This was solved by Ling Xiang Ning, Allan, age 13 from Raffles Institution, Singapore and Tony Cardell and John Lesieutre, age 14 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)