Example $4^{\circ}$ If a perfect square number has the last 3 digits the same and not 0, find the minimum value of this number.

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Solution: Since the last two digits of a perfect square can only be even 0, even 1, even 4, even 9, $\overline{25}$, or odd 6, the last three digits of a perfect square can only be 444. Since 444 is not a perfect square, we have $n \geqslant 1444$. And $1444=38^{2}$ is a perfect square, so the smallest perfect square is 1444.