How many times per day, and at what time(s) do the hour and minute hands on a clock form EXACTLY a 60 degree angle?

The times the situation occurs come in pairs, equally before and after the two hands meet (0-degree separation). As the two hand meet 22 times a day, that situation happens every 65 5/11 minutes, starting at 12:00. The 60-degree situation happens 1/6 of this time interval, both before and after each 0-degree separation. That's 10 10/11 minutes before and after noon, and repeated every 65 5/11 after each preceding occurrence.

 0:10  10/11
 0:54  6/11
 1:16  4/11
 2:00 
 2:21  9/11
 3:05  5/11
 3:27  3/11
 4:10  10/11
 4:32  8/11
 5:16  4/11
 5:38  2/11
 6:21  9/11
 6:43  7/11
 7:27  3/11
 7:49  1/11
 8:32  8/11
 8:54  6/11
 9:38  2/11
 10:00 
 10:43  7/11
 11:05  5/11
 11:49  1/11
 12:10  10/11
 12:54  6/11
 13:16  4/11
 14:00 
 14:21  9/11
 15:05  5/11
 15:27  3/11
 16:10  10/11
 16:32  8/11
 17:16  4/11
 17:38  2/11
 18:21  9/11
 18:43  7/11
 19:27  3/11
 19:49  1/11
 20:32  8/11
 20:54  6/11
 21:38  2/11
 22:00 
 22:43  7/11
 23:05  5/11
 23:49  1/11

Posted by Charlie on 2007-03-23 09:07:51