A precise clock has lost its number plate. How many hours could it be if the hands are perpendicular to each other?
We know that the minute hand moves through a $360^{\circ}$ angle in 60 minutes, while the hour hand moves through a $30^{\circ}$ angle in the same time. Therefore, the speed ratio is $12:1$. This means that in 1 minute, the minute hand moves $6^{\circ}$, and the hour hand moves one-twelfth of that, which is $0.5^{\circ}$. We need to examine when the difference in their paths will be $90^{\circ}$, or an odd multiple of it. (In the case of an even multiple, the difference would be $180^{\circ}$, meaning the two hands would be in a straight line.) Starting from the same initial position, i.e., from 0 o'clock, the condition for this is given by the equation

$$
6 x - 0.5 x = (2k + 1) 90
$$

Thus, the hands will first be perpendicular to each other after $x = 16 \frac{4}{11}$ minutes. In the next case, $x = 3 \cdot 16 \frac{4}{11} = 49 \frac{1}{11}$ minutes, and so on until the multiple of $16 \frac{4}{11}$ reaches $360^{\circ}$. Let's summarize our results in a table.

| $k$ | hour | minute | $k$ | hour | minute |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 0 | $16 \frac{4}{11}$ | 23 | 6 | $16 \frac{4}{11}$ |
| 3 | 0 | $49 \frac{1}{11}$ | 25 | 6 | $49 \frac{1}{11}$ |
| 5 | 1 | $21 \frac{9}{11}$ | 27 | 7 | $21 \frac{9}{11}$ |
| 7 | 1 | $54 \frac{6}{11}$ | 29 | 7 | $54 \frac{6}{11}$ |
| 9 | 2 | $27 \frac{3}{11}$ | 31 | 8 | $27 \frac{3}{11}$ |
| 11 | 3 | - | 33 | 9 | - |
| 13 | 3 | $32 \frac{8}{11}$ | 35 | 9 | $32 \frac{8}{11}$ |
| 15 | 4 | $5 \frac{5}{11}$ | 37 | 10 | $5 \frac{5}{11}$ |
| 17 | 4 | $38 \frac{2}{11}$ | 39 | 10 | $38 \frac{2}{11}$ |
| 19 | 5 | $10 \frac{10}{11}$ | 41 | 11 | $10 \frac{10}{11}$ |
| 21 | 5 | $43 \frac{7}{11}$ | 43 | 11 | $43 \frac{7}{11}$ |

From the table, we can also see that a certain position of the minute hand corresponds to two positions of the hour hand. In one case, the minute hand can be rotated $90^{\circ}$ in the positive direction to align with the hour hand, and in the other case, in the negative direction.

Therefore, the answer to the question of how many hours it could be is quite difficult to give, even if the size of the small and large hands can be clearly distinguished and we can somehow determine whether it is daytime or nighttime. If we cannot determine this, the number of possibilities increases from 22 to 44.