Question

Imagine a piece of square paper that measures 20 by 20 cm. You can make a box (with no lid) by cutting a square of the same size from each corner and folding up what's left to make a box. Keeping the lengths of each sides integers, what is the maximum volume box that can be made?

Answer 1

if the squares are of size x, then the volume is

v = x(20-2x)^2 = 4x(10-x)^2

Without benefit of calculus, you still know the general shape of the curve. Since we want integer values, just start checking.

x v
1 324
2 512
3 588
4 576

Now v is decreasing, so keeping all values integers, looks like max v is 588.