Let the radius of the inner circles be : r
Let us calculate the area of the dark zone, as follows:
Dark area = Area of square of side r - 2 times (area of square of side r - 1/ 4 area of inner circle)
Dark area = r exp 2 - 2 ( r exp 2 - 3.1416 * r exp 2 / 4 ) = r exp 2 * ( 3.1416 / 2 - 1)
Light area = 1/4 area of big circle - 1/2 area of inner circle - 1/2 area of inner circle + area of dark zone (intersection of two circles)
Light area = 1/4 * 3,1416 + (2 r) exp 2 - 3,1416 / 2 * r exp 2 - 3,1416 / 2 * r exp 2 + r exp 2 * (3,1416 / 2 - 1 ) = r exp 2 * (3,1416 / 2 -1)
Therefore, both areas are equal