the chess space and to become familiar with the three-dimensional kinematics and dynamics of the pieces, a number of checkmate positions follow. The 50 examples given are intended solely for practice. It is assumed that the isolated White King is to be checkmated. His position is either on a boundary level (α) or middle level (δ), and within that level on either a corner cell (a1), edge cell (e1), or central cell (e4).
A mate is naturally easier to achieve at the periphery of the chess space than at the centre. We label the White King’s position as follows:
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These space checkmate positions show above all that, under certain circumstances, the available piece material is quite sufficient — especially when the pawns’ effectiveness is enhanced by allowing them to move and capture in all directions, or when three or even five rows of pawns are deployed. In practical play, achieving checkmate is further aided by the presence of multiple friendly and enemy pieces, and by the possible appearance of promoted queens.
These rules underlie the following games, which are neither master nor model games, but serve purely for instruction and orientation.
Notation note: For moves on the same level (au niveau), the level designation may be omitted, as done in the third game. Check is indicated by †, board mate by ††, and space checkmate by ††† (or a cross with three bars). Representing a spatial position diagrammatically requires 8 ordinary diagrams stacked from bottom (α) to top (θ). Note: on levels β, δ, ζ, θ, the black corner cell is on the right.
One should not conclude from these intentionally brief games that space chess games are generally shorter than board chess games — quite the contrary. On average they last longer. Additional games based on other movement variants (no corner moves, space jumps of the Knight, one-directional pawn movement, conditional pawn promotion, etc.) are published in Deutsches Wochenschach.
Game chess is the prose, and problem chess the poetry of chess life. Just as the new space chess game greatly enriches practical play, cubic chess will also provide extraordinarily rich material for artistic compositions.
It is not the purpose of the present guide to deal with space chess problems. So long as the rules of play are not yet definitively established, such an undertaking would also be premature. The checkmate positions already communicated give an approximate picture of three-dimensional chess problems. They show that in particular the demand for beauty can be satisfied by space chess problems. Beyond this, an abundance of new ideas and the most surprising combinations await the creative artist.
The space chess game was first demonstrated by the author on the occasion of the International Chess Masters Tournament in Carlsbad (September 1907). This prompted a large number of articles in domestic and foreign newspapers and periodicals. The first scholarly work on the subject has been appearing since October 1907 in Deutsches Wochenschach (A. Stein’s Verlag, Potsdam). When judging that work, one should bear in mind that it represents the first draft of the new game. Birth pains everywhere can only be overcome gradually (cf. Game Variations). Support from prominent chess masters and a future conference will be difficult to do without. For the present, all those who have an interest in the progress of chess thought are invited to publish their views and experiences, or to communicate them to the author.
The space chess game is legally protected as D.R.G.M. 317321. A collapsible model is available, until further notice, through the intermediary of the author.