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Find all positive integers $n$, for which the grid always will cointain at least

Source: Moldova TST 2022

April 1, 2022
combinatorics

Problem Statement

Let nn be a positive integer. A grid of dimensions n×nn \times n is divided in n2n^2 1×11 \times 1 squares. Every segment of length 11 (side of a square) from this grid is coloured in blue or red. The number of red segments is not greater than n2n^2. Find all positive integers nn, for which the grid always will cointain at least one 1×11 \times 1 square which has at least three blue sides.
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