Understanding Clock Hand Coincidences

This question asks how many times the hour hand and the minute hand of a standard analog clock meet or coincide within a 24-hour period, which is considered a day.

To solve this classic clock problem, we need to understand the relative movement of the two hands.

Clock Hand Speeds

Let's first determine the speed of each hand in degrees per minute:

• The minute hand moves 360 degrees in 60 minutes.
• Speed of minute hand = $\frac{360^\circ}{60 \text{ minutes}} = 6^\circ/\text{minute}$.
• The hour hand moves 360 degrees in 12 hours, which is $12 \times 60 = 720$ minutes.
• Speed of hour hand = $\frac{360^\circ}{720 \text{ minutes}} = 0.5^\circ/\text{minute}$.

Relative Speed and Frequency

For the hands to coincide, the faster minute hand must "catch up" to the slower hour hand. The difference in their speeds is the relative speed:

Relative Speed = Speed of Minute Hand - Speed of Hour Hand

Relative Speed = $6^\circ/\text{minute} - 0.5^\circ/\text{minute} = 5.5^\circ/\text{minute}$.

The hands coincide every time the minute hand gains a full 360 degrees on the hour hand. The time taken for the minute hand to gain 360 degrees is:

Time for coincidence = $\frac{360^\circ}{\text{Relative Speed}} = \frac{360^\circ}{5.5^\circ/\text{minute}} = \frac{360}{11/2} \text{ minutes} = \frac{720}{11} \text{ minutes}$.

This means the hands coincide approximately every 65.45 minutes.

Coincidences in 12 Hours

In a 12-hour period, there are $12 \times 60 = 720$ minutes. If they coincided exactly every 60 minutes, they would meet 12 times. However, since the interval is slightly longer ($\frac{720}{11}$ minutes), they meet less than 12 times in 12 hours.

The number of times they coincide in 12 hours is:

Number of coincidences = $\frac{\text{Total minutes in 12 hours}}{\text{Time per coincidence}} = \frac{720 \text{ minutes}}{720/11 \text{ minutes/coincidence}} = 11 \text{ times}$.

The reason it's 11 and not 12 is that the coincidence that would theoretically occur between 11:00 and 12:00 happens exactly at 12:00. So, in the interval from 12:00 to 12:00, you have coincidences at 12:00, and then 10 more times scattered between the hours (roughly at 1:05, 2:11, ..., 10:55).

Coincidences in 24 Hours (A Day)

A day consists of 24 hours, which is two 12-hour cycles.

• In the first 12 hours (e.g., from 12:00 AM to 12:00 PM), the hands coincide 11 times. This includes the coincidence at 12:00 AM.
• In the second 12 hours (e.g., from 12:00 PM to 12:00 AM the next day), the hands coincide another 11 times. This includes the coincidence at 12:00 PM.

The coincidence at 12:00 PM (noon) is distinct from the coincidence at 12:00 AM (midnight). Therefore, in a full 24-hour period (e.g., from 00:00 to 24:00), the total number of times the hour and minute hands coincide is the sum of the coincidences in the two 12-hour periods:

Total coincidences in 24 hours = 11 (first 12 hours) + 11 (second 12 hours) = 22 times.

The hands coincide 22 times in a day.

Revision Table: Clock Hand Relative Positions Frequency

Additional Information: The 12 O'clock Exception

It's a common misunderstanding that the hands coincide 24 times in 24 hours. The key point of exception is the 12 o'clock mark. In a 12-hour cycle, the hands align at 12:00. The alignment that would typically occur between 11 and 12 does not happen within that hour interval; it occurs exactly at 12:00, which also serves as the alignment point for the 12-1 interval. This means the coincidence around 12 o'clock is 'shared' between two intervals, effectively reducing the count by one in each 12-hour period, leading to 11 coincidences per 12 hours and thus 22 coincidences in 24 hours.