A solid cube has been formed with 64 smaller cubes. How many smaller cubes will not be visible in any condition ?
- (1)8
- (2)6
- (3)4
- (4)2
Answer
Why
Correct — option (1), 8.
Step 1 — find the size of the big cube.
64 = 4 × 4 × 4, so each edge is 4 small cubes long.
Step 2 — decide which small cubes can never be seen.
A small cube can be seen only if one of its faces lies on the outside of the big cube. Turning the cube round brings every outer face into view, so the cubes never visible are those with no outer face: the inner core.
Step 3 — size the core.
Removing the outer layer takes one cube off each end of every row: 4 − 2 = 2.
Core = 2 × 2 × 2 = 8
Step 4 — check by counting the outside.
Corners, 3 faces showing: 8
Edges without corners, 2 faces: 12 × 2 = 24
Face centres, 1 face: 6 × 2² = 24
Outside total: 8 + 24 + 24 = 56
64 − 56 = 8 ✓
The idea to remember: in an n × n × n cube, (n − 2)³ small cubes are hidden inside.
Why the others are wrong
- (2)6 — Option (2) gives 6, the number of faces of a cube, not a number of small cubes.
The hidden cubes are the ones with no face on the outside. Peeling one layer off a 4 × 4 × 4 cube leaves a 2 × 2 × 2 core, which holds 8 small cubes.
- (3)4 — Option (3) gives 4, the number of small cubes in the centre of one face: (4 − 2)² = 4.
Each of those shows one face on the outside, so it can be seen. The hidden core is a cube, not a square: (4 − 2)³ = 8.
- (4)2 — Option (4) gives 2, the number of cubes between the two corners on one edge: 4 − 2 = 2.
Each of those shows two faces, so it is visible. The hidden core takes that 2 in all three directions: 2 × 2 × 2 = 8.
Concept
A large cube built from n × n × n equal small cubes sorts its pieces by how many faces show on the outside.
Corners show 3 faces, and a cube has 8 corners. Edge cubes other than corners show 2 faces: 12 edges × (n − 2). Face-centre cubes show 1 face: 6 faces × (n − 2)². The inner core shows none: (n − 2)³.
The four counts add up to n³; for n = 4, 8 + 24 + 24 + 8 = 64. The same counts serve when the big cube is painted and "shows a face" becomes "has a painted face".
RPSC's 2023 syllabus lists "Shapes and their sub sections" under Mental Ability in Reasoning & Mental Ability.
The four counts are the terms of the expansion (m + 2)³ = m³ + 6m² + 12m + 8, with m = n − 2: core, face centres, edges and corners.
The same layer-by-layer reasoning gives the volume of a hollow box: outer volume minus inner volume, as 64 − 8 = 56 small cubes form the outer shell here.
Key facts
- 64 = 4³, so the big cube is 4 small cubes along each edge.
- Hidden small cubes in an n × n × n cube: (n − 2)³; for n = 4, that is 8.
- Small cubes showing 3 faces: the 8 corners, for any big cube with n ≥ 2.
- Small cubes showing exactly 2 faces: 12(n − 2); exactly 1 face: 6(n − 2)².
- For n = 4: 8 corners + 24 edge cubes + 24 face centres + 8 hidden = 64.
For an n × n × n cube the hidden core is (n − 2)³; here n = 4.
Study next
Common traps
- Taking the core as a square. The hidden part is a 2 × 2 × 2 cube of 8 pieces, not the 2 × 2 = 4 at the centre of one face.
- Removing one layer from only one end. The outer shell covers both ends of every row, so each row loses 2 cubes: 4 − 2 = 2.
- Treating the base as hidden for good. The stem's "in any condition" allows the cube to be turned, so the face it rests on can be seen too.
A question can build a cube from small cubes and ask how many are hidden, as this one does, or paint the big cube and ask how many small cubes have 0, 1, 2 or 3 painted faces.
A question can also use a cuboid, where each edge length enters the count separately.
Related PYQs
UnlockIAS compared this question with questions from other RAS Prelims papers and found none similar enough to link.
Practice
- practice — not a real PYQ
A cube of side 5 cm is painted on all its faces and then cut into 125 cubes of side 1 cm. How many of the small cubes have no painted face ?
- (a)27
- (b)36
- (c)54
- (d)8
Answer(1) — The unpainted cubes form the core: (5 − 2)³ = 27. Option (2) is the number with exactly two painted faces, 12 × 3 = 36; option (3) is the number with exactly one, 6 × 9 = 54; option (4) is the corners, which have three. - practice — not a real PYQ
A solid cube made of 27 equal small cubes is painted on all its faces. How many small cubes have exactly two painted faces ?
- (a)8
- (b)6
- (c)12
- (d)1
Answer(3) — Here n = 3, and each of the 12 edges has 3 − 2 = 1 cube between its corners: 12 × 1 = 12. Option (1) is the corners (three painted faces), option (2) the face centres (one painted face), and option (4) the single hidden cube (none).