Number analogy — identify the rule and complete the pair. 9 : 8 :: 16 : ?

Introduction / Context:
Numeric analogies often exploit representations such as powers. Here, both 9 and 16 are perfect powers, suggesting a rule that swaps base and exponent to create the mapped value.

Given Data / Assumptions:

• Given pair: 9 maps to 8.
• Find value for: 16 maps to ?.
• Assume one consistent transformation applies to both items.

Concept / Approach:
Express each number as a^b. Then map it to b^a. This “base–exponent interchange” is a well-known puzzle device. For example, 9 = 3^2 and swapping gives 2^3 = 8.

Step-by-Step Solution:
Rewrite 9 as 3^2. Swap base and exponent to get 2^3 = 8. This matches the given 9 : 8 mapping, confirming the rule. Now apply to 16: rewrite 16 as 2^4. Swap base and exponent: 4^2 = 16.

Verification / Alternative check:
Check if simpler rules like subtracting 1 (9 → 8) would fit 16 → 15. That option exists but does not respect the “power swap” insight evident in 9 as 3^2 mapping to 2^3. The power-swap rule explains both items elegantly and uniquely.

Why Other Options Are Wrong:
27 or 18: inconsistent with 2^4 ↔ 4^2. 17 or 15: simple ±1 tricks do not generalize from the first mapping.

Common Pitfalls:
Stopping at 9 → 8 as “minus one.” Always test candidate rules on the second pair to ensure consistency.

Final Answer:
16