Cubes and Dice Notes for Reasoning

Cubes and Dice are two spatial reasoning topics that test how well you can imagine a three-dimensional object from a flat description. Questions on painted cubes and dice are common in SSC CGL, CHSL, railway exams and defence exams, and they appear in UPSC CSAT as well: CSAT has asked about cubes painted on their faces and cut into smaller cubes, as well as about the arrangement of numbers on dice. These chapters look tricky because you cannot see the object, but a few formulas and two or three dice rules are enough to solve almost every question in the exam.

Quick Facts at a Glance

• A cube has 6 faces, 12 edges and 8 corners (vertices).
• If a painted cube is cut into n × n × n small cubes, total small cubes = n³.
• Cubes with 3 faces painted = 8 (the corners), for n ≥ 2.
• Cubes with 2 faces painted = 12(n − 2).
• Cubes with 1 face painted = 6(n − 2)².
• Cubes with no face painted = (n − 2)³.
• In a standard die, opposite faces add up to 7.

Part A: Cubes

1. Painted Cube Formulas

Suppose a large cube is painted on all six faces and then cut into n³ identical small cubes (n cuts per edge means n pieces along each edge).

Check: 8 + 24 + 24 + 8 = 64. Always verify that the parts add up to n³.

Example: A cube of side 5 cm is painted on all faces and cut into 1 cm cubes. Here n = 5. Total = 125. Three faces painted = 8; two faces = 12 × 3 = 36; one face = 6 × 9 = 54; no face = 27.

2. Cubes Painted With Different Colours

Questions often paint opposite faces with the same colour or adjacent faces with different colours. Example: A cube is painted red on two opposite faces, blue on two opposite faces and green on the remaining two, then cut into 64 small cubes (n = 4).

• Cubes with all three colours = 8 corners (each corner touches one face of each pair).
• Cubes with only red and blue = the edges where red and blue faces meet, excluding corners: there are 4 such edges, each with n − 2 = 2 cubes, so 8 cubes.
• Cubes with only one colour, say red = the two red faces' centres: 2 × (n − 2)² = 8.

3. Cubes Painted on Some Faces Only

If not all faces are painted, count carefully using the structure. For instance, if only the top and bottom faces are painted and n = 3, then cubes with exactly one face painted = 2 × 9 = 18, and cubes with no paint = 9 (the middle layer). Draw the layers if needed.

4. Cutting a Cube With a Fixed Number of Cuts

• With k cuts, the maximum number of pieces is achieved by distributing cuts as evenly as possible among the three directions.
• Example: 6 cuts (2 in each direction) produce 3 × 3 × 3 = 27 pieces.
• Example: 7 cuts (3, 2, 2) produce 4 × 3 × 3 = 36 pieces.

Part B: Dice

5. Types of Dice

• Standard die: The sum of numbers on opposite faces is always 7 (1-6, 2-5, 3-4).
• General die: Opposite faces do not follow a fixed sum. You must work them out from the given positions.

6. Rules to Find Opposite Faces

Rule 1: One common face. If two positions of the same die show one face in common, rotate both views so the common face is in the same position (usually written first) and read the other faces in clockwise order. The faces at corresponding positions are opposite each other.

Rule 2: Two common faces. If two positions show two faces in common, the remaining third faces are opposite each other.

Rule 3: Adjacent faces cannot be opposite. Any two faces that appear together in a single view of the die are adjacent, so they cannot be opposite.

Example (two common faces): Position 1 shows 2, 3, 6; position 2 shows 2, 3, 5. Faces 2 and 3 are common, so 6 is opposite 5.

Example (adjacency): Three views of a die show (1, 2, 3), (1, 4, 5) and (2, 4, 6). Face 1 is adjacent to 2, 3, 4 and 5, so 1 must be opposite 6. Face 2 is adjacent to 1, 3, 4 and 6, so 2 is opposite 5. The remaining pair, 3 and 4, are opposite.

7. Open Dice (Unfolded Nets)

Sometimes a die is shown as a flat net of six squares. To find opposite faces in a net:

• In a straight row of squares, alternate squares are opposite (first and third, second and fourth).
• Two squares that touch each other along an edge are always adjacent, never opposite.
• In a standard "cross" net, the top and bottom squares of the vertical column are opposite, and the two side squares of the horizontal row are opposite.

Example: A net shows a row A, B, C, D with E above B and F below B. A and C are opposite; B and D are opposite; E and F are opposite.

8. Cubes and Dice in UPSC CSAT

CSAT questions usually test the painted-cube formulas or ask which face is opposite a given face, often with a small twist such as different colours on different faces. Since CSAT has negative marking, draw a quick sketch of the cube or list the adjacent faces before answering. These questions are fast to solve once the formulas are remembered, so they are worth attempting.

Common Mistakes to Avoid

1. Confusing the number of cuts with the number of pieces per edge.
2. Forgetting that corner cubes have three painted faces, not two.
3. Assuming every die is a standard die where opposite faces sum to 7.
4. Calling two faces opposite even though they appear together in one view.
5. Not checking that all painted-cube categories add up to n³.

Preparation Strategy

• Memorise the four painted-cube formulas and test them for n = 3, 4 and 5.
• Practise 15 painted-cube questions with different colour patterns.
• Practise 15 dice questions, including at least five open-net questions.
• Use a real die or make a paper cube to visualise faces; this helps more than any formula.

Practice Questions with Answers

1. A painted cube is cut into 27 small cubes. How many have exactly two faces painted? Answer: 12 × 1 = 12.
2. A painted cube is cut into 216 small cubes. How many have no face painted? Answer: n = 6, so 4³ = 64.
3. In a standard die, which number is opposite 3? Answer: 4.
4. Two views of a die show (A, B, C) and (A, B, D). Which face is opposite C? Answer: D.

Cuboids: Extending the Formulas

Exams sometimes use a cuboid instead of a cube. If a painted cuboid is cut into l × b × h small cubes, the formulas become:

• Three faces painted = 8 (corners).
• Two faces painted = 4[(l − 2) + (b − 2) + (h − 2)].
• One face painted = 2[(l − 2)(b − 2) + (b − 2)(h − 2) + (l − 2)(h − 2)].
• No face painted = (l − 2)(b − 2)(h − 2).

Example: A painted cuboid is cut into 4 × 3 × 3 = 36 small cubes. No face painted = 2 × 1 × 1 = 2. Two faces painted = 4(2 + 1 + 1) = 16. One face painted = 2(2 + 1 + 2) = 10. Three faces = 8. Total = 2 + 16 + 10 + 8 = 36 ✓.

Thinking in Layers

When a question confuses you, slice the big cube into horizontal layers. For n = 4, there are four layers of 16 small cubes each. The top and bottom layers have paint on one large face, while the middle two layers have paint only on their outer ring. Counting layer by layer is slower than formulas but almost impossible to get wrong, and it is especially useful for unusual painting patterns where formulas do not apply directly.

Related Notes

• Analogy, Classification and Odd One Out Notes
• Direction Sense and Seating Arrangement Notes
• Mensuration Notes: Area, Volume and Perimeter
• All Reasoning Notes

Frequently Asked Questions

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

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

How many small cubes have three faces painted?

Always 8, because only the corner cubes have three painted faces (for n of 2 or more).

What is the formula for small cubes with two faces painted?

12(n − 2), where n is the number of small cubes along one edge.

What is the formula for small cubes with one face painted?

6(n − 2)².

What is the formula for small cubes with no face painted?

(n − 2)³.

What is a standard die?

A die in which the numbers on opposite faces always add up to 7.

How do you find the opposite face when two views share two faces?

The remaining third faces in the two views are opposite each other.

Can two faces seen together be opposite?

No. Faces visible together in one view are adjacent, so they cannot be opposite.

How do you find opposite faces in an open die net?

In a straight row, alternate squares are opposite. Squares that share an edge are adjacent.

How many pieces do 6 cuts make in a cube?

At most 27 pieces, using 2 cuts in each of the three directions.

Are cube and dice questions asked in UPSC CSAT?

Yes. CSAT has asked painted-cube and dice-type questions in several years.

How can I visualise cubes better?

Make a paper cube or use a real die and practise rotating it while solving questions.