# Count the edge cubes: visual puzzle and solution | SwaVid

A cube is painted on every face, then cut into 27 equal cubes. How many small cubes have exactly two painted faces? See the complete step-by-step solution.

Canonical: https://www.swavid.com/puzzles/painted-cube

Source: https://www.swavid.com/puzzles/painted-cube

# Count the edge cubes

## 12 cubes.

### Corners have three faces

### Edges have two faces

### Check the total

SwaVid visual puzzle

A cube is painted on every face, then cut into 27 equal cubes. How many small cubes have exactly two painted faces?

Detailed solution

A small cube gets one painted face for each big-cube face it touches. Two painted faces means it sits on an edge but not a corner.

The 8 corner cubes touch three faces each.

A cube has 12 edges. On a 3×3×3 cube, each edge has one cube between its two corners.

8 corners + 12 edges + 6 face-centres + 1 hidden centre accounts for all 27.

Counting all cubes on the edges gives 20, but that includes the 8 corners, which have three painted faces.

- 1 Corners have three faces The 8 corner cubes touch three faces each. 8 cubes × 3 faces
- 2 Edges have two faces A cube has 12 edges. On a 3×3×3 cube, each edge has one cube between its two corners. 12 edges × 1 cube = 12
- 3 Check the total 8 corners + 12 edges + 6 face-centres + 1 hidden centre accounts for all 27. 8 + 12 + 6 + 1 = 27

## Key Links

- [All puzzles](https://www.swavid.com/puzzles)
- [Browse all puzzles](https://www.swavid.com/puzzles)