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.

What's most interesting about the problem is that the inside is a quantity which doesn't depend on N, raised to the N power.

From this, it looks like the recoloring procedure has a certain independence. Whatever the inside is, you just multiply it another time to get the next step.

The reason this happens is not immediately clear to me, after thinking about it for a couple of minutes.

Posted by Gamer on 2008-06-08 21:43:26