A standard six-faced die has opposite faces that always sum to $7$ (i.e., 1 is opposite 6, 2 is opposite 5, 3 is opposite 4). If the die is rolled such that face '3' is on top and face '2' is facing North, which face is facing South?

(A) 5

(B) 4

(C) 6

(D) 1

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Geometric Invariant:** On any standard cubical die, opposite faces sum to 7: * Face 1 is opposite Face 6. * Face 2 is opposite Face 5. * Face 3 is opposite Face 4. 2. **Stage 2: Directional Alignment:** North and South are opposite geographic directions on the horizontal plane. 3. **Stage 3: Face Deduction:** If North is Face 2, South must be the face directly opposite to 2. 4. **Stage 4: Opposite Value:** $7 - 2 = 5$. 5. **Stage 5: Distractor Forensics:** Face 4 is on the bottom (opposite Face 3). Face 1 and 6 occupy East and West.

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