Cubes – MBA CET questions with solutions

Approach: Count a stepped pyramid layer by layer, including hidden supporting cubes.

Solution:
The figure is a corner pyramid: each visible upper cube rests on cubes behind/below it.
Top layer: 1 cube
Middle layer: 2 visible + 1 hidden = 3 cubes (1 + 2)
Bottom layer: 3 visible + 3 hidden = 6 cubes (1 + 2 + 3)
Total = 1 + 3 + 6 = 10

Cetking shortcut: A triangular pyramid with n steps has triangular-number layers: 1 + 3 + 6 + … ; for 3 steps = 10.

Answer: E

Approach: In a cube net, faces in a straight line with one face between them are opposite.

Solution:
Net: column K–Q–M–J, with L and N on either side of Q.
Column: K opposite M, Q opposite J (alternate faces).
Row: L – Q – N → L opposite N.
Face opposite N = L.

Why the other options are wrong:
A (K), D (M), E (Q): share an edge with N when folded.
B (J): J is opposite Q, and adjacent to N.

Cetking shortcut: Skip one face in any straight row/column of the net → opposite pair.

Answer: C

Approach: Box from a net — find which faces are adjacent and where the marks lie after folding.

Solution: In the net, the narrow strip with the small black mark lies between the cross-design face and the large plain face, and the mark is at its left end. The plain face and the cross-design face are opposite each other.
Box 1 shows the plain face in front with a plain narrow strip on top and an end flap — a valid view.
Box 2 has the black mark at the right end of the top strip; folding keeps it at the left end → not possible.
Boxes 3 and 4 show the cross-design face with a black mark on an end flap, but the mark is on a long strip, not on a flap → not possible.

Cetking shortcut: Track one distinctive mark (the black rectangle) — its position rules out most options.

Answer: A

Approach: Painted cuboid — cubes touching three different colours.

Solution: The block is 6 × 4 × 1 → 6 × 4 × 1 = 24 unit cubes in a single layer.
Because the height is 1, every cube has green on its top and bottom (the two 6 × 4 faces).
Black = the two 4 × 1 end faces (4 cubes each, 8 in all). Red = the two 6 × 1 side faces (6 cubes each).
Corner cubes (4): green, green, red, black. Edge non-corner cubes: green, green + one of red/black. Inner cubes: (6 − 2) × (4 − 2) = 8, green on two faces only.
Red, green and black together only where a red side meets a black end — the 4 corners → 4.

Why the other options are wrong:
A 16 — border cubes, but most have only two colours.
B 8 — inner cubes (green only) or black cubes.
D 12, E 24 — overcounts.

Cetking shortcut: Three different colours on one small cube = corner cube; a flat block has 4 such corners.

Answer: C

Approach: Number of unit cubes = volume ÷ 1.

Solution: 6 × 4 × 1 = 24 cubes of side 1 cm.

Answer: A

Approach: Count coloured faces per position.

Solution: The block is 6 × 4 × 1 → 6 × 4 × 1 = 24 unit cubes in a single layer.
Because the height is 1, every cube has green on its top and bottom (the two 6 × 4 faces).
Black = the two 4 × 1 end faces (4 cubes each, 8 in all). Red = the two 6 × 1 side faces (6 cubes each).
Corner cubes (4): green, green, red, black. Edge non-corner cubes: green, green + one of red/black. Inner cubes: (6 − 2) × (4 − 2) = 8, green on two faces only.
4 coloured faces = green, green, red, black → only the corner cubes → 4.
(Edge cubes have 3 coloured faces; inner cubes have 2.)

Why the other options are wrong:
B 8 — inner cubes, which have only 2 coloured faces.
C 12 — edge cubes (3 coloured faces): 2 × 4 + 2 × 2 = 12.
D, E — no position gives these.

Answer: A

Approach: Green on two sides only = cubes not on the border.

Solution: The block is 6 × 4 × 1 → 6 × 4 × 1 = 24 unit cubes in a single layer.
Because the height is 1, every cube has green on its top and bottom (the two 6 × 4 faces).
Black = the two 4 × 1 end faces (4 cubes each, 8 in all). Red = the two 6 × 1 side faces (6 cubes each).
Corner cubes (4): green, green, red, black. Edge non-corner cubes: green, green + one of red/black. Inner cubes: (6 − 2) × (4 − 2) = 8, green on two faces only.
Border cubes = 24 − inner = 24 − (4 × 2) = 16.
Only green (top and bottom) = inner cubes = (6 − 2) × (4 − 2) = 8.

Why the other options are wrong:
A 4 — corner cubes.
C 12 — edge non-corner cubes.
D 16 — all border cubes.

Cetking shortcut: Inner cubes of an l × b one-layer block = (l − 2)(b − 2).

Answer: B

Approach: Remove cubes that carry black (all of them carry green).

Solution: The block is 6 × 4 × 1 → 6 × 4 × 1 = 24 unit cubes in a single layer.
Because the height is 1, every cube has green on its top and bottom (the two 6 × 4 faces).
Black = the two 4 × 1 end faces (4 cubes each, 8 in all). Red = the two 6 × 1 side faces (6 cubes each).
Corner cubes (4): green, green, red, black. Edge non-corner cubes: green, green + one of red/black. Inner cubes: (6 − 2) × (4 − 2) = 8, green on two faces only.
Black-and-green cubes = cubes on the two black end faces = 4 + 4 = 8.
Remaining = 24 − 8 = 16.

Why the other options are wrong:
A 8 — that is the number removed.
B 12, D 20, E 10 — wrong count of black cubes.

Cetking shortcut: Since every cube is green, "black and green" simply means "black": 2 ends × 4 cubes = 8.

Answer: C

Approach: Cube cutting — n³ cubelets; 'set of faces' means a pair of opposite faces.

Solution:
343 = 7³ → the cube is cut into 7 slices along each of the 3 directions.
7 slices need 6 cuts in one direction.
Total minimum cuts = 6 × 3 = 18.

Why the other options are wrong:
D (7): is the number of slices per edge, not cuts.
A, C, E: do not equal 3 × (n − 1).

Cetking shortcut: Minimum cuts for n × n × n = 3(n − 1).

Answer: B

Approach: Cube cutting — n³ cubelets; 'set of faces' means a pair of opposite faces.

Solution:
343 = 7³ → n = 7.
Each of the 3 colours is on one pair of opposite faces, so every edge of the big cube joins two faces of different colours.
Corner cubelets touch 3 faces → 3 colours (8 of them).
Edge cubelets excluding corners touch exactly 2 faces → exactly 2 colours.
Per edge: n − 2 = 5 cubelets; 12 edges → 12 × 5 = 60.

Why the other options are wrong:
A (49): is one face's total (7 × 7).
D (25): one face's inner cubelets (5 × 5).
B (55), E (45): miscount of edges.

Cetking shortcut: Exactly two colours (opposite faces same colour) = 12(n − 2).

Answer: C

Approach: In a straight strip of faces in a net, alternate faces are opposite.

Solution:
The vertical strip is 1, 2, 3, 4 → 1 is opposite 3 and 2 is opposite 4.
6 (left of 3) and 5 (right of 4) are the remaining pair → 5 is opposite 6.
So the face opposite 4 is 2.

Answer: B

Approach: In a net, faces separated by one face in a straight line are opposite.

Solution:
Column 1-2-3-4: 1 is opposite 3, 2 is opposite 4.
6 (left of 3) and 5 (right of 4) fold onto opposite sides → 5 opposite 6.
So the number opposite 4 is 2.

Why the other options are wrong:
1 / 3: both adjacent to 4 (1 and 3 are opposite each other).
5 / 6: they are opposite each other, so adjacent to 4.

Cetking shortcut: Skip-one rule: in any straight line of the net, alternate faces are opposite.

Answer: B