Example 3 Determine whether the following equations have real solutions. If they do, find the real solutions.
(1) $\sum_{0}^{5}\left[2^{i} x\right]=12345$;
(2) $\sum_{0}^{5}\left[2^{i} x\right]=123456$.
Solution: (1) $\because \sum_{0}^{5} 2^{i}=63$,
$$
\therefore p=\left[\frac{12345}{63}\right]=195 \text{. }
$$
$\because 12345=195 \times 63+60$,
$\therefore$ take $k=31<2^{5}$, then $31+\left[\frac{31}{2}\right]+\left[\frac{31}{2^{2}}\right]+\left[\frac{31}{2^{3}}\right]+\left[\frac{31}{2^{4}}\right]=57$.
$$
\sum_{0}^{5} 2^{i} p+\sum_{0}^{5}\left[\frac{k}{2^{i}}\right]
$$

is of the form.
Therefore, the equation $\sum_{0}^{5}\left[2^{i} x\right]=12345$ has no real solution.
$$
\begin{array}{l}
\text{(2) } \because \sum_{0}^{5} 2^{i}=63, \\
\therefore p=\left[\frac{123456}{63}\right]=1959. \\
\because 123456=1959 \times 63+39, \\
\therefore \text{ take } k=21<2^{5} \text{, then } \\
21+\left[\frac{21}{2}\right]+\left[\frac{21}{2^{2}}\right]+\left[\frac{21}{2^{3}}\right]+\left[\frac{21}{2^{4}}\right]=39.
\end{array}
$$
$\therefore$ take $k=21<2^{5}$, then
$\therefore 123456$ can be expressed as
$$
\sum_{0}^{5} 2^{i} p+\sum_{0}^{5}\left[\frac{k}{2^{i}}\right]
$$

is of the form.
Therefore, $\sum_{0}^{5}\left[2^{i} x\right]=123456$ has a real solution, and its solution is
$$
1959 \frac{21}{32} \leqslant x<1959 \frac{22}{32}.
$$

To this point, all the problems mentioned earlier have been solved.