What times on a clock form a straight line?

Answer 1

Starting with the hands in a straight line on the same side of the centre, consider how the hands move:

At 12 o'clock they are in a straight line on top of each other. In 1 hour, the minute hand has moved a full circle, but the hour hand has moved forward a bit, so they are in a straight line again when the minute hand a moved a bit more. This will be repeated for each hour, but when the hour hand reaches the 12 again, the hands will have been in line 11 times in the 12 hours. So the hands are in line every 12/11 hours, or 1 hour 5 mins 27 3/11 seconds, giving the times as (rounded to nearest second):

12:00:00, 1:05:27, 2:10:55, 3:16:22, 4:21:49. 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05, 10:54:33

When considering the hands in line on opposite side of the centre similar logic can be applied as before and so it is known that the hands will line up every 1 hour 5 mins 27 3/11 seconds, thus when the hands are lined up opposite the centre, the times will be (to the nearest second):

6:00:00, 7:05:27, 8:10:55, 9:16:22, 10:21:49, 11:27:16, 12:32:44, 1:38:11, 2:43:11, 3:49:05, 4:54:33

Answer 2

12:00, 3:15, 1:05, 4:20...ETC.

Answer 3

69 past 12