Problem

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Tags: induction, invariant, geometry, geometric transformation, combinatorics proposed, combinatorics



Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. (Hint: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)