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Good board filled with 0s and 1s

Source: Kazakhstan international contest 2006, Problem 3

January 22, 2006
combinatorics proposedcombinatorics

Problem Statement

Let m≥n≥4 m\geq n\geq 4 be two integers. We call a m×n m\times n board filled with 0's or 1's good if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in 3×3 3\times 3 sub-boards is the same; 3) the sum of all the numbers in 4×4 4\times 4 sub-boards is the same. Find all m,n m,n such that there exists a good m×n m\times n board.
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