A big cube of 20 cm side, having all of its sides green coloured, is cut into…
2023
A big cube of 20 cm side, having all of its sides green coloured, is cut into small cubes of side 4 cm. How many such small cubes will have only one coloured face?
- A.
27
- B.
36
- C.
54
- D.
8
Show answer & explanation
Correct answer: C
When a cube is painted on all outer faces and then cut into n equal divisions per edge, the resulting small cubes fall into four categories by how many painted faces they expose: corner cubes (3 painted faces) = 8 always; edge cubes excluding corners (2 painted faces) = 12(n-2); face-centre cubes excluding edges (1 painted face) = 6(n-2)2; and fully interior cubes (0 painted faces) = (n-2)3.
Each edge of the 20 cm cube is cut into 4 cm segments, giving n = 20/4 = 5 divisions per edge.
Cubes with exactly one coloured face are the face-centre cubes, excluding the border row along every edge: count = 6(n-2)².
Substitute n = 5: 6(5-2)² = 6(3)² = 6 × 9 = 54.
Total small cubes = n3 = 53 = 125. Summing every category confirms this: corners 8 + edges 12(n-2) = 36 + face-centres 6(n-2)2 = 54 + interior (n-2)3 = 27 gives 8 + 36 + 54 + 27 = 125 — matching the total and confirming 54 face-centre cubes.