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Source: Latvia TST for Baltic Way 2020 P6

October 22, 2020
combinatorics

Problem Statement

For a natural number n≥3n \ge 3 we denote by M(n)M(n) the minimum number of unit squares that must be coloured in a 6×n6 \times n rectangle so that any possible 2×32 \times 3 rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural n≥3n \ge 3 the number M(n)M(n) can be expressed as M(n)=pn+kn3M(n)=p_n+k_n^3, where pnp_n is a prime and knk_n is a natural number?
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