At what time is the hour hand, minute hand and second hand exactly 120 degrees apart?

Question

Consider 12:20:40. The second hand and minute hand is not exactly 120 degrees. The minute hand would have moved by an additional 4 degrees in 40 seconds, thus the difference is only 116 degrees between the second and minute hand.

If there isn't an exact answer, at what time will hands be as close as possible to 120 degrees?

None of the answers so far are correct. Bandf's answer 2:54:34 has 123.4 degrees between the minute and second hand. This is 3.4 degrees away from 120 degrees.
One possible solution with none of the hands having more than 1 degree difference from 120 degrees is 5:49:09.
Hour vs minute: 120.325 degrees
minute vs second: 119.1 degrees
hour vs second: 120.575 degrees
The worst error here from 120 degrees is 0.9 degrees.
Maybe someone can find a solution more accurate than 5:49:09

Answer 1

There are no times at which the 3 hands are exactly 120 degrees apart but there are several where they are close.

Let's start thinking about the hour and minute hands only (leaving the second hand out of the picture for now).

If you think about these two hands, there are 11 times when they will be exactly in alignment, starting at 12:00 and then every 1 hr, 5 minutes, 27.2727 seconds after that.

(Note: 12 hours * 60 minutes * 60 seconds / 11 divisions = 3927.2727 seconds between occurrences)

Similarly, there are 11 times when the minute hand will be 120 degrees ahead of the hour hand and 11 times when the minute hand will be 120 degrees behind (or 240 degrees ahead) of the hour hand.

It turns out that each occurrence (0 degrees, 120 degrees, 240 degrees) is evenly spaced and happens every 21 minutes, 49.0909 seconds. (33 times in 12 hours).

(Note: 3927.2727 / 3 = 1309.0909 seconds which is 21 minutes 49.0909 seconds.)

We eliminate the 0 degree case, leaving 22 occurrences in a 12 hour period where the hands are +/- 120 degrees apart.

However, the second hand at each of these times is never exactly at the right spot.

In the following table, the first column is the time when the hour and minutes are exactly 120 degrees apart. The second column is when the minute hand is 120 degrees from the second hand. The last column is the number of seconds difference (rounded).

12:21:49 12:21:41 8
12:43:38 12:43:23 15

1:27:16 1:27:47 30
1:49:05 1:49:29 23

2:32:44 2:32:52 8
2:54:33 2:54:34 1 <-- closest

3:38:11 3:38:18 7
4:00:00 4:00:20 20

4:43:38 4:43:23 15
5:05:27 5:05:45 17

5:49:05 5:49:09 3
6:10:55 6:10:50 4

6:54:33 6:54:14 19
7:16:22 7:16:36 14

8:00:00 8:00:20 20
8:21:49 8:21:41 8

9:05:27 9:05:25 2
9:27:16 9:27:07 9

10:10:55 10:10:30 24
10:32:44 10:32:52 8

11:16:22 11:16:36 14
11:38:11 11:38:18 7

The closest time is at 2:54:34, as shown in the attached picture.

Answer 2

What about 4:40:00 or 8:20:00

It can only be done when the second hand is at 12 or:00

It would also depend on the movement of the dial. Some minute/hour hands do not move incrementally as the second/minute hand travels around the clock. Smaller wind up clocks and electronic clocks have small enough gears to progress smoothly between time marks while larger mechanical clocks progress sharpley from mark to mark .

Answer 3

First try to find out the relative speed of the hands. then, you will get the idea.

minutes hand:
360 degrees--1 hour
1 min--360/60=6degrees
1 sec-----6/60=(1/10)degrees

seconds hand:
1 min---360 degrees
1 sec---360/60=6 degrees

Relative speed of minutes hand and seconds hand is 5 1/10 degrees per second.(=51/10 degrees per sec).

at 12:20, seconds hand shows 12, and minutes hand shows 4. difference is exactly 120 degrees.

if u want to find another time after 12:20 at which the difference is 120 degrees, then the seconds hand should be 120 deg forward.

240/(51/10)=47.0588.......sec

at 12:20:47.05888, the difference will be 120 deg.

Answer 4

04:00:00. The second hand is at 12. The minute hand is at 12 and the hour hand is at 4. Or 08:00:00.

Answer 5

I have to be honest, I couldn't answer this but having read knarsimhaswa's answer, I can assure you he is correct! Having said that, I haven't actually checked the arithmetic but the method he has used is spot on.

Well done!

Answer 6

3:00.30
Three O'clock and 30 seconds after.

Answer 7

There's no exact solution.

Answer 8

there can be no such time