Cubes and Dice Notes for SSC Exams

Cubes and dice questions are of two kinds: a painted cube cut into small cubes, and a dice shown in two or three positions for which you find the opposite faces. Both kinds use simple rules. These cubes and dice notes for SSC give the formulas, the opposite-face rules and solved examples.

Facts About a Cube

Painted Cube Cut Into Small Cubes

A cube painted on all faces is cut into n × n × n equal small cubes (n equal parts on each edge).

The total number of small cubes is n × n × n. For n = 3 the total is 8 + 12 + 6 + 1 = 27.

The minimum number of cuts to make n × n × n small cubes from a big cube is 3(n − 1). For 27 cubes it is 6 cuts, and for 64 cubes it is 9 cuts.

Opposite Faces of a Dice

• On a standard dice, the opposite faces add up to 7: 1 and 6, 2 and 5, 3 and 4.
• If two faces are common in two positions of a dice, the remaining faces of the two positions are opposite to each other.
• A face which is seen next to another face cannot be opposite to it.

Example 1. Position I shows 1, 2 and 3. Position II shows 1, 3 and 5. Which face is opposite to 2?
The faces 1 and 3 are common. The remaining faces are 2 and 5, so 5 is opposite to 2.

Example 2. Three positions show (1, 2, 3), (1, 4, 2) and (6, 2, 3). Find the opposite pairs.
From the first two positions, 1 and 2 are common, so 3 and 4 are opposite. From the first and third positions, 2 and 3 are common, so 1 and 6 are opposite. The last pair is 2 and 5. The pairs are 3–4, 1–6 and 2–5.

Opposite Faces in an Unfolded Cube (Net)

• In a row of four faces, the first and the third faces are opposite, and the second and the fourth faces are opposite.
• Two faces that are separated by exactly one face in a row or a column are opposite.
• Faces that are next to each other in the net are always adjacent on the cube.

Example 3. A net has a row of four faces A, B, C and D. A face E is above C and a face F is below C. Which face is opposite to E?
E and F are separated by C, so F is opposite to E. Also A is opposite to C, and B is opposite to D.

More Solved Examples

Example 4. A cube is painted on all sides and cut into 27 equal small cubes. How many cubes have exactly 3 painted faces?
The corners of a cube are 8, so the answer is 8.

Example 5. A cube is painted on all sides and cut into 64 equal small cubes. How many have exactly 2 painted faces?
12 × (4 − 2) = 24.

Quick Tips

• Learn the painted cube table by heart for n = 3, 4 and 5.
• Use the common faces rule for dice, and the "one face in between" rule for nets.
• On a standard dice, the opposite faces add up to 7.
• Draw a rough cube and mark the faces to avoid confusion.

Quick Revision Points

• A cube has 6 faces, 12 edges and 8 vertices.
• 3 painted faces = 8, 2 faces = 12(n−2), 1 face = 6(n−2) squared, 0 faces = (n−2) cubed.
• Cuts needed = 3(n − 1).
• Two common faces mean the remaining faces are opposite.
• In a row of faces, the first and the third are opposite.

Practice Questions With Answers

1. How many faces, edges and vertices does a cube have?
   Answer: 6 faces, 12 edges and 8 vertices
2. What is opposite to 1 on a standard dice?
   Answer: 6
3. A cube painted on all faces is cut into 27 equal small cubes. How many have exactly 3 painted faces?
   Answer: 8
4. In the same 27 cubes, how many have exactly 2 painted faces?
   Answer: 12
5. In the same 27 cubes, how many have exactly 1 painted face?
   Answer: 6
6. In the same 27 cubes, how many have no painted face?
   Answer: 1
7. A painted cube is cut into 64 small cubes. How many have exactly 2 painted faces?
   Answer: 24
8. What is the minimum number of cuts to make 27 small cubes from a big cube?
   Answer: 6
9. In a net with a row of four faces A, B, C and D, which face is opposite to B?
   Answer: D
10. Two positions of a dice show (1, 2, 3) and (1, 3, 5). Which face is opposite to 2?
    Answer: 5

How to Practise This Chapter

Practise figure questions daily on paper, because you must see the figures to learn them. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Non Verbal Reasoning notes page, and also read the Reasoning notes and Verbal Reasoning notes. Read the previous topic, Embedded Figures, and also try the Mathematics notes for solid shapes.

Frequently Asked Questions

How many faces, edges and corners does a cube have?

A cube has 6 faces, 12 edges and 8 corners.

What is the sum of the opposite faces on a standard dice?

It is 7, for example 1 and 6, 2 and 5, 3 and 4.

How many small cubes have 3 painted faces?

Always 8, the corners of the cube.

How many small cubes have 2 painted faces?

It is 12 times (n minus 2) for an n by n by n cube.

How many small cubes have 1 painted face?

It is 6 times (n minus 2) squared.

How many small cubes have no painted face?

It is (n minus 2) cubed.

How many cuts are needed for 27 small cubes?

At least 6 cuts, as 3 times (3 minus 1) is 6.

How do I find the opposite faces in a net?

Faces separated by exactly one face in a row or column are opposite.

How do I find the opposite faces from two dice positions?

If two faces are common, the remaining face of each position are opposite.

Is this topic asked in SSC exams?

Yes. Cubes and dice are asked often in SSC reasoning papers.

How much time should I take for one question?

About 40 seconds.

How many questions come from this topic in SSC exams?

Usually one or two questions, but the number changes from year to year.