In this guide
Dice and cube questions feel hard because you can't hold a die in the exam hall. You don't need to. Almost every dice question asks one thing, which faces are opposite, and almost every cube question is a counting formula. Learn five rules and the topic becomes routine.
How they appear in SSC CGL
These questions come in the General Intelligence and Reasoning section of Tier 1 and the Reasoning module of Tier 2. The count varies from shift to shift. The formats:
| Format | What you are shown |
|---|---|
| Two or more views of one die | Find the face opposite a given number |
| Open net | A flat unfolded pattern; find opposite faces or the folded cube |
| Painted cube | A cube painted and cut; count small cubes by painted faces |
| Painted cuboid | The same with unequal sides |
| Colour-painted cube | Different colours on different faces; count by colour |
Since you can't see the pictures here, every view below is written as top, front, right: the three faces you see when you look at a die from above, in front and a little to the right.
Rule 1: the standard die
On a standard die, opposite faces add up to 7: 1–6, 2–5 and 3–4. Use this only when the question says the die is standard or the views clearly follow it. Many questions use non-standard dice on purpose.
Rule 2: faces seen together are never opposite
Any two faces visible in the same view share an edge. Opposite faces can never be seen together. This rule alone often knocks out two options.
Rule 3: one common face
If two views share one face in the same position, the faces in matching positions around it are opposite.
Worked example 1. View 1: top 1, front 2, right 3. View 2: top 1, front 4, right 5.
- Face 1 is on top in both. The die has only been turned about the vertical axis.
- A quarter turn one way would bring 3 to the front. A quarter turn the other way would bring 2 to the right. Neither happened, so it is a half turn, and each side face has moved to the opposite side.
- So 2 is opposite 4 and 3 is opposite 5.
- The only faces left are 1 and 6, so 1 is opposite 6.
Rule 4: two common faces
If two views share two faces, the third faces in the two views are opposite each other.
Worked example 2. View 1: top 6, front 2, right 4. View 2: top 2, front 6, right 5.
Faces 6 and 2 appear in both views, so they share an edge. Exactly two faces touch both 6 and 2, and those two faces must be opposite each other (they sit on either end of that edge). Here they are 4 and 5. So 4 is opposite 5.
Notice what this does not tell you. With these two views alone, the other pairs are not fixed. If the options ask about 1 or 3, you need another view.
Rule 5: the face adjacent to four others
If one face has been seen next to four different numbers across all the views, it touches all four side faces, so the one number it has never been seen with is opposite it.
Worked example 3. View 1: top 2, front 1, right 3. View 2: top 3, front 5, right 2. View 3: top 5, front 6, right 2. Find all three opposite pairs.
- Face 2 appears in all three views, next to 1, 3, 5 and 6. The only number never seen with it is 4. So 2 is opposite 4.
- Views 2 and 3 share two faces, 5 and 2. By Rule 4, their third faces, 3 and 6, are opposite. So 3 is opposite 6.
- The remaining pair is 1 and 5.
Dice nets
A net is the flat, cross-like pattern that folds into a cube. Two rules find the opposite faces:
- In any straight line of squares, faces with exactly one square between them are opposite (first and third, second and fourth).
- In a Z shape of four squares (two squares in one row, the next two stepped down by one), the two end squares are opposite.
Worked example 4. A net has a row of four squares, 1-2-3-4, left to right. Square 5 sits above square 2 and square 6 sits below square 2. In the row, 1 and 3 are one apart, and so are 2 and 4. Squares 5 and 6 are in the same column with 2 between them. Opposite pairs: 1–3, 2–4 and 5–6.
Worked example 5. A net has a row of four squares, 2-3-4-5. Square 1 sits above the first square (2), and square 6 sits below the last square (5). In the row: 2–4 and 3–5. The two end flaps, 1 and 6, fold over to become the top and bottom. Opposite pairs: 2–4, 3–5 and 1–6.
Painted cubes
A cube is painted on all six faces and then cut into n × n × n equal small cubes.
| Small cubes with | Where they are | Number |
|---|---|---|
| Three faces painted | Corners | 8 |
| Two faces painted | Edges, not corners | 12(n − 2) |
| One face painted | Face centres | 6(n − 2)² |
| No face painted | Hidden inside | (n − 2)³ |
| At least one face painted | Everything but the inside | n³ − (n − 2)³ |
Worked example 6. A painted cube is cut into 64 equal small cubes, so n = 4.
- Three faces: 8.
- Two faces: 12 × 2 = 24.
- One face: 6 × 2² = 24.
- None: 2³ = 8.
- Check: 8 + 24 + 24 + 8 = 64.
If the four counts don't add to n³, one formula was misapplied.
Painted cuboids
For a painted cuboid cut into a × b × c small cubes (each side at least 2):
- Three faces: 8.
- Two faces: 4[(a − 2) + (b − 2) + (c − 2)].
- One face: 2[(a − 2)(b − 2) + (b − 2)(c − 2) + (c − 2)(a − 2)].
- None: (a − 2)(b − 2)(c − 2).
Worked example 7. A painted cuboid is cut into 5 × 4 × 3 = 60 small cubes. Here a − 2 = 3, b − 2 = 2 and c − 2 = 1.
- Two faces: 4(3 + 2 + 1) = 24.
- One face: 2(3 × 2 + 2 × 1 + 1 × 3) = 2 × 11 = 22.
- None: 3 × 2 × 1 = 6.
- Check: 8 + 24 + 22 + 6 = 60.
Different colours on different faces
Suppose a cube has red on top and bottom, blue on front and back, and green on left and right, and is cut into 27 cubes (n = 3).
- Cubes with red and blue only lie on the four edges where a red face meets a blue face, excluding corners: 4 edges × (n − 2) = 4.
- Cubes with red only are the centre cube of each red face: 2 × (n − 2)² = 2.
- Every corner cube shows all three colours: 8.
Common mistakes
| Mistake | Fix |
|---|---|
| Assuming a die is standard | Use "sum 7" only if stated or clearly consistent |
| Reading faces in a different order in each view | Fix one convention (top, front, right) and stick to it |
| Using Rule 4 to settle every pair | Two common faces fix only one pair |
| Using 12n or 6n² in painted cubes | Use (n − 2): drop the corner cube at each end |
| Skipping the total check | The counts must add to n³ or abc |
Practice
- On a standard die, which number is opposite 2?
- View 1: top 3, front 1, right 2. View 2: top 3, front 4, right 6. Which number is opposite 1?
- A painted cube is cut into 125 equal cubes. How many have exactly two painted faces?
- In the same case, how many have no painted face?
- A painted cube is cut into 216 equal cubes. How many have only one painted face?
- A net is a staircase: squares A and B side by side in the top row; C below B with D to its right in the middle row; E below D with F to its right in the bottom row. Which square is opposite A?
- A painted cuboid is cut into 4 × 3 × 2 small cubes. How many have exactly two painted faces?
- A cube painted on all faces is cut into 27 cubes. How many have at least one painted face?
Answers
- 5. 2 + 5 = 7.
- 4. Face 3 is on top in both views. A quarter turn would put 2 in front or 1 on the right; neither happened, so it is a half turn. 1 is opposite 4 (and 2 is opposite 6).
- 36. n = 5, so 12 × 3 = 36.
- 27. 3³ = 27.
- 96. n = 6, so 6 × 4² = 96.
- D. A, B, C, D form a Z, so the ends A and D are opposite. By the same rule, B–E and C–F.
- 12. 4[(2) + (1) + (0)] = 12. Check: 8 corners + 12 + one-face 4 + inside 0 = 24.
- 26. 27 − 1³ = 26. Only the centre cube is unpainted.
What to do next
- Write 1 to 6 on the faces of any small box and check Rules 3, 4 and 5 by hand once.
- Memorise the painted-cube table and always finish with the total check.
- Solve 10 dice and 10 cube questions from past papers, writing every view as top, front, right before you think.
- Then read non-verbal reasoning and counting figures, the other two visual topics.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
Get the next SSC CGL guide by email
New guides every week. No spam, unsubscribe any time.