Steinhaus chessboard theorem. (English) Zbl 0977.52026

Kulpa, W.; Socha, L.; Turzański, M.

Acta Univ. Carol., Math. Phys. 41, No. 2, 47-50 (2000).

Summary: The following statement is due to H. Steinhaus [Kalejdoskop matematyczny, PWN Warszawa 1954]. Consider a chessboard with some “mined” squares on. Assume that the king cannot go across the chessboard from the left edge to the right one without meeting a mined square. Then the rook can go across the chessboard from upper edge to the lower one moving exclusively on mined squares.
According to W. Surówka several proofs of the Steinhaus Theorem seem to be incomplete or use induction on the size of the chessboard [W. Surówka, Pr. Nauk. Uniw. Slask. Katowicach Ann. Math. Silesianae 1399(7), 57-61 (1993; Zbl 0807.52018)]. In this note we generalize the Steinhaus Theorem assuming that the chessboard (= square) is divided into arbitrary polygons (not necessarily squares) and we show an algorithm allowing to find rook’s or king’s route between chosen opposite edges of the chessboard.

MSC:

52C20

Tilings in \(2\) dimensions (aspects of discrete geometry)

Keywords:

Steinhaus theorem; chessboard

Citations:

Zbl 0807.52018