The other day I wanted to figure out a way to calculate every time on the clock where the minute hand and the hour hand lie directly over one another. I think I've got it figured out but I'm not 100% sure. Let's think of the time as x:y, x being the hour and y being the minute. Even though seconds are not units of distance, we can measure the distance of one full rotation as 60 "segments." Now for every 12 that the minute hand moves, the hour hand moves 1, so the ratio is 12:1. For every full number the hour hand advances, it is actually 5 minutes or segments. So combining that fact with the 12:1 ratio I came up with the equation y = 5x + y/12 and to get only one y in the equation y - y/12 = 5x 12(y - y/12) = 12(5x) 12y - y = 60x (11y)/11 = (60x)/11 y = 60x/11 Now let's say we want to figure out the specific time between 1:00 and 2:00. The hour will be 1 so we'll plug in 1 to the equation y = 60(1)/11 y = 60/11 y = 5.45 So if x=1 and y=5.45 the time will be 1:05.45 with the decimal being a fraction of a minute. To turn that into seconds we'll just multiply the .45 by 60 to get 27.0 seconds. If my calculations are correct, the hour hand and the minute hand will lie directly over one another at approximately 1:05 and 27.3 seconds and the other times can all be calculated by substituting numbers 1-11 for x. The times in hours:minutes.seconds would be 12:00, 1:05.27.0, 2:10.54.6, 3:16.21.6, 4:21.49.2, 5:27.16.2, 6:32.43.8, 7:38.10.8, 8:43.38.4, 9:49.05.4, 10:54.33.0 To show something interesting, say we want to figure out what time this will happen at the hour 11. This will not happen because it will get to 12:00 but let's plug in 11 anyway y = (60 x 11)/11 y = 660/11 y = 60 This will leave us with the time 11:60, which is 12:00, so it all works out nicely.