How many times a day do the clock hands overlap?
The clock hands puzzle: how many times in 24 hours do the hour and minute hands of an analogue clock overlap, and at exactly what times? Nearly everyone says 24. The answer is 22, and the relative-speed argument that proves it is worth owning.
An ordinary analogue clock has an hour hand and a minute hand, both sweeping smoothly rather than ticking. At midnight they lie exactly on top of each other.
How many times in the next 24 hours will the two hands overlap again, and at what exact times?
The snap answer is "once an hour, so 24". It is wrong, and the reason it is wrong is more interesting than the count. Between 12:00 and 1:00 the hands overlap at 12:00 only; by the time the minute hand comes round again at 1:00, the hour hand has moved on, and the catch-up happens a few minutes later. So the overlaps are not spaced by an hour. Work out what they are spaced by.
Before opening hints, write down the speed of each hand in degrees per minute. The whole puzzle is one subtraction and one division from there.
Stuck? Open hints one at a time
Each hint gives away one more layer. Stop the moment something clicks.
Hint 1 a nudge
The minute hand covers 360 degrees in 60 minutes: 6 degrees per minute. The hour hand covers 360 degrees in 12 hours: 0.5 degrees per minute.
Hint 2 the key move
Think of the minute hand chasing the hour hand. It gains 5.5 degrees every minute, and an overlap happens each time it has gained a full 360 degrees. How many minutes is that, and how many times does it fit into a day?
The solution
Reveal the full solution
Twenty-two times, spaced 65 minutes and 27.27 seconds apart.
The minute hand moves at 6 degrees per minute and the hour hand at 0.5, so the minute hand gains 5.5 degrees per minute on the hour hand. An overlap occurs every time that gain reaches 360 degrees, which takes 360 divided by 5.5, that is 720/11 minutes, or 65 minutes 27 and 3/11 seconds. A day is 1440 minutes, and 1440 divided by 720/11 is exactly 22. So there are 22 overlaps in 24 hours, 11 in each 12-hour half, at times 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, and 10:54:33, then the same cycle again. There is no overlap in the 11 o'clock hour on either cycle, which is the overlap the "24" answer thinks exists.
The exact-time list follows from the same fraction: the k-th overlap after midnight happens at k times 720/11 minutes. The interviewer's favourite check is asking for the time of the fifth overlap; a candidate who has the formula answers 5:27:16 in a few seconds, and one who has only the count starts sketching clock faces.
What this puzzle is really testing
Relative motion, which is reasoning in the frame of the slower object. Treating the hour hand as stationary and the minute hand as moving at 5.5 degrees per minute turns a two-body problem into a one-body one. That reframing is exactly the trick behind fast and slow pointers: in the slow pointer's frame, the fast pointer advances one step per iteration, so it must land on the slow one within a single loop of the cycle. The Floyd cycle mathematics deep-dive does the full argument, and it is the clock puzzle with pointers instead of hands.
Follow-ups to expect
How many times are the hands at right angles? Every 180 degrees of relative gain, so 44 times a day. Directly opposite? 22. What is the angle between the hands at 3:15? The minute hand is at 90 degrees, the hour hand at 90 plus 7.5, so 7.5 degrees. And the version with a second hand, asking when all three coincide: only at 12:00, twice a day, which takes a genuine argument about rational multiples to prove.