Combinatorics · Invariants · Parity · Board colouring · Transformation games
Problem 5, 2006
A number is written in every cell of an \(8 \times 8\) table. A move consists of choosing any \(3 \times 3\) square of the table (nine cells) or any \(4 \times 4\) square (sixteen cells) and increasing by \(1\) each of the numbers standing in the chosen square.
Is it true that every starting table can be turned, by a sequence of such moves, into a table in which all the numbers are even?
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Serbian Regional Competition (Okruzno takmicenje) 2006, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source