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I first worked it backwards. I start with f(x,y) = (x^2 - y^2)/8. Then I substitute x+2y for x and x-2y for y. Do the algebra and get xy. This proves the converse of the problem.
Now run it forwards: f(x+2y, x-2y) = xy = 8xy/8 = ( (x^2 + 4xy + 4y^2) - (x^2 - 4xy + 4y^2) ) /8 = ( (x+2y)^2 - (x-2y)^2) )/8. Substitute x for x+2y and y for x-2y and get f(x,y) = ( x^2 - y^2)/8.
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