Reasoning : Logical Reasoning & Analytical Ability

Q 216 / 325

UPSC CSE Prelims 2016

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:

EXPLANATION

Correct Option (b)

The problem involves a large cube, painted on all its faces, which is subsequently cut into smaller cubes of equal size. The side of each small cube is specified as one-fourth the side of the large cube. This implies that the large cube is divided into 4 segments along each of its edges.

  • Let 'n' represent the number of divisions along each edge of the large cube. In this scenario, n = 4.
  • The total number of small cubes formed from the large cube is n³ = 4³ = 64.
  • Cubes with only one face painted are those located exclusively on the central part of each face of the original large cube. These cubes do not touch any edges or corners of the large cube.
  • For a cube divided into 'n' segments along each edge, the number of small cubes with only one face painted on a single face is determined by the formula (n-2)².
  • Substituting n = 4 into the formula, the number of such cubes per face is (4-2)² = 2² = 4.
  • Since a cube possesses 6 faces, the total number of small cubes with exactly one face painted is calculated as 6 × (n-2)² = 6 × 4 = 24.

Incorrect Options:

  • Option (a) 32: This value does not align with the standard combinatorial calculations for cubes with only one face painted under the given division parameters. It may result from an erroneous application of formulas or a miscalculation.

  • Option (c) 16: This number represents the total count of small cubes on a single face of the large cube (4 × 4 = 16). It fails to account for the contributions from all six faces of the large cube, which is necessary to determine the total number of one-face painted cubes.

  • Option (d) 8: This value consistently represents the number of corner cubes in any cube division, as a cube always has 8 corners. Corner cubes have three faces painted, not just one, making this option irrelevant to the question asked.