International Mathematics Competition
for University Students
2026

IMC2026: Day 2, Problem 10

Problem 10. An infinite chessboard of size \(\displaystyle d>0\) is obtained by colouring the interiors of the squares of an infinite square grid of side length \(\displaystyle d\) alternately white and black following the usual chessboard pattern. The points belonging to the grid lines have neither colour and the grid may be translated and rotated arbitrarily in the plane.

Is it true that for any finite set of points \(\displaystyle p_1,\ldots,p_n\) in the plane, there exist \(\displaystyle d\in(0,1)\) and an infinite chessboard of size \(\displaystyle d\) such that all the points \(\displaystyle p_i\) lie in white squares?

David Hruška, Czech Academy of Sciences, Prague

  Solution