The painted cube

Hello, and welcome to Myriora Maths! A big cube is painted all over. Then it's cut into twenty-seven small cubes. How many small cubes have three painted faces, two, one, or none at all?

Three painted faces: those are the corners. A cube has eight corners, so eight small cubes have three painted faces.

Two painted faces: those are on the edges, but not the corners. A cube has twelve edges, and each edge has one middle cube. So twelve small cubes have two painted faces.

One painted face: that's the middle of each face. A cube has six faces, with one middle each. So six small cubes have one painted face.

And no paint at all? Just the one cube hidden in the very middle. Check: eight plus twelve plus six plus one is twenty-seven. Every cube is counted!

For a bigger cube, look in the same places. Cut a cube into four by four by four. The corners: always eight. Each edge now has two middle cubes: twelve times two is twenty-four. Each face has a two by two middle: six times four is twenty-four. And the hidden inside is two by two by two: eight.

A common mistake is counting the corners as edge cubes too. Each edge has three cubes, so some people say twelve times three is thirty-six. But the end cubes of each edge are corners, with three painted faces! Only the middle ones count: twelve.

Your turn! A painted cube is cut into five by five by five small cubes. How many small cubes have exactly one painted face? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! The middle of each face is three by three: nine cubes. There are six faces. Six times nine is fifty-four!

Quick review. Corners have three painted faces, and there are always eight. Edge middles have two. Face middles have one. And the hidden inside has none. Check that they add up to the total. See you next time!