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Maximum number of black squares [Iran Second Round 2017, P3]

Source: Iran National Olympiad 2017, Second Round, Problem 3

April 20, 2017

combinatorics

Problem Statement

Let

n

be a natural number divisible by

3

. We have a

n \times n

table and each square is colored either black or white. Suppose that for all

m \times m

sub-tables from the table (

m > 1

), the number of black squares is not more than white squares. Find the maximum number of black squares.