Number analogy with perfect squares — find the missing term. 25 : 37 :: 49 : ?

Introduction / Context:
This analogy uses perfect squares and a consistent transformation. Such puzzles typically involve adding a fixed expression tied to the square’s root. We need to discover the rule from the first pair and apply it to the second.

Given Data / Assumptions:

• First pair: 25 maps to 37.
• Second pair: 49 maps to ?.
• Both 25 and 49 are perfect squares: 5^2 and 7^2.

Concept / Approach:
Try a root-based rule. Notice that 25 = 5^2 and 37 equals (5 + 1)^2 + 1 because (6^2) + 1 = 36 + 1 = 37. If the pattern is “(root + 1)^2 + 1,” apply it to 7^2 (i.e., 49).

Step-by-Step Solution:
Identify the root of 25: r = 5. Compute (r + 1)^2 + 1 = (5 + 1)^2 + 1 = 6^2 + 1 = 36 + 1 = 37. Now apply to 49 with r = 7. Compute (7 + 1)^2 + 1 = 8^2 + 1 = 64 + 1 = 65.

Verification / Alternative check:
Consider other plausible patterns like adding the square’s root or adding a prime. None reproduce 25 → 37 as neatly as the “(r + 1)^2 + 1” rule and simultaneously give an integer for 49.

Why Other Options Are Wrong:
41, 56, 60, 53: these would require unrelated or inconsistent adjustments that do not generalize from 25 → 37 and fail to form a clean rule for 49.

Common Pitfalls:
Overfitting rules to a single pair without confirming they extend to the second pair. Always test consistency.

Final Answer:
65