Math · Stage 6

Painted-cube colouring counts

Dip a big cube in paint, let it dry, then saw it into little cubes. Some little cubes caught paint on 3 sides, some on 2, some on 1, and some stayed clean. Four tidy rules tell you exactly how many of each — for a cube of any size.

✦ Math📘 Stage 6 · S6⏱ ~14 min🧒 Ages 10–12★ Free to start

I can count the painted faces of a cut-up cube.

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Kavi Paints a Cube 🖌️

Kavi dipped a big 3×3×3 block in teal paint 🖌️, let it dry, then sawed it into 27 little cubes 🧊. “The eight corner cubes caught paint on 3 sides! The edge cubes got 2, the face-middles got 1 — and one little cube, hidden right in the centre, stayed completely clean.” Kavi grinned: “I never have to look — four rules count them for me, for a cube of any size.”

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What your child will learn

1Four kinds of little cubeWhen you paint a big cube and cut it up, every little cube is one of four kinds, by how many of its sides caught paint. Drop each little cube in the right box →

2A cube has 8 corners, 12 edges, 6 facesWhy those four rules? Because a cube's own shape never changes: it always has the same corners, edges and faces. Tap each part →

3The painted-cube rulesPaint an n×n×n cube, then cut it into n³ little cubes. On each of the 12 edges the pieces between the two corners number n − 2; on each of the 6 faces the inner square is (n − 2) × (n − 2); and the hidden core is an (n − 2)×(n − 2)×(n − 2) cube. Tap each rule →

4A 3×3×3 cube, cut into 27Here are the four counts for a painted 3×3×3 cube. Read how many little cubes fall in each group — and watch them total 27.

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A peek inside

A 3×3×3 cube is painted all over, then cut into 27 little cubes. How many little cubes have exactly 2 painted faces?

2-face cubes sit on the edges (not the corners): 12 × (n − 2) = 12 × (3 − 2) = 12 × 1

Try this at home: 3 painted faces: a 4×4×4 cube is painted, then cut up → how many little cubes have 3 painted faces? __

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

Start the free lesson →⬇ Download PDF

Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.