QB_PH_GA_023: Analytical & Spatial Aptitude

A solid wooden cube of side $3\text{ cm}$ is painted red on all its six exterior faces and then sliced into identical smaller cubes of side $1\text{ cm}$. How many of these smaller cubes have EXACTLY TWO faces painted red?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Standard Cube Dissection Formulas** for an $n \times n \times n$ cube ($n = 3$): - Total cubes $= n^3 = 27$ - 3 faces painted (corners) $= 8$ - 2 faces painted (edge centers) $= 12(n - 2)$ - 1 face painted (face centers) $= 6(n - 2)^2$ - 0 faces painted (interior core) $= (n - 2)^3$ 2. **Computation for 2 Faces Painted**: $N_{\text{2-faces}} = 12(n - 2) = 12(3 - 2) = 12 \times 1 = 12$ 3. **Sum Check**: $8 + 12 + 6(1) + 1(1) = 8 + 12 + 6 + 1 = 27$ (Consistent). 4. **Official Answer**: 12.