The magic square below shows that all negative products can be achieved by assigning all cells with negative numbers.
Out of the 9 negative numbers in the squares, a maximum of how many can change their sign to positive in such a way that all the products remain unchanged?
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Each row, column and diagonal must have either 1 or 3 negative signs for the product to stay negative, so there must be at least 1 negative sign per row, and thus at least 3 negative signs in total. The maximum possible number of sign changes is then 6, which can be achieved in the following way:
+ + − + − + − + +
Each row/column has to have a different pairing of sign changes, for otherwise one of the columns/rows would have 3 positive signs, and there cannot be a sign change of the centre square, for otherwise one of the diagonals would have 3 positive signs.
Afterword: For a 4 by 4 grid we can change a maximum of 12 squares, as follows:
+ − + + + + + − + + − + − + + +
Theory: For an n × n grid, n > 2 , we can change a maximum of n 2 − n negative signs to plus signs.