A popular toy among math lovers and others is made by arranging $27$ small cubes into a larger $3 \times 3 \times 3$ cube and then painting the outside.

(The outside is usually painted in a variety of colors, but for our purposes in today’s challenge, we can just consider one.)

Consider the smaller cubes — how many of them are painted on none of their faces? On one face, two faces, or three faces? Are any painted on more than three faces? How would our answers change if we made a bigger cube?

For answers to these questions, keep reading. Or, skip to today’s challenge for a bigger cube.