Cut out a square of size x by x on each corner. Thus the length of the box is 0.5 - 2x and the length of the box is 0.5 - 2x and the height of the box is x. Thus the volume of the box will be:

V = (0.5 - 2x)*(0.5 - 2x)*x = 4x^3 - 2x^2 + 0.25x

We want the maximum value of this function.

I presume you know Calculus? We wish to set V'(x) = 0

So

V'(x) = 12x^2 - 4x + 0.25 = 0

Thus x = 1/4 and x = 1/12.

Obviously the x = 1/4 solution is ridiculous because we would be cutting the entire length of the board off in cutting the corners. So the only solution is x = 1/12.

Thus V(1/12) = 1/108 = 0.009289 m^3

-Dan