I've had this problem in my head for a long time, but I didn't feel motivated to solve it until today at Chinese school when I saw it actually happen: At exactly what times do the hands on a clock align on top of each other?

At first, I did it using a brute force method- plotting cosine functions and finding intersections. Then I gave it some more thought, and I figured there had to be some kind of pattern. After some more analysis, I postulated that if one hand moves around the clock N times in the time it takes another to go around once, then the hands will align in exactly (N-1) locations. In addition, these (N-1) locations will be equally spaced around the clockface. Therefore, if you evaluated 60n/11 (for n = integers from 0 to 10, inclusive), you'd find the minutes at which the hands align on a real 12-hour clock. I checked it, and the numbers matched up with the intersections from the cosine graphs. I still don't get how or why it works this way though... weird.

Man, it took me like 2 hours of number crunching (mostly as mental math) to verify my hypothesis. Haha, how dorky is that? That this is the kind of thing I sit around thinking about. =P Well, anyways, in case you ever wanted to know, here are the times at which the hands on a clock align (rounded to the nearest minute): 12:00, 1:05, 2:11, 3:16, 4:22, 5:27, 6:33, 7:38, 8:44, 9:49, and 10:55.