This was solved by Ling Xiang Ning, Allan from Raffles Institution, Singapore and Tony Cardell and John Lesieutre from State College Area High School, Pennsylvania, USA.

This is Allan's solution: "The first part of the animation shows that a square of size a2 subtracted off a square of size b2 would result in a rectangle of sides (b - a)(a + b - a + a) = (b - a)(b + a). We can see that the difference of the two squares b2 - a2 is the same area as the area of a rectangle (b - a)(b + a).
Therefore b2 - a2 = (b-a) (b+a) "

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