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beetles creeping on neighbouring cells on a n x n square

Source: 2011 Belarus TST 1.4

November 8, 2020
combinatorics

Problem Statement

Given a n×nn \times n square table. Exactly one beetle sits in each cell of the table. At 12.0012.00 all beetles creeps to some neighbouring cell (two cells are neighbouring if they have the common side). Find the greatest number of cells which can become empty (i.e. without beetles) if a) n=8n=8 b) n=9n=9
Problem Committee of BMO 2011
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