exupero's blog

Hour and minute hand alignment

On an analog watch the minute hand often aligns with the much slower hour hand. To see exactly when requires some basic algebra:

where h is the hour and m is the minute in the hour. Setting θminute equal to θhour gives

For each hour, then, we can calculate exactly when the two hands align:

12 o'clock12:00:00
1 o'clock1:05:27
2 o'clock2:10:54
3 o'clock3:16:21
4 o'clock4:21:49
5 o'clock5:27:16
6 o'clock6:32:43
7 o'clock7:38:10
8 o'clock8:43:38
9 o'clock9:49:05
10 o'clock10:54:32
11 o'clock

The later the hour, the bigger the head start the hour hand has and the longer it takes the minute hand to catch up. Note that a time is missing for 11 o'clock. At 11 o'clock the hour hand starts so far ahead that the minute hand doesn't catch up until the start of the next hour, right at 12 o'clock. Between 11 o'clock and 12 o'clock the minute hand never overlays the hour hand.

One other way the minute hand can align with the hour hand is to point the directly opposite it, 180° ahead or behind the hour hand:

which gives us a second column:

12 o'clock12:00:0012:32:43
1 o'clock1:05:271:38:10
2 o'clock2:10:542:43:38
3 o'clock3:16:213:49:05
4 o'clock4:21:494:54:32
5 o'clock5:27:16
6 o'clock6:32:436:00:00
7 o'clock7:38:107:05:27
8 o'clock8:43:388:10:54
9 o'clock9:49:059:16:21
10 o'clock10:54:3210:21:49
11 o'clock11:27:16

Notice the oddity this time: between 5 and 6 o'clock the hour hand and minute hand never point exactly opposite each other. Just as they get close, the clock rolls over to 6 o'clock.