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It has often occured to me that many simple things surrounding me can be modeled by mathmatical equations. One day, out of pure boredom, I happened to look down at my watch and noticed that the hour and the minute hands were directly lined up. Quickly thinking about this I was curious as to how many times this occured on a daily basis, and specificly at what times. My first intiution was that it of course occured 11 times and the time were in multiples of 5. So it would occur at 1:05, 2:10, 3:15, etc.. However, this did not prove to be correct since therre is a slight adjustment for the fraction of the hour that has elapsed.

Calculating the times

1. Derive equations for each hand as a function of theta.

2. We set the two angles equal to each other.

3. We solve for M (minutes) in terms of H (hours) because H has to be a counting whole number from 1-12.

4. We plug in 1-12 to find the exact times at which the hands should cross.

{1:05, 2:11, 3:16, 4:22, 5:27, 6:33, 7:38, 8:44:, 9:49, 10:54, 12:00}

5. As we see, the time is actually slightly larger than every 5 minutes.

 

There are many other derivations that stem from this, such as, when are the hands antiparallel, and when do they form right angles.

1. Derive equations for each hand as a function of theta.

2. We set the two angles equal to each other plus our offset of pi (pi/2 for right angles, pi for antiparallel).

3. We solve for M (minutes) in terms of H (hours). There are two solutions, one for H < 5 and one for H > 5 because we don't want to have negative minutes.

4. We plug in 1-12 to find the exact times at which the hands are antiparallel (or at right angles if we had used pi/2). The only difference between this and when the hands were parallel is the scalar 360/11 which could be varied to find any angle between the two hands.

{1:38, 2:44, 3:49, 4:54, 6:00, 7:05, 8:11, 9:16, 10:22, 11:27, 12:33}