Complete the relation with embedded clue: 7 : 56 :: 6 : 9 : ? (Hint: use n * (n + k) where k is indicated on each side.)

Introduction / Context:
Some analogies insert an extra hinting number to signal a simple polynomial-like product. Here, the right side includes “6 : 9 : ?,” suggesting the operation uses both 6 and 9, analogous to how 56 relates to 7 and its adjacent value 8 on the left.

Given Data / Assumptions:

• Left: 7 : 56 — observe 56 = 7 * (7 + 1).
• Right: 6 : 9 : ? — interpret as ? = 6 * (6 + 3) because 9 is 6 + 3.

Concept / Approach:
Use the pattern n * (n + k), where k is the increment shown alongside n. On the left, k = 1 (implicit, since 56 = 7 * 8). On the right, k = 3 (explicit via 9), so compute 6 * (6 + 3) = 6 * 9.

Step-by-Step Solution:

Verification / Alternative check:
Quick multiplication confirms 54. The provided options include 54, matching neatly.

Why Other Options Are Wrong:
They correspond to incorrect multipliers or squaring attempts.

Common Pitfalls:
Trying to force symmetric +1 increments on both sides; the right-hand clue 9 explicitly sets +3.

Final Answer:
54