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Snakes on a chessboard

Source: Own. Malaysian APMO CST 2024 P4

February 24, 2024
combinatorics

Problem Statement

Ivan has a n×nn \times n board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let dnd_n be the maximum number of black squares possible, prove that there exist some real constants aa, bb, c≥0c\ge 0 such that; an2−bn≤dn≤an2+cn.an^2-bn\le d_n\le an^2+cn.
Proposed by Ivan Chan Kai Chin
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