Then, either the king can cross the board from left to right without meeting a mined square, or the rook can cross the board from top to bottom moving only on mined squares.Gale[2] proved a variant of the theorem in which the tiles on the chessboard are hexagons, as in the game of Hex.
Kulpa, Socha and Turzanski[1] prove a generalized variant of the chessboard theorem, in which the board can be partitioned into arbitrary polygons, rather than just squares.
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[4] present the proof of Tkacz and Turzanski to the n-dimensional chessboard theorem, and use it to prove the Poincare-Miranda theorem.
Suppose by contradiction that an n-dimensional function f, satisfying the conditions to Miranda's theorem does not have a zero.
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.