Number analogy — find the missing term. 144 : 10 :: 169 : ?

Introduction / Context:
This is a numeric analogy question. You are given one completed pair and one incomplete pair. The task is to uncover the rule that maps the first number of a pair to the second and then apply the same rule to the second pair to find the missing value.

Given Data / Assumptions:

• Pair 1: 144 maps to 10.
• Pair 2: 169 maps to ? (unknown).
• We assume a single consistent rule applies to both pairs.

Concept / Approach:
Square numbers and their square roots are common in such analogies. Notice 144 is 12^2 and 169 is 13^2. A simple transformation such as f(n) = sqrt(n) − k is plausible when n is a perfect square.

Step-by-Step Solution:
Recognize 144 = 12^2, so sqrt(144) = 12. Since 144 maps to 10, the rule can be: output = sqrt(n) − 2, because 12 − 2 = 10. Apply the same rule to 169: sqrt(169) = 13. Compute: 13 − 2 = 11.

Verification / Alternative check:
Try other simple operators such as adding digits of the square root or halving, but none fit as cleanly. The minus-two rule fits both pairs perfectly when the inputs are perfect squares.

Why Other Options Are Wrong:
14: would require adding 2, contradicting the first pair. 13 or 12: would correspond to no consistent offset that matches 144 → 10. 9: implies subtracting 4 from 13, which breaks the first mapping.

Common Pitfalls:
Ignoring that 144 and 169 are perfect squares; trying unrelated digit tricks often leads to inconsistencies across pairs.

Final Answer:
11