No
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 threedimensional
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, ac42 and ac42 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 selfexplanatory. 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.
Summary
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.
Tony Thomas
February 2005
