Cubes and Painted Cubes made clear.
Separate the geometry of cutting a cube from questions about which faces touch. For painted-cube counts, classify small cubes by corners, edges, faces and interior.
Key concepts in Cubes and Painted Cubes
Equal subdivision
Cut each edge of a large cube into n equal parts to obtain n³ small cubes.
Three painted faces
If all six outside faces were painted before cutting and n≥2, the 8 corner cubes each have three painted faces.
Two painted faces
Edge cubes excluding corners: 12(n−2).
One or zero painted faces
Face interiors: 6(n−2)². Completely unpainted cubes: (n−2)³. These formulas assume n≥2 and all six faces painted.
Where a small cube sits: n=4
| Location | Painted faces | Count |
|---|---|---|
| Corner | 3 | 8 |
| Edge, excluding corners | 2 | 24 |
| Face interior | 1 | 24 |
| Inside the large cube | 0 | 8 |
| Total | 64 |
A reliable solving method
- Confirm which outside faces are painted and how many subdivisions occur along each edge.
- Classify by location rather than attempting to count a drawing.
- Apply the matching formula with its conditions.
- Add the categories and check the result equals n³.
Step-by-step solved examples
A painted cube is cut into 27 equal cubes. How many have two painted faces?
- 27 = 3³, so n=3.
- Two-face cubes lie on 12 edges, excluding their corners.
- 12(3−2) = 12 cubes.
A painted cube is cut into 64 equal cubes. Count unpainted cubes.
- 64 = 4³, so n=4.
- Remove one outer layer from both ends of each dimension.
- Interior count = (4−2)³ = 8.
Check all categories for n=3.
- Three-face cubes = 8; two-face cubes = 12.
- One-face cubes = 6(1)² = 6; unpainted = 1.
- 8+12+6+1 = 27, matching the total.
Cubes and Painted Cubes: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Five subdivisions on each edge produce how many cubes?
Show answer and explanation — question 1
Answer D: 125
5³ = 125.
All faces painted, n=4: three-face cubes?
Show answer and explanation — question 2
Answer B: 8
The eight corners each have three painted faces.
All faces painted, n=4: two-face cubes?
Show answer and explanation — question 3
Answer C: 24
12(4−2) = 24.
All faces painted, n=4: one-face cubes?
Show answer and explanation — question 4
Answer C: 24
6(4−2)² = 24.
All faces painted, n=5: unpainted cubes?
Show answer and explanation — question 5
Answer C: 27
(5−2)³ = 27.
All faces painted, n=2: cubes with no painted face?
Show answer and explanation — question 6
Answer A: 0
Every small cube is a corner; there is no interior.
Frequently asked questions
Does a small cube on a face always have one painted face?
No. Face corners have three, and non-corner edge cubes have two; only face interiors have one.
Can I identify opposite faces from one cube view?
A single view shows three mutually adjacent faces. Additional labelled views or a net are needed to determine unseen opposite pairs.
Confirm which outside faces are painted and how many subdivisions occur along each edge. Add the categories and check the result equals n³.