The clock and its hands

1. The hour hand completes one revolution in 12 hours, while the minute hand completes one revolution in 1 hour. This means that the minute hand "advances" the hour.
2. Angular speeds: Minute hand: 360° per hour = 6° per minute.
3. In 24 hours: 11 × 2 = 22 overlaps (not counting the one at 12:00 twice).

Answer: 22 times a day.
Explanation:
The hour hand completes one revolution in 12 hours, while the minute hand completes one revolution in 1 hour. This means that the minute hand "advances" the hour.
Angular velocities:

• Minute hand: $360°$ per hour = $6°$ per minute
• Schedule: $30°$ per hour = $0.5°$ per minute
• Relative speed: $6 - 0.5 = 5.5°$ per minute

For the minute hand to reach the hour from one overlay to the next, it must gain $360°$:

$$\text{Tiempo} = \frac{360°}{5.5°/\text{min}} = \frac{720}{11} \text{ minutos} \approx 65.45 \text{ minutos}$$

In 12 hours there are:

$$\frac{12 \times 60}{720/11} = \frac{720 \times 11}{720} = 11 \text{ superposiciones}$$

In 24 hours: $11 \times 2 = 22$ overlaps (not counting the one at 12:00 twice).