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

a^{2}

subtracted off a square of size

b^{2}

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

b^{2} - a^{2}

is the same area as the area of a rectangle

(b - a) (b + a)

.
Therefore

b^{2} - a^{2} = (b - a) (b + a)