A solid wooden cube of edge length $4\text{ cm}$ is painted green on all six exterior faces and then sliced into 64 smaller unit cubes of side $1\text{ cm}$. How many of these smaller unit cubes have EXACTLY two faces painted green?
Options:
(A) 8
(B) 16
(C) 24
(D) 32
Answer Key:
C
Step-by-Step Solution & Explanation
For an $n \times n \times n$ painted cube sliced into unit cubes ($n = 4$):
• Cubes with 3 faces painted (corners) $= 8$
• Cubes with 2 faces painted lie along the 12 edges, excluding the 2 corner vertices per edge:
$N_{\text{2-face}} = 12 \times (n - 2) = 12 \times (4 - 2) = 12 \times 2 = 24$
• Cubes with 1 face painted $= 6(n - 2)^2 = 6(4) = 24$
• Cubes with 0 faces painted (interior) $= (n - 2)^3 = 2^3 = 8$
Total $= 8 + 24 + 24 + 8 = 64$. Therefore, exactly 24 unit cubes have 2 faces painted.