Clock hands

I always liked the analogue clock. I tried to wear digital, and could not get used to the clock in the Windows tray - for my brain, this means an additional digital-to-analog conversion, so instead of them I have Analog Clock .
Once, in one entertaining book, I read the condition of the problem, which stated that if you swap the minute and hour hands, you will get an abracadabra in most cases. But there are such states of arrows, when their conversion leads to a real result, the trivial solution in this case is midday or midnight. And what are the other "temporary pairs" on the dial? Then I closed the problem book and began to solve.

It is easy to know that in a minute the minute hand passes 6 °, and the hour hand - 0.1 °.
Let the time on some hours - x seconds, and on the other - y seconds. Then for the first hours, x / 120 is the angle that the hour hand passes, and x / 10 is the angle that the minute hand passes during time x . But since the minute hand at the same time makes as many full turns as many hours have passed, it is necessary to make an amendment - x / 10 - 360m , where m is the number of whole hours. Similar picture for the second hours.

Now we have a system of equations:
x / 120 = y / 10 - 360m;
y / 120 = x / 10 - 360n;
n, m є [0.11].
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Where do we find that
x = 302. (097902) · (12n + m);
y = 302. (097902) · (12m + n).

Thus, we have expressed time on both hours through the extra minute minute turnovers. Since the correction factors for both clocks are in the set of integers from 0 to 11, it remains only to construct a square matrix of 12 x 12, the diagonal of which will contain the time when the two arrows match, and at the same time serve as the axis of symmetry for finding the pairs of times.

0:00:00	1:00:25	2:00:50	3:01:15	4:01:40	5:02:05	6:02:31	7:02:56	8:03:21	9:03:46	10:04:11	11:04:36
0:05:02	1:05:27	2:05:52	3:06:17	4:06:42	5:07:07	6:07:33	7:07:58	8:08:23	9:08:48	10:09:13	11:09:39
0:10:04	1:10:29	2:10:54	3:11:19	4:11:44	5:12:10	6:12:35	7:13:00	8:13:25	9:13:50	10:14:15	11:14:41
0:15:06	1:15:31	2:15:56	3:16:21	4:16:46	5:17:12	6:17:37	7:18:02	8:18:27	9:18:52	10:19:18	11:19:43
0:20:08	1:20:33	2:20:58	3:21:23	4:21:49	5:22:14	6:22:39	7:23:04	8:23:29	9:23:54	10:24:20	11:24:45
0:25:10	1:25:35	2:26:00	3:26:26	4:26:51	5:27:16	6:27:41	07:28:06	8:28:31	9:28:57	10:29:22	11:29:47
0:30:12	1:30:37	2:31:02	3:31:28	4:31:53	5:32:18	6:32:43	7:33:08	8:33:33	9:33:59	10:34:24	11:34:49
0:35:14	1:35:39	2:36:05	3:36:30	4:36:55	5:37:20	6:37:45	7:38:10	8:38:36	9:39:01	10:39:26	11:39:51
0:40:16	1:40:41	2:41:07	03:41:32	4:41:57	5:42:22	6:42:47	7:43:13	8:43:38	9:44:03	10:44:28	11:44:53
0:45:18	1:45:44	2:46:09	3:46:34	4:46:59	5:47:24	6:47:49	7:48:15	8:48:40	9:49:05	10:49:30	11:49:55
0:50:20	1:50:46	2:51:11	03:51:36	4:52:01	5:52:26	6:52:52	7:53:17	8:53:42	9:54:07	10:54:32	11:54:57
0:55:23	1:55:48	2:56:13	Three to four	Three to four	5:57:28	6:57:54	7:58:19	8:58:44	9:59:09	10:59:34	12:00:00

The attentive reader will notice that the pairs of time are every 5: 02.09 minutes, and every 65:45 minutes there is a combination of arrows.
The result: 66 pairs, 11 combinations of hands, and 2 coincidences, because 0 and 43200 seconds on the clock are one and the same.