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configuration of several checkers on grid paper is boring

Source: Israel Joseph Gillis Mathematical Olympiad 1998 p3

February 16, 2020
combinatoricscombinatorial geometry

Problem Statement

A configuration of several checkers at the centers of squares on a rectangular sheet of grid paper is called boring if some four checkers occupy the vertices of a rectangle with sides parallel to those of the sheet. (a) Prove that any configuration of more than 3mn/43mn/4 checkers on an m×nm\times n grid is boring. (b) Prove that any configuration of 2626 checkers on a 7×77\times 7 grid is boring.