1090.72 – Cube, Orange, Made and Unmade

One thousand unit cubes are fastened together to form a large cube with edges of 10 units. The big cube is painted bright orange. After the paint is dry, the cube is taken apart into the original one thousand little cubes which are scattered on your table top. How many of these unit cubes now have at least one orange face?

Solution

We need to count the number of cubes on the surface of the big cube, so we could count the number of cubes along the 12 edges (being careful not to double count the 8 vertex cubes) and add to that the number of cubes in the interior of each of the 6 faces…

But wait! There’s an easier way. What if we count the cubes that we have NO orange sides, that is, the cubes that are NOT on the surface of the big cube, and subtract that number from 1000? The interior of the 10 $\times$ 10 $\times$ 10 cube is an 8 $\times$ 8 $\times$ 8 cube containing 512 little cubes with no orange side.

So $1000 - 512 = 488$ cubes that have at least one orange side.

Possible class discussion if you have a few odd minutes: what would be a good way to fasten the cubes together if you know ahead of time that you're going to take them apart?