Annales Mathematicae Silesianae, Nr 7 (1993), s. 57-61
The paper includes purely cobinatorial proof of a theorem that implies the following theorem, stated by Hugo Steinhaus in [6, p.35]: consider
a chessboard (rectang:ular, not necessarily square) with some "mined" squares
on. Assume that the king, while moving in a.ccordance with the chess rules, cannot go across the chessboard from the left edge to the right one without meeting a mińed square. Then the rook can go· across' the chessboard from
the upper edge to the lower one moving exclusively ·on mined squares. All proofs of the Steinhaus theorem already published (see  and remarks on sorhe proofs in [7, p.211]) are incómplete, except the hexagonal variant proved
by Gale in .
Steinhaus theorem was thought in [6, p.269] as the lemma in the proof
of the Brouwer nxed point theorem for the square ( cf. Saskin  and Gale
). lt can alsó be used as the lemma for the mountain climbing theorem
of Homma  (see Mioduszewski ). In this paper the Steinhaus theorem
is used in the proof of a discrete analogue of the Jordan curve theorem (see Stout , where different proof is stated; cf. also Rosenfeld ).