# How many times?

You have a high-precision watch where all the hands start at noon all pointing exactly to the 12 at the top. All the hands move in atomic ticks where there's no intermediary position in between ticks. The second hand ticks every second. The minute hand every minute and the hour hand every 12 minutes. They do so for 12 hours until they return to their intial position at midnight. How many times do they exactly divide the face of the watch into 3 equal sections (i.e. there's a $120\unicode{xb0}$ angle between each of the hands)?

24 times

Because

In every hour interval there is a possibility with the order being hours-minutes-seconds and a possibility with hours-seconds minutes. Because for every given minute-hour position the second reaches every other position and for every hour there is always a 120 degree position between hour and minute and a 240 degree position so there are 12*2=24 moments when it is so.

To check I also did a simulation in Java with this code: http://pastebin.com/8Zg2p3YR. The result:

0:21:41
0:43:23
1:27:47
1:49:29
2:32:52
2:54:34
3:38:58
3:59:39
4:0:40
4:43:3
5:5:45
5:49:9
6:10:50
6:54:14
7:16:56
7:59:19
8:0:20
8:21:1
9:5:25
9:27:7
10:10:30
10:32:12
11:16:36
11:38:18

• Yup, the only trick is there's no trick! As in the case with a completely analog watch (no solution) and other hand configuration puzzles where one or more of the positions get skipped. Mar 20, 2016 at 20:32
• @PaulEvans yeah at first I thought that maybe there was a trick with positions getting skipped when different hands tick at the same time or something, therefore I also did a simulation to be sure I was correct
– Ivo
Mar 20, 2016 at 20:34
• Well done! I purposely didn't make it no-computers so you had that option. Mar 20, 2016 at 20:39
• I hate to say it, but the alignment of the numbers and varying number of didgits for the minutes and seconds annoys me as well as makes it hard to read. I assume its Java's Fault.
– Ryan
Jun 13, 2016 at 18:42