Problem 11.3. The teacher wrote the number 1818 on the board. Vasya noticed that if a multiplication sign is written between the hundreds and tens digits, the value of the resulting expression is a perfect square $\left(18 \times 18=324=18^{2}\right)$. What is the next four-digit number after 1818 that has the same property?
Answer: 1832.

Solution. Since we need to find the nearest four-digit number, let's try to find it in the form $\overline{18 a b}$. Then the number $18 \cdot \overline{a b}=3^{2} \cdot(2 \cdot \overline{a b})$ must be a perfect square. From this, it follows that $2 \cdot \overline{a b}$ must also be a perfect square. Clearly, $\overline{a b} \in\{19,20,21 \ldots, 31\}$ do not meet this condition, but $\overline{a b}=32$ does. Therefore, the answer to the problem is the number 1832.

Remark. The same answer could have been obtained by proving that $\overline{a b}=2 s^{2}$ for some natural number $s>3$.