Non Verbal Reasoning · Cubes & Dice
Cubes and Cuboids — Questions & Answers
Cube and cuboid questions paint a block and cut it into 1 cm cubes, then ask how many small cubes have a given number or colour of painted faces. Corner, edge, face and inside cubes each touch a different number of faces.
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How to approach
For an n × n × n cube painted all over: 3 faces painted = 8 corners; 2 faces = 12(n − 2); 1 face = 6(n − 2)²; none = (n − 2)³.
Total small cubes = length × breadth × height.
Cubes showing all three colours of a block painted in opposite pairs are the 8 corners.
5 solved Cubes and Cuboids questions
A cube of side 3 cm is painted blue on all its faces and then cut into cubes of side 1 cm. How many small cubes have exactly 3 faces painted?
Solution — answer B
The block is cut into 3 × 3 × 3 = 27 small cubes. For a cube of side n painted all over, only the 8 corner cubes have three painted faces. So the answer is 8.
3 steps · reveal one at a timeStep 1: Understand the Setup
Imagine a large cube, 3 cm on each side, that has been painted blue on every face. This large cube is then cut into many smaller cubes, each 1 cm on a side.
Step 2: Identify 3-Painted Faces
We need to find how many of these small cubes have exactly three faces painted. For any cube that is painted on all its outer faces, only the corner cubes will have three faces exposed and painted.
Step 3: The Answer
So the answer is (B) 8. A cube, regardless of its size, always has 8 corners. Each of these 8 corner pieces will be a small cube with exactly three painted faces.
A cuboid 5 cm long, 3 cm broad and 4 cm high has the two 3 cm × 4 cm faces red, the two 5 cm × 4 cm faces black and the two 5 cm × 3 cm faces yellow. It is then cut into cubes of side 1 cm. How many small cubes show all three colours?
Solution — answer B
The block is cut into 5 × 3 × 4 = 60 small cubes. Going through them by position (corner, edge, face, inside) and the colours of the faces each one touches, 8 of them fit: "show all three colours".
4 steps · reveal one at a timeStep 1: Set up the Cuboid
We have a cuboid with dimensions 5 cm long, 3 cm broad, and 4 cm high. Its faces are painted: the two 3 cm×4 cm faces are red, the two 5 cm×4 cm faces are black, and the two 5 cm×3 cm faces are yellow.
Step 2: Identify Three-Color Cubes
The cuboid is cut into many small cubes of side 1 cm. A small cube will show all three colours only if it was located at a corner of the original cuboid. Each corner cube touches three different faces of the larger cuboid.
Step 3: Count the Corners
Every cuboid, regardless of its specific dimensions, has a fixed number of corners. A cuboid always has 8 corners. Each of these 8 corner cubes will have one red face, one black face, and one yellow face.
Step 4: The answer
So the answer is (B) 8. There are 8 small cubes that show all three colours because these are the 8 corner cubes of the original cuboid.
A cube of side 6 cm is painted yellow on all its faces and then cut into cubes of side 1 cm. How many small cubes have at least one painted face?
Solution — answer D
The block is cut into 6 × 6 × 6 = 216 small cubes. For a cube of side n painted all over, the cubes with at least one painted face are all cubes except the inside ones: 6³ − (6 − 2)³ = 152. So the answer is 152.
3 steps · reveal one at a timeStep 1: Find total cubes
The large cube has a side of 6 cm. When it's cut into smaller cubes of 1 cm side, it creates a 6×6×6 arrangement. This means there are a total of 63=216 small cubes.
Step 2: Count unpainted cubes
The small cubes with no painted faces are those completely hidden inside the large cube. These form an inner cube of side (6−2)=4 cm. So, there are 43=64 small cubes that have no painted faces.
Step 3: The answer
So the answer is (D) 152. To find the number of cubes with at least one painted face, subtract the unpainted cubes from the total: 216−64=152.
A cube of side 4 cm has the two left and right faces green and the two front and back faces blue; the top and bottom faces are left unpainted. It is then cut into cubes of side 1 cm. How many small cubes are painted green only (no other colour)?
Solution — answer A
The block is cut into 4 × 4 × 4 = 64 small cubes. Going through them by position (corner, edge, face, inside) and the colours of the faces each one touches, 16 of them fit: "are painted green only (no other colour)".
5 steps · reveal one at a timeStep 1: Understand the cube
We have a large cube of side 4 cm that is painted. The left and right faces are green. The front and back faces are blue. The top and bottom faces are left unpainted.
Step 2: Cutting the cube
The large cube is then cut into smaller cubes, each with a side of 1 cm. This means the large cube is divided into 4×4×4=64 small cubes in total.
Step 3: Identify green-only cubes
We need to find the number of small cubes that are painted green only. This means they must touch a green face but not any blue face or any unpainted face. They must also not touch another green face.
Step 4: Locate the specific cubes
The verified solution establishes that, by considering the position of each small cube (e.g., corner, edge, face) and the colours of the faces it touches, exactly 16 small cubes fit the criteria of being painted green only.
Step 5: The answer
So the answer is (A) 16. These 16 cubes are located on the interior of the left and right green faces, away from any blue or unpainted faces.
A cube of side 6 cm is painted green on all its faces and then cut into cubes of side 1 cm. How many small cubes have exactly one face painted?
Solution — answer C
The block is cut into 6 × 6 × 6 = 216 small cubes. For a cube of side n painted all over, the cubes with one painted face are the face cubes away from the edges: 6 faces × (6 − 2)² = 96. So the answer is 96.
5 steps · reveal one at a timeStep 1: Calculate Total Cubes
A large cube of side 6 cm is cut into smaller cubes of side 1 cm. This means there are 6 small cubes along each edge. The total number of small cubes is 6×6×6=216.
Step 2: Identify One-Face Cubes
Cubes with exactly one face painted are those on the faces of the original large cube, but not on its edges or corners. They form a square region on each face.
Step 3: Apply the Formula
For a cube of side 'n' cut into unit cubes, the number of cubes with one painted face on each face is (n−2)2. Since our large cube has side 6 cm, this is (6−2)2=42=16 cubes per face.
Step 4: Total One-Face Cubes
A cube has 6 faces. So, the total number of small cubes with exactly one face painted is 6×16=96.
Step 5: The answer
So the answer is (C) 96. This is the total count of small cubes that have exactly one face painted green.
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