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Vidhya from Kensri School in India sent in a very well reasoned solution:

When we subtract an even number from an odd number, or vice versa, the difference is always odd. So if we fill up odd numbers and even numbers alternately, if there are an even number of squares, the differences will all be odd. But there is no solution (in other words the differences cannot all be odd) if there is an odd number of squares.