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Assuming √2 is rational, √2=a/b where a and b are integers and the fraction is in the lowest terms. Then a²=2b², so there are 2 squares with integer sides such that 
Placing the 2 smaller squares on the larger one, we see that the sum of the areas of the corner squares must equal the area of the black central one. We reached a contradiction since we assumed a and b are the smallest integers such that a²=2b²!
