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Table m x n

Source: Baltic Way 2004, problem 11

November 20, 2004
algorithmcombinatorics unsolvedcombinatorics

Problem Statement

Given a table m×nm\times n, in each cell of which a number +1+1 or −1-1 is written. It is known that initially exactly one −1-1 is in the table, all the other numbers being +1+1. During a move, it is allowed to chose any cell containing −1-1, replace this −1-1 by 00, and simultaneously multiply all the numbers in the neighbouring cells by −1-1 (we say that two cells are neighbouring if they have a common side). Find all (m,n)(m,n) for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial −1-1 stands.
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