Steinhaus chessboard theorem

Tkacz and Turzanski^[3] 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^[4] 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.

• A different theorem of Steinhaus, related to arranging rooks on a chessboard, that can be proved using Hall's marriage theorem.^[5]