## 11. Ones and Twos

Write down an even number of ones in a row. Subtract from the resulting number a similar number composed of a series of twos, which is half as long as the first series of ones. In doing so, you will get a perfect square. (For example, $1111-22=1089=33^{2}$.) Could you explain why?
$$
\begin{aligned}
\underbrace{11 \ldots 1}_{2 n \text { digits }}-\underbrace{22 \ldots 2}_{n \text { digits }} & =\underbrace{11 \ldots 1}_{n} \underbrace{11 \ldots 1}_{n}-2(\underbrace{11 \ldots 1}_{n})= \\
& =\underbrace{11 \ldots 1}_{n} \underbrace{00 \ldots 0}_{n}-\underbrace{11 \ldots 1}_{n}= \\
& =\underbrace{11 \ldots 1}_{n}(\underbrace{100 \ldots 0-1)=}_{n} \\
& =\underbrace{11 \ldots 1}_{n} \underbrace{99 \ldots 9}_{n})= \\
& =\underbrace{(11 \ldots 1}_{n}) \cdot 9 \cdot(\underbrace{11 \ldots 1}_{n})= \\
& =3^{2}(\underbrace{(1 \ldots \ldots 1}_{n})^{2}=(\underbrace{33 \ldots 3}_{n})^{2}
\end{aligned}
$$

Example: $11-2=9=3^{2}$.