Every time in a 12-hour cycle is listed on the right. Click one to load it.
A clock face is 360°, so each of the 12 hour marks is 30° apart and each minute mark is 6° apart. The minute hand moves 6° per minute. The hour hand moves 0.5° per minute — thirty degrees an hour, spread across sixty minutes. Almost every mistake in these problems comes from forgetting that second number and leaving the hour hand parked on its mark.
At 3:30 the minute hand is at 30 × 6 = 180°. The hour hand is not at 90° — it has spent half an hour creeping toward 4, so it sits at (3 × 30) + (30 × 0.5) = 105°. The difference is 75°, not the 90° most people answer. The shortcut |30H − 5.5M| gives |90 − 165| = 75 in one step.
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.