3. Find a natural number $n$, such that $\frac{n(n+1)}{2}$ is a three-digit number with identical digits.
Solution. From $\frac{n^{2}}{2}<\frac{n(n+1)}{2} \leq 999$, it follows that $n<\sqrt{2 \cdot 999} \approx 44.7$. Every three-digit number with identical digits is divisible by $111=3 \cdot 37$. Therefore, $37|n$ or $37 \mid(n+1)$. However, since $n \leq 44$, it follows that $n=37$ or $n=36$. By direct verification, it is found that only $n=36$ satisfies the condition of the problem.