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paint a finite number of cells black in an infinite white plane of square cells

Source: Tournament of Towns, Senior A-Level Paper, Spring 2020 , p7

June 3, 2020
Coloringcombinatoricstable

Problem Statement

Consider an infinite white plane divided into square cells. For which kk it is possible to paint a positive finite number of cells black so that on each horizontal, vertical and diagonal line of cells there is either exactly kk black cells or none at all?
A. Dinev, K. Garov, N Belukhov