Answer to old chestnut: clock hands:

Some people want to say that it happens once an hour, but that's not quite right. After it happens at 12:00, the next occurrence is not at exactly 1:05, but a short bit afterward, because the hour hand moves 1/12 of the way from 1 to 2 in the time that the minute hand moves from 12 to 1. Each hour, the amount of this error increases, until we find that for the meeting of hands at "11:00", the hour hand has moved all the way to 12:00 before the minute hand catches up. As a result, there are 11 such meetings in a 12 hour period, or 22 in a day.

The exact angles of the hands (in degrees, starting with 0 at 12:00 and going clockwise) are given by these equations, where H is the hour of the day (0 for 12), and M is the minutes into that hour, including fractions:

hour angle = 30 H + M/2
minute angle = 6 M

We want combinations of H and M that make these two angles equal. That is,

30 H = 5.5 M, or

M = 60/11 H

So, we have solutions at 0:0 (12:00), 1:05 5/11 (that is, 5/11 minute after 1:05:00), 2:10 10/11, 3:16 4/11, 4:21 9/11, 5:27 3/11, 6:32 8/11, 7:38 2/11, 8:43 7/11, 9:49 1/11, and 10:54 6/11.

Other questions have closely related answers. The hands point in opposite directions the same number of times (22). Since the hands travel at constant rates, they pass that position once between every two times the hands meet. Likewise, they meet at right angles twice as often, or 44 times, because they do so twice between each meeting of the hands.

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