Andy and Bobby have unique age!

Let Andy's age be $\overline{ab}$ and Bobby's age be $\overline{cd}$ . If we write their ages consecutively, we have $\overline{abcd} = m^{2}$ . Twenty three years from now, Andy's age will be $\overline{ab} + 23$ and Bobby's age be $\overline{cd} +23$ . If we rewrite their ages, we have $\overline{abcd} + 2323 = n^{2}$ .

We have $m^{2} + 2323 = n^2 \Rightarrow n^{2} - m^{2} = 2323$

Factorize, we have $(n+m)(n-m) = 101 \times 23$ . $\Rightarrow \color{#3D99F6} \text{We know that} \space n+m > n-m$

We have $n+m = 101$ and $n-m =23$ . Solving, we have $n = 62$ and $m= 39$ .

We note that $\quad \begin{aligned} m^{2} & = & \overline{abcd} \\ 1521 & = & \overline{abcd} \end{aligned}$ .

Observing, we have $\overline{ab} = 15$ and $\overline{cd} = 21$ .

The sum of their ages is $\boxed{36}$ .