What are the clock times when the hour and the minute hand would overlap or align with each other? Obviously, 12:00:00 is one of the times when this happens. However, I realized that it should happen every hour and so it should be solved 11 more times.

I've been intrigued how to solve this problem for sometime now but haven't found the time to really delve into it. Luzon is currently on "Enhanced Community Quarantine" due to COVID-19 at the moment and I’m looking for something to solve then remembered this puzzle.

Step 1 - Plot a radian chart around the clock

The way I was able to solve this problem is to plot a radian chart around a clock. Remember that half a circle is π radians and a full circle is 2π radians. Each hour can be represented by \( 1π/6 \) (1 am/pm), \( 2π/6 \) (2 am/pm), \( 3π/6 \) (3 am/pm), \( 7π/6 \) (7 am/pm) and so on and so forth then reduced to its simplest form.

Step 2 - Calculate the rate of movement of the hour and minute hands per second

One revolution or \(2 \pi \) of the second hand is 60 seconds which can be expressed by:

$$\frac{2 \pi}{60 seconds} = \frac{\pi}{30 seconds}$$

One revolution or \(2 \pi \) of the minute hand in seconds is (60 seconds/minute)(60 minutes) = 3,600 seconds which can be expressed by:

$$\frac{2 \pi}{3,600 seconds} = \frac{\pi}{1,800 seconds}$$

One revolution or \(2 \pi \) of the hour hand is (60 seconds/minute)(60 minutes/hour)(12 hours) = 43,200 seconds which can be expressed by:

$$\frac{2 \pi}{43,200 seconds} = \frac{\pi}{21,600}$$

Step 3 - Compute time when hour and minute align for a specific hour of the day

Remember that hour and minute hand aligns 12 times where the obvious one when it aligns is 12:00:00. Now for this particular task I'll compute for after 1:00 am/pm and 8:00 am/pm.

Compute hour and minute alignment after 1:00 am/pm

Now that we know the radian value for each hour around an analog clock i.e. 1:00 am/pm is \( \frac{\pi}{6} \) and rate of change in seconds of the hour hand is \( \frac{\pi}{1,800 seconds} \) and minute hand is \( \frac{\pi}{21,600} \) then we can compute the hour and minute alignment after 1:00 am/pm.

Since the hour hand we're computing (1 am/pm) starts at \( \frac{\pi}{6} \) (refer to the radian chart) then we will add it as the starting position plus the rate of change of the hour hand. For the minute hand, we add \( 0 \) just to signify that the starting position is from 00:00:00 then add its rate of change. Then, we can finally express this problem using a formula below where \( x \) is the number of seconds where the hour hand equals (aligns with) the minute hand:

$$\frac{\pi}{6}\pi + \frac{\pi}{21,600/second}x = 0 + \frac{\pi}{1,800/second}x$$

Solving for \( x \) using algebra we get \( x = 327.27 seconds \) or 5 minutes and 27 seconds or 01:05:27 in hh:mm:ss format.

Compute hour and minute alignment after 8:00 am/pm

Since the hour hand we're computing (8 am/pm) starts at \( \frac{4}{3}\pi \) (refer to the radian chart) then we will add it as the starting position plus the rate of change of the hour hand. For the minute hand, we add \( 0 \) just to signify that the starting position is from 00:00:00 then add its rate of change. Then, we can finally express this problem using a formula below where \( x \) is the number of seconds where the hour hand equals (aligns with) the minute hand:

$$\frac{4}{3}\pi + \frac{\pi}{21,600/second}x = 0 + \frac{\pi}{1,800/second}x$$

Solving for \( x \) using algebra we get \( x = 2,618.18 seconds \) or 43 minutes and 38 seconds or 08:43:38 in hh:mm:ss format.

The General Formula

The general formula of this problem is :

$$ S + \frac{\pi}{21,600/second}x = 0 + \frac{\pi}{1,800/second}x$$

Where \( S \) is the starting position of the hour hand and \( x \) is the number of seconds from the starting position where the hour and minute hand aligns or overlaps. Using algebra we can simplify to:

$$ x = \frac{(21,600)(S)}{11 \pi} $$

Where \( S \) is starting position of the hour hand in radians. \( x \) will be in seconds, so it's just a matter of converting it in minutes. For example, in the previous example, \( x = 2,618.18 seconds \) can be converted to 43 minutes and 38 seconds. We know that the starting position is 8 am/pm so the exact time is 8:43:38 in hh:mm:ss format.

Now compute for other hour times.