1 : 36 :: 2 : ? — Detect the transformation that maps 1 to 36 and apply it to 2 to find the missing value.

Introduction / Context:
We seek a simple arithmetic-to-power pattern. 36 suggests 6^2. If 1 maps to 36, then the base likely became 6. That hints at 'add 5, then square' as the rule.

Given Data / Assumptions:

• 1 → 36 implies (1 + 5)^2 = 6^2 = 36.
• Apply the same operation to 2.

Concept / Approach:
Transform n by adding 5 and then squaring the result: f(n) = (n + 5)^2.

Step-by-Step Solution:
1) Confirm: f(1) = (1 + 5)^2 = 36.2) Compute: f(2) = (2 + 5)^2 = 7^2 = 49.

Verification / Alternative check:
Other nearby ideas (square, cube directly) do not map 1 → 36; the '+5 then square' fits cleanly and generalizes.

Why Other Options Are Wrong:

• 69/70: Not perfect squares from (2 + 5).
• 81: Would be (2 + 7)^2; inconsistent with the 1 → 36 mapping.
• 64: 8^2; again, mismatched to the discovered rule.

Common Pitfalls:
Guessing 6^2 → 36 without identifying the consistent offset for the second mapping. Always codify the rule before applying it.

Final Answer:
49