Kulpa, W. (PL-SILS) AMS Author Profile; Socha, L. (PL-POLSL-ITR) AMS Author Profile; Turzański, M. (PL-SILS) AMS Author Profile
Source
Acta Universitatis Carolinae. Mathematica et Physica (Acta Univ. Carolin. Math. Phys.) (20000101), 41, no.~2, 47-50. ISSN: 0001-7140 (print).
Subject
52 Convex and discrete geometry -- 52C Discrete geometry 52C05 Lattices and convex bodies in $2$ dimensions
Language
English
Online Access
초록
Summary: ``The following result is due to Hugo Steinhaus [see {\it Mathematical snapshots}, Translated from the Polish, Reprint of the third (1983) English edition, Dover, Mineola, NY, 1999; MR1710978 (2000h:00002)]: Consider a chessboard with some `mined' squares. 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 the upper edge to the lower one moving exclusively on mined squares. \par ``According to W. Surówka\ [Ann. Math. Sil. No.~7 (1993), 57--61; MR1271184 (95a:52023)], several proofs of the Steinhaus theorem seem to be incomplete or use induction on the size of the chessboard. 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 one to find the rook's or the king's route between chosen opposite edges of the chessboard.''