Let's find all numbers of the form

$$
\underbrace{a a a \ldots a}_{n \text{ times}} \underbrace{b b b \ldots b}_{n \text{ times}}
$$

that are perfect squares.
Solution. If $n=1$, then we are talking about two-digit square numbers, which are: 16, 25, 36, 49, 64, 81. For the rest, we assume that $n \geq 2$. When a square number is divided by 4, the remainder can be 0 or 1 (it cannot be 2 or 3). The remainder when dividing by 4 is the same as the remainder when dividing the number formed by the last two digits of the number (written in the decimal system) by 4. A square number can only end in 0, 1, 4, 5, 6, or 9, and these are the only possible values for $b$. If the value of $b$ is 1, 5, or 9, then the remainder when the number is divided by 4 is the same as the remainder of 11, 55, or 99, which is 3, while if $b=6$, the remainder is the remainder of 66, which is 2; this shows that $b$ can only be 0 or 4. If $b=4$, then $n$ can be at most 3: since 10000 is divisible by 16, the remainder when a natural number is divided by 16 is the same as the remainder when the number formed by the last four digits is divided by 16. If the last four digits of the number are all 4, then the remainder is 12, so the number is divisible by 4 but not by 8, and its quarter (which is also a square number) gives a remainder of $12 / 4=3$ when divided by 4, which is impossible. For the case $n=2, b=4$, trying the 9 possible values of $a$, only $7744=88^{2}$ is suitable, and for the case $n=3, b=4$, we do not get a solution. Finally, let's assume that $b=0$. Then clearly $a \neq 0$, so our square number is exactly divisible by the $n$-th power of 10; therefore, $n$ must be even. The $n$-th root of our square number (which is also a square number) is $\underbrace{a a a \ldots a}_{n \text{ times}}$, and examining its remainder when divided by 4 and 16, as seen before, and considering the evenness of $n \geq 2$, we find that $a=4$ and $n=2$; however, with this, $\underbrace{\text { aaa ...a }}=44$, which is not a square number. Since there are no more solutions, the suitable square numbers are: $16,25,36,49$, ${ }^{n} \text{times}$

$64,81,7744$.