Chapter 5 · Topic 5.2 · Prelims + Mains2 of 3
General Mental Ability: Cubes: cuts, painted faces and weighing
Key points
- Cuts: n parts along an edge need n − 1 cuts; 64 = 4 × 4 × 4 needs 3 + 3 + 3 = 9 cuts (CSAT 2024, b).
- Painted n × n × n cube: 3 faces = 8 corners; 2 faces = 12 × (n − 2); 1 face = 6 × (n − 2)²; none = (n − 2)³.
- CSAT 2016: side one-fourth → 4 × 4 × 4 → one face painted = 24 (b). CSAT 2025: 60 = 3 × 4 × 5, 9 cuts, 6 unpainted (c).
- Weighing (CSAT 2026): 7 cubes, one lighter: 2 weighings (a).
2 PYQs 2009–2018
Prelims box: facts and traps
- Count corners, edges, faces, core; weigh in threes.
Cubes: cuts, painted faces and weighing
Like cutting a square cake for a party: the corner pieces get icing on three sides, the middle pieces get none.
The four kinds of small cube (n = number of parts along one edge):
- 3 faces painted: the 8 corners.
- 2 faces painted: each of the 12 edges has (n − 2) middle cubes, so 12 × (n − 2).
- 1 face painted: each of the 6 faces has an inner (n − 2) × (n − 2) square, so 6 × (n − 2)².
- No face painted: the inner core, (n − 2)³.
| Small cubes with | Count for a 4 × 4 × 4 cube | How to see it |
|---|---|---|
| 3 faces painted | 8 | The 8 corners |
| 2 faces painted | 24 | 12 edges × 2 middle cubes |
| 1 face painted | 24 | 6 faces × inner 2 × 2 |
| No face painted | 8 | Inner 2 × 2 × 2 |
Battle IAS worked example (total 8 + 24 + 24 + 8 = 64). Check: the four counts add to n³.
CSAT 2024, Q5: What is the least possible number of cuts required to cut a cube into 64 identical pieces?
- 64 = 4 × 4 × 4: each direction has 4 parts, needing 3 cuts. Three directions: 3 + 3 + 3 = 9.
- Answer (b).
What is the least possible number of cuts required to cut a cube into 64 identical pieces?
- 8
- 9
- 12
- 16
Show answer
Answer: (b) · UPSC official key
CSAT 2016, Q59: A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is:
- Side one-fourth means 4 × 4 × 4.
- On each face, the inner 2 × 2 = 4 cubes have only that face painted.
- 6 faces × 4 = 24. Answer (b).
A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is :
- 32
- 24
- 16
- 8
Show answer
Answer: (b) · UPSC official key
CSAT 2025, Q15: A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which statements are correct?
- Spread the cuts across the three directions: 60 = 3 × 4 × 5.
- Cuts = 2 + 3 + 4 = 9, so statement I is correct.
- Unpainted pieces = (3 − 2) × (4 − 2) × (5 − 2) = 1 × 2 × 3 = 6, so statement II is correct.
- Answer (c).
A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct? I. The minimum number of cuts is 9. II. The number of smaller pieces which are not painted on any face is 6. Select the correct answer using the code given below:
- I only
- II only
- Both I and II
- Neither I nor II
Show answer
Answer: (c) · UPSC official key
CSAT 2026, Q51: Seven cubes are identical in shape. Six weigh the same and one is lighter. Using a balance, what is the minimum number of attempts needed to find the odd cube with certainty?
- Put 3 cubes on each pan.
- If they balance, the 7th cube is the light one (1 weighing).
- If not, take the lighter three and weigh 1 against 1. The lighter pan holds it; if they balance, it is the cube left out.
- So 2 weighings always work. Answer (a).
Seven cubes are identical in shape. Out of these, the weight of each of the six cubes is equal and the weight of the remaining cube is less than the weight of any other cube. A balance is used to identify the lightest cube. What is the minimum number of attempts required to distinguish the odd cube with certainty?
- 2
- 3
- 4
- 1
Show answer
Answer: (a) · UPSC provisional key (final key awaited)
What UPSC asks: Prelims (CSAT): n − 1 cuts; corners, edges, faces, core; unequal cuts like 3 × 4 × 5. Next likely angle: odd-coin weighing, always in groups of three.
Mains box: analysis and answer points
- Not a Mains topic; CSAT reasoning only.
Practice
Practice MCQs (8)
A cube painted on all faces is cut into 27 identical smaller cubes. How many smaller cubes have paint on exactly one face?
Answer (c) & explanation
Step 1: 27 = 3 × 3 × 3, so each edge is cut into 3 parts. Step 2: A cube with exactly one painted face sits in the centre of a face, away from all edges. Step 3: Each face of a 3 × 3 × 3 cube has exactly one such centre square, and there are 6 faces, so the answer is 6.
A solid cube is painted on all its faces and then cut into 64 identical smaller cubes. How many of the smaller cubes have paint on exactly two faces?
Answer (a) & explanation
Step 1: 64 small cubes means each edge is cut into 4 parts (4 × 4 × 4 = 64). Step 2: A small cube has exactly two painted faces only if it lies on an edge of the big cube but is not a corner. Step 3: Each edge has 4 small cubes, of which 2 are corners, leaving 4 − 2 = 2 middle cubes. Step 4: A cube has 12 edges, so 12 × 2 = 24.
In a class of 40 students, 25 like tea, 20 like coffee and 8 like neither. How many students like both tea and coffee?
Answer (a) & explanation
Step 1: Students who like at least one drink = 40 − 8 = 32. Step 2: Adding tea and coffee lovers counts the 'both' group twice: 25 + 20 = 45. Step 3: The extra over 32 is the double count: 45 − 32 = 13. So 13 students like both.
In a group of 60 people, 35 read a Hindi newspaper, 30 read an English newspaper and 12 read both. How many people read only the English newspaper?
Answer (c) & explanation
Step 1: The 30 English readers include the 12 who read both. Step 2: Only-English readers = 30 − 12 = 18. Step 3: As a cross-check, only-Hindi = 35 − 12 = 23, and 23 + 12 + 18 = 53, which is within 60, leaving 7 who read neither.
A cube is painted red on two opposite faces and blue on the remaining four faces. It is then cut into 64 identical smaller cubes. How many smaller cubes have both red and blue paint on them?
Answer (b) & explanation
Step 1: 64 = 4 × 4 × 4. A small cube shows both colours only if it touches a red face and a blue face, that is, it lies on the border of a red face. Step 2: The border of a 4 × 4 face has 4 + 4 + 2 + 2 = 12 small cubes. Step 3: There are two red faces, and their borders do not overlap (they are on opposite sides). Step 4: Total = 2 × 12 = 24.
A solid block measuring 4 cm × 3 cm × 2 cm is painted on all its faces and then cut into 1 cm cubes. How many of these cubes have paint on exactly one face?
Answer (a) & explanation
Step 1: A cube with exactly one painted face lies inside a face, away from that face's border. Step 2: On a face of size p × q, such cubes form a (p − 2) × (q − 2) patch. Step 3: The two 4 × 3 faces give 2 × 1 = 2 each, that is 4. The 4 × 2 faces give 2 × 0 = 0 and the 3 × 2 faces give 1 × 0 = 0. Step 4: Total = 4 + 0 + 0 = 4.
In a town, 70% of the residents like cricket and 60% like football. Every resident likes at least one of the two games. What percentage of the residents like both games?
Answer (c) & explanation
Step 1: Since everyone likes at least one game, the union is 100%. Step 2: Adding the two percentages counts the 'both' group twice: 70 + 60 = 130. Step 3: The excess over 100 is the overlap: 130 − 100 = 30%.
In a survey of 200 people, 90 read newspaper X, 80 read Y and 70 read Z. 30 read both X and Y, 25 read both Y and Z, 20 read both X and Z, and 10 read all three. Consider the following statements:
1. 25 people read none of the three newspapers.
2. 45 people read exactly two newspapers.
3. 130 people read exactly one newspaper.
4. 50 people read only X.
How many of the above statements are correct?
Answer (c) & explanation
Step 1: Union = 90 + 80 + 70 − 30 − 25 − 20 + 10 = 175, so none = 200 − 175 = 25: statement 1 is correct. Step 2: Exactly two = (30 + 25 + 20) − 3 × 10 = 45: statement 2 is correct. Step 3: Exactly one = 175 − 45 − 10 = 120, not 130: statement 3 is wrong. Step 4: Only X = 90 − 30 − 20 + 10 = 50: statement 4 is correct. Step 5: Three statements are correct.
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Prelims PYQs on this topic (2)
A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?
Answer (a) & explanation
12 small cubes have exactly two painted faces (a). Step 1: a cube with exactly two painted faces sits on an edge of the big cube, but not at a corner. Step 2: the 3 cm cube is cut into 3 × 3 × 3 = 27 small cubes. Each edge has 3 small cubes. The 2 at the ends are corners (three painted faces), so 1 in the middle has two. Step 3: a cube has 12 edges, so 12 × 1 = 12. Option (b), 8, is the corner count. Option (c), 6, is the number with one painted face. Option (d), 4, does not fit.
Tip: for an n-cube, two-face cubes = 12 × (n − 2). For n = 3 that is 12.
Answer as per the official UPSC answer key.
The outer surface of a 4 cm × 4 cm × 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm × 1 cm × 1 cm small cubes. How many small cubes do not have painted faces?
Answer (a) & explanation
8 small cubes have no painted face (a). Step 1: the 4 × 4 × 4 cube is cut into 64 small cubes. Only the outer layer is painted. Step 2: the unpainted cubes form an inner block. Remove one layer from each side: 4 − 2 = 2 cubes along each edge. Step 3: the inner block is 2 × 2 × 2 = 8 cubes. Option (c), 24, is the number with exactly one painted face. Option (b), 16, and option (d), 36, do not fit.
Tip: for an n × n × n painted cube, unpainted = (n − 2)³, one face = 6(n − 2)², two faces = 12(n − 2), and three faces = 8. Learn these four formulas.
Answer as per the official UPSC answer key.
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In 150 words, explain the significance of General Mental Ability with one recent example.
Flashcards (5)
Cuts to make 4 parts along one edge?
3.
Cuts for 64 identical cubes?
9.
In a 4 × 4 × 4 cube, cubes with one face
painted? 24.
Cubes with 3 faces painted, any size?
8 (the corners).
Fewest weighings to find a lighter cube
among 7? 2.
Next topic · 5.3 →General Mental Ability: Venn diagrams: both, neither and exactly one