The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?

• Let Mr. X's age last year be $n^2$ and his age next year be $m^3$.

• Since his age next year is two years older than last year, the equation is $m^3 - n^2 = 2$.

• The only integer solution for this equation is $n = 3$ and $m = 3$, meaning his age last year was $3^2 = 9$ and next year it will be $3^3 = 27$.

• This implies his current age is 10 (since last year was 9) or 26 (since next year is 27). Wait, if last year was 9, current is 10, next year is 11 (not 27). Let's re-evaluate the timeline.

• If age last year = $n^2$ and age next year = $m^3$, then $m^3 - n^2 = 2$. The solution is $n=3, m=3$. So age last year = 9, age next year = 27. This is a contradiction in time (9 vs 27). Let's look for $m^3 - n^2 = 2$ where the ages are close. Actually, the standard problem implies $Age_{last} = n^2$ and $Age_{next} = m^3$. The difference is 2 years. $m^3 - n^2 = 2$. The only small integer solution is $3^3 - 3^2 = 27 - 9 = 18 \neq 2$. Let's check $m^3 - n^2 = 2$ again. $1^3 - (-1)^2 = 0$. $2^3 - 2^2 = 4$. $3^3 - 5^2 = 27-25=2$. So $m=3, n=5$. Age last year = $5^2 = 25$. Age next year = $3^3 = 27$. This fits perfectly (25 last year, 26 current, 27 next year).

• So, Mr. X is currently 26 years old.

• We need to find the least number of years he must wait for his age to become a cube again. His current age is 26. The next perfect cube after 26 is $3^3 = 27$ (which is next year, so he waits 1 year? No, the question asks for the *next* time it becomes a cube *again* after the instance described, or simply the next cube age. Usually, 'again' implies the subsequent cube. The cubes are 1, 8, 27, 64, 125... He is 26. Next cube is 27 (1 year). But 27 is 'next year' in the premise. Does 'wait' imply future beyond next year? Or is the answer based on the next cube after 27? Let's check the options. If he waits 1 year, he is 27. If the question implies the *next* cube after the one mentioned in the 'next year' scenario, we look at the cube after 27, which is 64. $64 - 26 = 38$. This matches option (b).