Question

How many cubes will have only one colour?

• A. 10
• B. 12
• C. 14
• D. 18
• A. 1
• B. 2
• C. 4
• D. 8
• A. 34
• B. 24
• C. 16
• D. 12

Answer 1

The Correct Option is A

Given: A 4 cm × 3 cm × 3 cm block; four faces of size 4×3 are painted (two yellow and two red), and the remaining two faces of size 3×3 are painted green. The block is cut into 1 cm × 1 cm × 1 cm cubes.

Step 1 — Total number of small cubes:
The big block has volume = 4 × 3 × 3 = 36 cm³.
Each small cube has volume = 1 cm³.
Hence, the block is cut into 36 small cubes.

Step 2 — Formula for cubes with only one colour:
On a face of dimension A × B (in terms of small cubes), the number of cubes with exactly one painted face = (A − 2) × (B − 2).
This is because we remove the boundary cubes (edges and corners) and take only the face-centre cubes.

Step 3 — Apply to the 4 × 3 faces (painted yellow and red):
Interior cubes = (4 − 2) × (3 − 2) = 2 × 1 = 2 per face.
There are 4 such faces (2 yellow, 2 red).
Contribution = 4 × 2 = 8 cubes.

Step 4 — Apply to the 3 × 3 faces (painted green):
Interior cubes = (3 − 2) × (3 − 2) = 1 × 1 = 1 per face.
There are 2 such faces.
Contribution = 2 × 1 = 2 cubes.

Step 5 — Total cubes with only one colour:
= 8 (from yellow/red faces) + 2 (from green faces)
= 10 cubes.

Step 6 — Sanity check by classification:
• 3-colour corner cubes = 8.
• 2-colour edge cubes = 16.
• 1-colour face-centre cubes = 10.
• 0-colour interior cubes = 2.
Sum = 8 + 16 + 10 + 2 = 36 (matches total).

Final Answer:
The number of cubes with only one colour = 10.

Answer 2

Goal: Find the number of 1 cm × 1 cm × 1 cm cubes with no colour after cutting the painted 4 × 3 × 3 block.

Step 1 — Understand what “no colour” means:
A small cube has no colour iff it never touched any outer surface before cutting. In a rectangular block of L × B × H small cubes, these are the completely interior cubes obtained by peeling off a 1-cube thick layer from every face.

Step 2 — Direct interior-cubes formula:
Number of unpainted (no-colour) cubes = (L − 2) × (B − 2) × (H − 2).
Here L = 4, B = 3, H = 3 (in units of 1-cm cubes).
So, unpainted = (4 − 2) × (3 − 2) × (3 − 2) = 2 × 1 × 1 = 2.

Step 3 — Complementary check (optional but reassuring):
Total small cubes = 4 × 3 × 3 = 36.
Classify painted cubes by where they lie:
• Corners (3 colours): always 8.
• Edges (2 colours, excluding corners): for an edge of length n, it contributes (n − 2). The 4 × 3 × 3 block has four edges of length 4 and eight edges of length 3, so edges = 4×(4 − 2) + 8×(3 − 2) = 8 + 8 = 16.
• Face-centres (exactly 1 colour): interior on each painted face. For 4×3 faces (four faces): (4 − 2)×(3 − 2) = 2 per face → 4×2 = 8. For 3×3 faces (two faces): (3 − 2)×(3 − 2) = 1 per face → 2×1 = 2. Total 1-colour = 8 + 2 = 10.
Sum of painted (≥1 colour) = 8 (corners) + 16 (edges) + 10 (face-centres) = 34.
Hence unpainted = Total − Painted = 36 − 34 = 2 (matches).

Final Answer: 2 cubes have no colour.

Answer 3

Goal: Count the number of 1 cm × 1 cm × 1 cm cubes that have exactly two colours after cutting the painted 4 × 3 × 3 block.

Key Idea: A cube has exactly two colours iff it lies on an edge of the original block but is not a corner. Corner cubes carry three colours; face-centre cubes carry one; interior cubes carry none. Thus, “two-colour cubes” = “edge cubes excluding corners.”

Step 1 — Edge counting principle:
For any edge of length n (measured in small 1-cm cubes), the number of two-colour cubes on that edge is (n − 2), because we exclude the two corner cubes at the ends.

Step 2 — Identify the 12 edges and their lengths:
The block dimensions (in small cubes) are 4 × 3 × 3, so the 12 edges come in two types:
• 4 edges of length 4 (along the 4-cm side). Each contributes (4 − 2) = 2 two-colour cubes.
• 8 edges of length 3 (along the 3-cm sides). Each contributes (3 − 2) = 1 two-colour cube.

Step 3 — Total two-colour cubes (edge-sum):
Total = 4 × (4 − 2) + 8 × (3 − 2) = 4 × 2 + 8 × 1 = 8 + 8 = 16.

Step 4 — Sanity check by full classification (optional but reassuring):
Total cubes = 4 × 3 × 3 = 36.
• 3-colour (corners): 8.
• 2-colour (edges, excluding corners): 16 (as computed).
• 1-colour (face-centre cubes): On each face A × B, count (A − 2)(B − 2). For four 4×3 faces → 4 × (2 × 1) = 8; for two 3×3 faces → 2 × (1 × 1) = 2; total = 10.
• 0-colour (fully interior): (4 − 2)(3 − 2)(3 − 2) = 2 × 1 × 1 = 2.
Sum = 8 + 16 + 10 + 2 = 36 (matches total), so the count 16 is consistent.

Conclusion: The number of cubes with exactly two colours is 16.