Paul Klee

Thursday, July 30, 2009

Infinite Chess

The conventional game of chess provides more than enough challenges for even the strongest player, but many extensions to the game have been proposed. Among these are three-dimensional chess, of Star Trek fame, and designs using an infinite board. These latter attempts use the idea of potential infinity rather than absolute infinity. The designs presented below imply actually infinite domains as well as potentially infinite subspaces.

Extending chess to an infinite domain involves defining the required space and also redefining how the chessmen move within it. The conventional chessboard has 64 squares, which are indexed from a1 at the bottom left hand corner to h8 at the top right. This provides a convenient notation for recording the moves of a game. So, the rows are indexed by the first eight numbers and the files (columns) by the first eight letters, as shown below.

The board can be rendered unlimited simply by allowing the index to include all the finite numbers and all the finite combinations of letters. For example, a square such as g100 or ay39 would be legitimate. In this way the chessmen could move about in an infinite space, without changing the conventional rules by which they move.

A peculiarity of this extended index is that the bottom and left hand boundaries of the board are preserved. On such a board the white chessmen can move forward or to the right without limit but remain constrained to the left and behind. However, it is not clear where the opposing black pieces are to be located or whether they should be symmetrically constrained behind and to the left like the white pieces, which would clearly never do.

One solution is to confine the starting positions of the opposing armies to the dimensions of the standard board but relativise their position in the infinite plane. This can be done by extending the index to include negative values, analogous to the notation of the Cartesian plane. For example, ac42, -ac42,

ac-42 and -ac-42 would be legitimate and distinct locations, where ac represents 26 + 3 = 29 squares.

The extension to an infinite board would affect the powers of the chessmen differently. The queen, rook and bishop could make unlimited moves but the king, knight and pawn would be restricted to a single move, and so would gain limited freedom on the extended board. Their relative powers would be diminished accordingly.

An alternative scheme is to separate the white and black chessmen by an infinite space. The immediate consequence would be that no matter how far the queens, rooks or bishops moved according to their enhanced powers, they could never engage the enemy. To rescue the game from this impasse requires a further extension to the powers of the pieces and the pawns.

The solution is to allow all the chessmen to make infinite moves, from one domain to another, according to strict but familiar rules. These rules are as follows:

Rule 1: A man may either make a short (finite) move or a long (infinite) move but not both.

Rule 2: In making a long move, a man must move from one domain to another in the same manner as required by a short move.

Rule 3: A move from one domain to another preserves the finite position of the man.

The first rule is self-explanatory. The player may either make a move within the domain the man occupies or move the man to another domain, subject to rules 2 and 3.

The meaning and relation of the infinite domains needs to be explained before elaborating on rules 2 and 3. Each domain is a replication of the infinite space defined above. The domains are arranged in a square matrix, which must be sufficiently large to allow long moves as defined in rule 2. For example, a 5 x 5 matrix is necessary to allow the knights access to every domain. Any larger matrix could be adopted but, for aesthetic purposes, an 8 x 8 matrix of domains is ideal.

A system of notation can now be defined to locate the men within both domain (board) and its finite subspace. Each of the 64 infinite boards is indexed from A1 to H8, analogous to the indexing of the conventional chessboard. A double reference of the form XYxy locates an individual square within a domain. For example, the white king is located on the square E1e1 at the start of the game and the black king is on square E8e8.

The initial positions of the 32 men can now be described. The rule for setting up the board is simple. On the conventional board, the white queen sits on square d1: on the infinite board she sits on square D1d1. The trick is to duplicate the local reference in the board reference. The white queen’s pawn conventionally starts on d2, so it occupies D2d2 on the infinite matrix. An infinite bird’s eye view would show the initial set up to be identical to that of the conventional game.

Rules 2 and 3 can now be explained more fully. The white king’s knight begins on square G1g1. The knight is free to make a short move to either G1f3 or G1h3. In addition, the knight can make a short move to G1e2, because all the pawns start off in domain 2. The knight can make initial long moves to F3g1 or H3g1 but not to E2g1, because this square is occupied by the king’s bishop’s pawn.

A notable feature is that all the pieces can make unrestricted finite moves at the opening, because each one is alone in its domain. This allows the players to jump into a new domain from an unlimited number of positions. Like many art forms, it is the constraints rather than absolute freedom that leads to interesting works. No less so in the game of chess. For this reason the proposed game can be modified by restricting each domain to the usual finite 8 x 8 matrix of squares. The result is an extremely complex finite expansion of the traditional game of chess.

The diagram below shows some examples of long moves. A domain set of 3 x 3 boards has been used for compactness of presentation. It can be seen that knight, bishop and rook can reach across domains. The power of a pawn to take diagonally in a long move is also illustrated. The pawns power to move two squares on its first move allows it to make a double long move. The en passant rule is similarly preserved.


The extension of the game of chess to multiple domains generates a family of games, which may be either finite or infinite. This can be achieved by the addition of the three special rules for long moves and by adding a square or rectangular matrix of boards of one’s choice. The double notation allows the computerisation of the game. The implications for geometry and the theory of infinite number will not be considered here. Suffice to say that the examination of such models should provide useful insights in these areas of enquiry.

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