A classic age riddle with a time-shift: A says to B: “I am twice as old as you were when I was as old as you are.” The sum of their present ages is 63. Find the difference in their ages.

Introduction / Context:
This celebrated age riddle asks you to translate a nested time-shift statement into algebra. Let A be older than B. Define the time t years ago when A's age equalled B's current age; then interpret “you were” accordingly. Solving yields a fixed ratio of their present ages.

Given Data / Assumptions:

• Let present ages be A (older) and B.
• At time t = A − B years ago, A − t = B (A was as old as B is now).
• Statement: A = 2 * (B − t).
• Sum: A + B = 63.

Concept / Approach:
From t = A − B, substitute into A = 2(B − t) to get a linear relation between A and B. Combine with the sum to find both ages and hence their difference.

Step-by-Step Solution:

Verification / Alternative check:
t = A − B = 9. Nine years ago, B was 18; A now is 36 = 2 × 18, satisfying the riddle.

Why Other Options Are Wrong:
27/12/6 do not match the 4:3 ratio with sum 63.

Common Pitfalls:
Setting t incorrectly or mixing up which age corresponds to “you were” vs “I was.” Define t explicitly to avoid confusion.

Final Answer:
9 years