How many 1 inch cubes do you need to fill a cube that has an edge of 1 foot?

Answer 1

There are 12 inches in 1 foot, so there would be (12x12x12 =) 1728 cubes in a solid 1-foot cube. To hollow it out, we remove a cube of 10x10x10 from the middle (leaving just the cubes on the outside), so the answer is (1728 - 10x10x10 = 1728 - 1000 =) 728 cubes.

The other way to approach this is to consider the sides - a cube has 6 faces, each with (12x12 =) 144 cubes in it, so the answer seems like it should be (12x12x6 =) 864. This doesn't take into account the edges of the cube (which overlap between sides). There are 12 edges to the cube, each with 12 cubes in them, so we need to subtract these from the total, giving (864 - 12x12 = 864 - 144 =) 720. This is still not quite right, as we have now counted the corner cubes 3 times (as 3 faces meet at each corner), and then discounted them 3 times (as 3 edges meet at each corner), so the corner cubes are not currently accounted for. There are 8 corners, so 8 more cubes, so the true answer is (720 + 8 =) 728, which is the same answer as we got from the first method.

The answer (checked by two different methods) is therefore 728.