1
A cube painted red on all faces is cut into 27 identical smaller cubes. How many small cubes have three faces painted?
Correct answer: A. 8
Only the 8 corner cubes have three painted faces, whatever the size of the cube.
easy
Cubes, Cutting and Dice MCQs
Solved reasoning MCQs on painted cubes cut into smaller cubes, the minimum number of cuts, cube facts and standard dice, with the formulas that solve them.
20 MCQsUpdated 27 October 2026
This set covers painted cubes cut into smaller cubes, counting cubes with paint on three, two, one or no faces, the minimum number of cuts needed, the edges and faces of a cube, and the numbers on opposite faces of a standard die.
For a cube cut into n × n × n pieces, corners always give 8 cubes with three faces painted, and the inside gives (n − 2)³ with none. The explanations show each formula worked out.
- 2
A cube painted red on all faces is cut into 27 identical smaller cubes. How many small cubes have exactly two faces painted?
Correct answer: B. 12
Cubes with two painted faces lie on the edges but not at corners: 12 × (n − 2) = 12 × 1 = 12.
easy - 3
A cube painted red on all faces is cut into 27 identical smaller cubes. How many small cubes have exactly one face painted?
Correct answer: C. 6
One painted face means the centre of each face: 6 × (n − 2)² = 6 × 1 = 6.
easy - 4
A cube painted red on all faces is cut into 27 identical smaller cubes. How many small cubes have no face painted?
Correct answer: A. 1
The unpainted cubes form the inner core: (n − 2)³ = 1³ = 1.
easy - 5
A cube painted red on all faces is cut into 64 identical smaller cubes. How many small cubes have exactly two faces painted?
Correct answer: A. 24
Here n = 4 (4 × 4 × 4 = 64). Two faces painted: 12 × (n − 2) = 12 × 2 = 24.
medium - 6
A cube painted red on all faces is cut into 64 identical smaller cubes. How many small cubes have exactly one face painted?
Correct answer: B. 24
Here n = 4. One face painted: 6 × (n − 2)² = 6 × 4 = 24.
medium - 7
A cube painted red on all faces is cut into 64 identical smaller cubes. How many small cubes have no face painted?
Correct answer: C. 8
Here n = 4. No face painted: (n − 2)³ = 2³ = 8.
medium - 8
A cube painted red on all faces is cut into 125 identical smaller cubes. How many small cubes have exactly two faces painted?
Correct answer: D. 36
Here n = 5 (5 × 5 × 5 = 125). Two faces painted: 12 × (n − 2) = 12 × 3 = 36.
medium - 9
A cube painted red on all faces is cut into 125 identical smaller cubes. How many small cubes have exactly one face painted?
Correct answer: A. 54
Here n = 5. One face painted: 6 × (n − 2)² = 6 × 9 = 54.
medium - 10
A cube painted red on all faces is cut into 125 identical smaller cubes. How many small cubes have no face painted?
Correct answer: B. 27
Here n = 5. No face painted: (n − 2)³ = 3³ = 27.
medium - 11
A cube painted red on all faces is cut into 216 identical smaller cubes. How many small cubes have exactly one face painted?
Correct answer: C. 96
Here n = 6 (6 × 6 × 6 = 216). One face painted: 6 × (n − 2)² = 6 × 16 = 96.
hard - 12
A cube painted red on all faces is cut into 216 identical smaller cubes. How many small cubes have no face painted?
Correct answer: D. 64
Here n = 6. No face painted: (n − 2)³ = 4³ = 64.
hard - 13
A cube painted red on all faces is cut into 216 identical smaller cubes. How many small cubes have exactly two faces painted?
Correct answer: A. 48
Here n = 6. Two faces painted: 12 × (n − 2) = 12 × 4 = 48.
hard - 14
A cube painted red on all faces is cut into 125 identical smaller cubes. How many small cubes have at least one face painted?
Correct answer: B. 98
Total 125 minus the 27 unpainted inner cubes (3³) = 98.
hard - 15
What is the minimum number of straight cuts needed to cut a cube into 27 identical smaller cubes?
Correct answer: C. 6
Two cuts in each of the three directions divide every edge into 3 parts: 3 × 2 = 6 cuts.
medium - 16
What is the minimum number of straight cuts needed to cut a cube into 64 identical smaller cubes?
Correct answer: D. 9
Each edge must be split into 4 parts, which needs 3 cuts in each of the three directions: 3 × 3 = 9 cuts.
medium - 17
How many edges does a cube have?
Correct answer: A. 12
A cube has 6 faces, 12 edges and 8 vertices (corners).
easy - 18
On a standard die, the numbers on opposite faces always add up to 7. Which number is opposite 3?
Correct answer: C. 4
3 + 4 = 7. The opposite pairs are 1–6, 2–5 and 3–4.
easy - 19
On a standard die, what is the sum of the numbers on all six faces?
Correct answer: D. 21
1 + 2 + 3 + 4 + 5 + 6 = 21, which is also three opposite pairs of 7.
easy - 20
If the sum of the numbers on two opposite faces of a standard die is always 7, what is the total of the three faces you cannot see when the die rests on a table showing 2 on top, 1 in front and 3 on the right?
Correct answer: A. 15
The hidden faces are opposite the visible ones: 5 (opposite 2), 6 (opposite 1) and 4 (opposite 3); 5 + 6 + 4 = 15.
medium