Every time in a 12-hour cycle is listed on the right, to the hundredth of a second. Click one to load it, or click the twelve-hour chart below.
Drag either hand. The other one follows, as it would on a real clock.
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Everything above is about the hour hand and the minute hand. There is a third hand on the face, and it turns the question into a better one: the three angles between three hands, and how often any of them close to nothing.
The minute and second hands meet 59 times an hour — not 60. The minute hand is moving too, so the second hand has to make up a full lap plus the ground the minute hand covered, and it gains a lap only 59 times in the hour. The same argument in slower motion gives the familiar result for the hour and minute hands: 11 times in twelve hours, every 65 minutes and 27.3 seconds, which is why they overlap 22 times a day and not 24.
And the hour and second hands meet 719 times in twelve hours, for the same reason with the numbers swapped. All three counts are one less than the naive answer, always for the same cause: the hand being chased has not stood still.
The hour and minute hands align only at multiples of 12/11 hours after noon. At every one of those moments except the first, the fractional part puts the second hand somewhere else entirely — 27.3 seconds past for the first, 54.5 for the second, and so on, never zero again until the eleventh brings you back to twelve o'clock. So the three hands are exactly together at 12:00:00 and at no other time on the dial.
The near miss is the pleasing part. Searched at hundredth-of-a-second resolution across the twelve hours, the closest the three ever come away from noon is a spread of 0.50°, and it happens twice: at 8:43:43.65 and at 3:16:16.35. The two are mirror images. Run the clock backwards from twelve and every hand sits at minus its forward angle, so whatever the three do at a time t after noon they do in reverse at t before it; 3:16:16.35 is exactly as far after noon as 8:43:43.65 is before midnight. Half a degree is about a millimetre at the tip of the second hand on a 30 cm wall clock: it looks exactly right from across a room, and it is wrong to anything that measures it. The second field takes hundredths, so both buttons land exactly — and moving either by six tenths of a second throws the second hand two degrees off, which is a fair demonstration of how unforgiving the fast hand makes this question.
At 3:00:00 the hour hand is at 90°, the minute hand at 0° and the second hand at 0°. Two of the three agree perfectly and the answer to "are the hands together" is still no.
At the first hour–minute overlap after twelve — 1:05:27.3 — those two are both at 32.7°, but the second hand is at 163.6°, a third of the way round the dial. The two-hand question says "aligned"; the three-hand question says the hands are 130° apart at their widest.
That gap is the whole reason the three-hand coincidence is so rare. The second hand moves 720 times faster than the hour hand, so it is only ever in the right place by accident — and the arithmetic says the accident happens exactly once per twelve hours, at the moment everything starts from zero together.
A clock face is 360°: hour marks 30° apart, minute marks 6°. The minute hand moves 6° a minute; the hour hand 0.5° a minute.
Hour hand 30H + 0.5M, minute hand 6M: the difference simplifies to |30H − 5.5M|. Over 180°, subtract from 360° for the smaller angle.
The gap between the hands changes at 5.5° a minute. Every question (when do they meet, oppose, make right angles) is that one number divided into an angle.
Leaving it on its mark is the single most common error. At 3:30 it is halfway to the 4.
The hands meet every 720/11 minutes: 65 minutes 27.27 seconds, a little over an hour, not exactly one. So they overlap 11 times in 12 hours and 22 times a day, and there is no overlap in the 11 o’clock hour: the one that would fall there happens exactly at 12:00. Only two coincidences fall on a whole minute, both 12:00; every other is at an eleventh of a minute, why a clock never quite looks like it is doing what you expect.
Twice per overlap cycle the hands pass through 90°, and there are 22 cycles a day. The same reasoning gives 44 times a day for any angle strictly between 0° and 180°, and only 22 for the two extremes, where the way in and the way out are the same moment. Solving per hour instead silently drops every solution past 180° and undercounts right angles as 17 rather than 22 in twelve hours.
Two hands always make two angles that add to 360°. Convention takes the smaller one; if a question asks for the reflex angle it wants the other: both are shown.
An idealised analogue clock: both hands sweep continuously, so the hour hand is already part-way to the next hour at twenty past. Many quartz movements step the minute hand once a second and some step the hour hand in five-minute jumps, which changes the angle at every moment except exactly on the hour.
Take the absolute difference between the hour hand at 30H + 0.5M and the minute hand at 6M, which simplifies to |30H − 5.5M|. If that exceeds 180°, subtract from 360° for the smaller angle between the hands.
It covers 360° in 12 hours — 30° per hour — and 30 ÷ 60 = 0.5° per minute. Leaving the hour hand on its mark is the single most common error in these problems.
75°, not 90°. The minute hand is at 180° and the hour hand has moved halfway from 3 to 4, reaching 105°.
Every 720/11 minutes — 65 minutes and 27.27 seconds. That is 11 overlaps in 12 hours and 22 in a day, not 12 and 24, because the minute hand must make up a whole lap on a target that is itself moving.
44. The hands pass through 90° twice in each overlap cycle, and there are 22 cycles a day.
The minute hand travels 360° an hour and the hour hand 30°, so it gains 330° an hour and laps the hour hand every 12/11 of an hour — 65 minutes and 273⁄11 seconds.
That is eleven times in twelve hours, not twelve. There is no overlap somewhere between 11 and 12, because the eleventh lands exactly on 12:00 — the hands catch up just as the cycle restarts. The same is true of the opposite case: eleven oppositions, and twenty-two right angles.
Only two of these fall on a whole minute, and both are 12:00. Every other coincidence happens at an eleventh of a minute, which is why a clock face never quite looks like it is doing what you expect.
| # | Hands together | Hands opposite |
|---|---|---|
| 1 | 12:00:00.000 | 12:32:43.636 |
| 2 | 1:05:27.273 | 1:38:10.909 |
| 3 | 2:10:54.545 | 2:43:38.182 |
| 4 | 3:16:21.818 | 3:49:05.455 |
| 5 | 4:21:49.091 | 4:54:32.727 |
| 6 | 5:27:16.364 | 6:00:00.000 |
| 7 | 6:32:43.636 | 7:05:27.273 |
| 8 | 7:38:10.909 | 8:10:54.545 |
| 9 | 8:43:38.182 | 9:16:21.818 |
| 10 | 9:49:05.455 | 10:21:49.091 |
| 11 | 10:54:32.727 | 11:27:16.364 |
Exact values, as elevenths of an hour rather than rounded clock times.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.