The Steinhaus chessboard theorem is the following theorem, due to

Hugo Steinhaus Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Univers ...

Consider a
chessboard A chessboard is a game board used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During p ...
on which some cells contain
landmines A land mine, or landmine, is an explosive weapon often concealed under or camouflaged on the ground, and designed to destroy or disable enemy targets as they pass over or near it. Land mines are divided into two types: anti-tank mines, whic ...
. Then, either the
king King is a royal title given to a male monarch. A king is an Absolute monarchy, absolute monarch if he holds unrestricted Government, governmental power or exercises full sovereignty over a nation. Conversely, he is a Constitutional monarchy, ...
can cross the board from left to right without meeting a mined square, or the
rook Rook or rooks may refer to: Games *Rook (chess), a piece in chess that moves horizontally and vertically * Rook (card game), a trick-taking card game People, characters, individuals *a rookie, a rook * Russell Rook, Baron Rook (The Lord Rook; 21 ...
can cross the board from top to bottom moving only on mined squares.

Two-dimensional variants

Gale A gale is a strong wind; the word is typically used as a descriptor in nautical contexts. The U.S. National Weather Service defines a gale as sustained surface wind moving at a speed between .

proved a variant of the theorem in which the tiles on the chessboard are hexagons, as in the game of

Hex Hex usually refers to: * A curse or supposed real and potentially supernaturally realized malicious wish * Hexadecimal, a base-16 number system often used in computer nomenclature Hex, HEX, or The Hex may also refer to: Magic * Hex sign, a b ...

. In this variant, there is no difference between king moves and rook moves. Kulpa, Socha and Turzanski prove a generalized variant of the chessboard theorem, in which the board can be partitioned into arbitrary polygons, rather than just squares. They also give an algorithm for finding either a king route or a rook route.

n-dimensional variants

Tkacz and Turzanski generalize the chessboard theorem to an n-dimensional board:

Consider a grid of n-dimensional cubes. Color each cube with one of ''n'' colors 1,...,''n''. Then, there exists a set of cubes all colored ''i'', which connect the opposite grid sides in dimension ''i''.

Ahlbach present the proof of Tkacz and Turzanski to the ''n''-dimensional chessboard theorem, and use it to prove the Poincare-Miranda theorem. The intuitive idea is as follows. Suppose by contradiction that an ''n''-dimensional function f, satisfying the conditions to Miranda's theorem does ''not'' have a zero. In other words, for each point x, there is at least one coordinate ''i'' for which ''f_i''(x) is nonzero. Let us color each point x with some color ''i'' for which ''f_i''(x) is nonzero. By the Steinhaus chessboard theorem, there exists some ''i'' for which there is a path of points colored ''i'' connecting the two opposite sides on dimension ''i''. By the Poincare-Miranda conditions, ''f_i''(x)<0 at the start of the path and ''f_i''(x)>0 at the end of the path, and the function is continuous along the path. Therefore, there must be a point on the path on which ''f_i''(x)=0 - a contradiction.

References

{{reflist Fixed-point theorems

Two-dimensional variants

n-dimensional variants

See also

References