When are the two hands of a 12-hour clock in the same straight line except at 12 o'clock?

Answer 1

There are 21 clock times other than 12 o'clock when the hands "line up" in a straight line. In 10 of these, the hands are overlapping, and in the other 11, they are directly opposite each other on the clock face.

Hands together

For each 24-hour period, the hands on a clock face are in the same place 22 times, twice at the times designated "noon" and "midnight". The other 20 times are between the hours, when the hour hand has moved part of the way between the hours before being overtaken by the minute hand, The times are between 1 and 2, 2 and 3, and so forth, and finally between 10 and 11. There are none between 12 and 1, or 11 and 12.

The 10 clock times in minutes and seconds are defined by the formula M = H * (60/11) or the formula S = H * (3600/11). These can be defined as decimals or whole minutes and seconds where H is between 1 and 10.

Clock times (ten besides 12 o'clock) :

1:05:27 (1:05.4545 minutes, rounded down)

2:10:54 (2:10.9090 minutes, rounded down from 54.54 seconds)

3:16:21 (3:16.3636 minutes, rounded down from 21.82 seconds)

4:21:49 (4:21.8181 minutes, rounded down from 49.09 seconds)

5:27:16 (5:27.2727 minutes, rounded down from 16.36 seconds)

6:32:43 (6:32.7272 minutes, rounded down from 43.63 seconds)

7:38:10 (7:38.1818 minutes, rounded down from 10.91 seconds)

8:43:38 (8:43.6363 minutes, rounded down from 38.18 seconds)

9:49:05 (9:49.0909 minutes, rounded down from 5.45 seconds)

10:54:32 (10:54.5454 minutes, rounded down from 32.72 seconds)

Hands Opposite

The opposite times are when the hands are offset by 6 hours from the above.

Clock times (ten plus 6 o'clock exactly) :

12:32:43

1:38:10

2:43:38

3:49:05

4:54:32

6:00:00

7:05:27

8:10:54

9:16:21

10:21:49

11:27:16