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black squares in 2015 x2015 board

Source: Mathematics Regional Olympiad of Mexico Center Zone 2015 P3

November 12, 2021
combinatoricsgridColoring

Problem Statement

A board of size 2015×20152015 \times 2015 is covered with sub-boards of size 2×22 \times 2, each of which is painted like chessboard. Each sub-board covers exactly 44 squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly 20152015 black squares visible. What is the maximum number of visible black squares?
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