Correct Option (A)
The problem requires determining the number of small cubes that do not have any painted faces. These are the cubes located entirely within the interior of the larger cube, untouched by the initial painting process.
For a large cube of side length 'N' units, when it is uniformly sliced into smaller cubes of 1 unit side length, the number of small cubes with no painted faces can be calculated using the formula (N-2)³.
Given that the large cube has a side length of 4 cm, N = 4.
Therefore, the number of unpainted cubes = (4-2)³ = 2³ = 8.
Incorrect Options:
The other options are incorrect as they do not represent the number of small cubes with no painted faces.
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Option B (16): This value does not correspond to the number of interior cubes in a 4x4x4 configuration. It is not derived from the formula for unpainted cubes.
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Option C (24): This value typically represents the number of cubes with exactly one painted face (calculated as 6 × (N-2)²) or exactly two painted faces (calculated as 12 × (N-2)) in a cube of side N. For N=4, cubes with one painted face = 6 × (4-2)² = 6 × 2² = 24. Cubes with two painted faces = 12 × (4-2) = 12 × 2 = 24. Neither of these categories consists of cubes with no painted faces.
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Option D (36): This value does not correspond to the number of unpainted cubes, nor does it represent any standard category of painted cubes (one, two, or three faces painted) in this type of problem.