Consider the following, seemingly simple question: how many times does the minute hand overlap the hour hand on an analog clock?
Well, the rotation rates are just ![2 \pi/\unit[1]{hr}](/images/2008/12/02/hrsminshandsIL1.png)
for the minute hand, and ![2 \pi/\unit[12]{hr}](/images/2008/12/02/hrsminshandsIL2.png)
for the hour hand. The minute hand moves 12 times as fast. I could simply write down the angular position of the hands as a function of time, which would just be
but thinking better of it, I realize that I need to reset the minute hand every time it reaches the 12-o-clock mark by subtracting 1 revolution, which is 
. Thus,
for positive integer 
. Equating these and solving, I get
![t=\frac{2 \pi n}{\omega_m - \omega_h} = \frac{12 \cdot \unit[60]{s}}{11} \, n](/images/2008/12/02/hrsminshandsDSP3.png)
Now, I can enumerate these in Matlab:
>> floor([12*[0:21]'/11 mod(12*60*[0:21]'/11,60) mod(12*60*[0:21]'/11,1)*60])
to which Matlab spits out
| 0:00:00 | 12:00:00 |
| 1:05:27 | 13:05:27 |
| 2:10:54 | 14:10:54 |
| 3:16:21 | 15:16:21 |
| 4:21:49 | 16:21:49 |
| 5:27:16 | 17:27:16 |
| 6:32:43 | 18:32:43 |
| 7:38:10 | 19:38:10 |
| 8:43:38 | 20:43:38 |
| 9:49:05 | 21:49:05 |
| 10:54:32 | 22:54:32 |
The next entry would be midnight of the next day, which doesn't count. Two columns of 11 rows, that's 22. Twenty two! Mind blowing. Since the minute hand never crosses the hour hand during the 11-o-clock hour, which happens twice per day, that happens two fewer times than hours in a day.