08 : 66 :: ? : 38 — Decode the rule mapping the two-digit number to the right-side value and find the missing left-side number.

Introduction / Context:
Coded analogies sometimes compress a transformation into a compact mapping. Here, “08” maps to 66. We must infer a simple numeric rule that produces 66 from 8 (ignoring the leading 0) and then reverse-apply it to obtain a left number that would yield 38 on the right.

Given Data / Assumptions:

• 08 → 66.
• We need x → 38 using the same rule.
• Leading zeros in two-digit notation may not change the actual magnitude of x.

Concept / Approach:
A neat fit is: f(n) = n^2 + 2. Test on n = 8: 8^2 + 2 = 64 + 2 = 66 (works). Now solve for n in f(n) = 38: n^2 + 2 = 38 → n^2 = 36 → n = 6 (taking the positive root as per standard digit use). Thus, the missing left-side number is 6 (which could also appear as 06 in two-digit form).

Step-by-Step Solution:
1) Hypothesize f(n) = n^2 + k; fit k using 8 → 66.2) 8^2 = 64; 66 - 64 = 2 → k = 2.3) Solve n^2 + 2 = 38 → n^2 = 36 → n = 6.

Verification / Alternative check:
Other simple rules (e.g., n*8 + something) are less tidy and fail to generalize cleanly. The square-plus-constant rule explains both mappings succinctly.

Why Other Options Are Wrong:

• 2, 12, 19: f(2) = 6; f(12) = 146; f(19) = 363; none equals 38.

Common Pitfalls:
Forgetting that a leading zero may be stylistic and not part of the arithmetic. Focus on the operative digit.

Final Answer:
6