Answer is 132 white boxes.
The designs are increasing by 2 x 2 each time.
Design 1 is 2 x 2,
Design 2 is 4 x 4,
Design 3 is 6 x 6,
Design 4 will be 8 x 8
Design 5 will be 10 x 10
Design 6 will be 12 x 12 for a total of 144 boxes.
The diagonal of shaded boxes will be 12 shaded.
So 144 - 12 = 132 white boxes.
