14 : 9 :: 26 : ? — Identify the numeric rule that maps the first number to the second (state your reasoning) and apply the same rule to find the missing term for 26.

Introduction / Context:
In number analogies, the first pair reveals the hidden rule, which must then be applied to the second pair. Here, 14 maps to 9. We must discover a consistent transformation and use it to compute the image of 26.

Given Data / Assumptions:

• The mapping is deterministic and single-valued: one input gives one output.
• 14 is mapped to 9 in the first pair.
• We seek the corresponding output for 26 using the same rule.
• Standard transformations considered in such questions include squares, cubes, digit operations, and proximity to perfect powers.

Concept / Approach:
The value 9 is the largest perfect square less than 14. This suggests the rule 'map any number to the nearest lower perfect square' (also described as 'greatest perfect square <= n', but excluding equality here since 14 is not itself a square). We test the idea on the second number, 26.

Step-by-Step Solution:
1) List perfect squares around 14: 9 (3^2), 16 (4^2). The lower square is 9 → matches the given mapping.2) List perfect squares around 26: 25 (5^2), 36 (6^2). The lower square is 25.3) Therefore, by the same rule, 26 maps to 25.

Verification / Alternative check:
Check for competing rules (digit sums, products, factorials). None of those standard manipulations land exactly on 9 from 14 as directly as the 'nearest lower perfect square' rule does. Applying it to 26 consistently gives 25.

Why Other Options Are Wrong:

• 16/27/28/21: These are not the greatest perfect squares below 26; only 25 fits the rule.

Common Pitfalls:
Choosing the 'nearest' square (which could be 25 or 36) instead of the 'nearest lower' square. The first pair (14 → 9) clarifies it must be 'lower'.

Final Answer:
25