The interior of the square [0,1] x [0,1] is initially coloured white.

Four random numbers: u,v,x,y in the range [0,1] are selected and the points inside the rectangle formed by the corners (u,v), (x,y), (u,y), (x,v) are recoloured: areas painted white are repainted red, and areas painted red are repainted white.

This recolouring procedure is repeated N times. Show that the expectation value of the red area is given by the formula:

1/2 - 1/2 ∫ dA ∫ dB {1 - 2 A (1-A) B (1-B)}^N

where both integrals go from 0 to 1.

A really interesting problem, and perhaps we should be trying to work out this integral in special cases?

However, I do agree. The definition of the box using random coordinates is very clear, and the probability should be 4 times that given, so that the final integral will show a coefficient of '8' rather than '2'.