The two hands of a clock are identical, so we cannot tell which one indicates the hour and which one indicates the minutes. Are there any clock positions where the time can still be read unambiguously? How can we characterize the hand positions that cannot occur?
Considering that the clock face is divided into 60 parts, let's calculate the rotation of the hands in this "minute time" unit (although the rotation of the small hand represents 12 times more). We will call such a hand rotation 1 min. If $\alpha$ is the rotation of the minute hand, $\beta$ is the rotation of the hour hand, and $k$ hours have passed, that is, $k$ times $\beta$ is 5 min, with the remainder being less than 5 min $\left(k=\left[\frac{\beta}{5}\right]\right)$, then $\alpha=12 \beta-k \cdot 60 \mathrm{~min}$. In other words, on the clock, if the small hand is rotated 11 times more than its rotation from the "12", it will exactly cover the large hand.

If now both hands are the same, then only such hand positions can occur where one of the hands, when rotated 11 times more than its rotation from the "12", exactly covers the other. If this happens only with one of the hands, then the clock position is unambiguous: the hand that is rotated further is the hour hand, the other is the minute hand. If, however, the cover occurs when both hands are rotated 11 times more, then it is undecidable what time the clock shows, except when both point exactly to the "12". (Which is the "large" and which is the "small" among the two identical hands, of course, we cannot know in this case either.) This happens when not only $\alpha=12 \beta-k \cdot 60$, but also $\beta=12 \alpha-l \cdot 60$ holds, so $\alpha=144 \alpha-m \cdot 60 \min , \beta=144 \beta-n \cdot 60 \min (m=k+12 l, n=l+12 k)$, or $\alpha=m \frac{60}{143}, \beta=n \frac{60}{143}$.

Thus, we have found that on a clock, 143 times in 12 hours, at equal intervals, i.e., approximately every 5 minutes and 21 seconds, hand positions occur where the positions of the two hands can be interchanged. If, on the other hand, we randomly draw two hands on a clock face, it is as rare as a white raven to find a possible clock position (unless we set a well-known clock position intentionally).

Solved it well (3 points): Fried E, Gehér L., Kővári T.

Solved it well except for the ambiguous cases (2 points): Gacsányi S., Haris B., Kánya J.