The age of a person (in years) in $2025$ is a perfect square. His age (in years) was also a perfect square in $2012$. His age (in years) will be a perfect cube $m$ years after $2025$. Determine the smallest positive value of $m$.

Step 1: Let the person's age in $2025$ be $x=k^2$ and his age in $2012$ be $y=j^2$ for some positive integers $k$ and $j$. The difference in years between $2025$ and $2012$ is $13$. Thus:

Step 2: Since $13$ is a prime number, the only integer factor pairs are $(13,1)$:

Adding these equations gives $2k=14⟹k=7$, and subtracting gives $2j=12⟹j=6$.

• Age in $2025$: $x=7^2=49$ years.

• Age in $2012$: $y=6^2=36$ years.

Step 3: The age in $2025+m$ must be a perfect cube: $49+m=c^3$ for some integer $c$.
To find the smallest positive $m$, we look for the smallest perfect cube strictly greater than $49$.
The perfect cubes are $1,8,27,64,125,\dots$
The smallest cube greater than $49$ is $64=4^3$. Thus:

So the smallest positive value of $m$ is $15$.