Sitting on a bus one day in Iran, as we were driving through the desert, I was looking at a digital 7-segment clock at the front of the bus. Near the clock was a convex mirror, used by the driver to allow him to see everyone in the back. In noticed the reflection of the clock in the mirror, and it amused me that sometimes the value displayed on the mirror image of the clock would still be a valid time.

This got me thinking about how many times are valid when reversed horizontally in a mirror (e.g. 00:51 = 12:00). I also wondered how many times are valid when reversed vertically in a mirror (e.g. 11:06 = 11:09), when rotated by 180 degrees (e.g. 10:20 = 05:01) and when the number order is reversed (e.g. 00:32 = 23:00). And, of course, for each of these how many are valid given a 12 hour clock (12:00 to 11:59) as well as a 24 hour clock (00:00 to 23:59).

The tables below are the fruits of my labour. I'm sure there must be a better way of describing this, with animation and maybe a 7 segment display font!

24 Hour Clock

Time	Vertical	Horizontal	Rotated	Reversed
00:00	00:00	00:00	00:00	00:00
00:01	00:01	10:00	10:00	10:00
00:02	00:05			20:00
00:03	00:03
00:05	00:02	20:00	20:00
00:08	00:08
00:10	00:10	01:00	01:00	01:00
00:11	00:11	11:00	11:00	11:00
00:12	00:15			21:00
00:15	00:12	21:00	21:00
00:18	00:18
00:20	00:50	05:00	05:00	02:00
00:21	00:51	15:00	15:00	12:00
00:22	00:55			22:00
00:23	00:53
00:25	00:52
00:28	00:58
00:30	00:30			03:00
00:31	00:31			13:00
00:32	00:35			23:00
00:33	00:33
00:35	00:32
00:38	00:38
00:40				04:00
00:41				14:00
00:50	00:20	02:00	02:00	05:00
00:51	00:21	12:00	12:00	15:00
00:52	00:25
00:53	00:23
00:55	00:22	22:00	22:00
00:58	00:28
01:00	01:00	00:10	00:10	00:10
01:01	01:01	10:10	10:10	10:10
01:02	01:05			20:10
01:03	01:03
01:05	01:02	20:10	20:10
01:08	01:08
01:10	01:10	01:10	01:10	01:10
01:11	01:11	11:10	11:10	11:10
01:12	01:15			21:10
01:15	01:12	21:10	21:10
01:18	01:18
01:20	01:50	05:10	05:10	02:10
01:21	01:51	15:10	15:10	12:10
01:22	01:55			22:10
01:23	01:53
01:25	01:52
01:28	01:58
01:30	01:30			03:10
01:31	01:31			13:10
01:32	01:35			23:10
01:33	01:33
01:35	01:32
01:38	01:38
01:40				04:10
01:41				14:10
01:50	01:20	02:10	02:10	05:10
01:51	01:21	12:10	12:10	15:10
01:52	01:25
01:53	01:23
01:55	01:22	22:10	22:10
01:58	01:28
02:00	05:00	00:50	00:50	00:20
02:01	05:01	10:50	10:50	10:20
02:02	05:05			20:20
02:03	05:03
02:05	05:02	20:50	20:50
02:08	05:08
02:10	05:10	01:50	01:50	01:20
02:11	05:11	11:50	11:50	11:20
02:12	05:15			21:20
02:15	05:12	21:50	21:50
02:18	05:18
02:20	05:50	05:50	05:50	02:20
02:21	05:51	15:50	15:50	12:20
02:22	05:55			22:20
02:23	05:53
02:25	05:52
02:28	05:58
02:30	05:30			03:20
02:31	05:31			13:20
02:32	05:35			23:20
02:33	05:33
02:35	05:32
02:38	05:38
02:40				04:20
02:41				14:20
02:50	05:20	02:50	02:50	05:20
02:51	05:21	12:50	12:50	15:20
02:52	05:25
02:53	05:23
02:55	05:22	22:50	22:50
02:58	05:28
03:00	03:00			00:30
03:01	03:01			10:30
03:02	03:05			20:30
03:03	03:03
03:05	03:02
03:08	03:08
03:10	03:10			01:30
03:11	03:11			11:30
03:12	03:15			21:30
03:15	03:12
03:18	03:18
03:20	03:50			02:30
03:21	03:51			12:30
03:22	03:55			22:30
03:23	03:53
03:25	03:52
03:28	03:58
03:30	03:30			03:30
03:31	03:31			13:30
03:32	03:35			23:30
03:33	03:33
03:35	03:32
03:38	03:38
03:40				04:30
03:41				14:30
03:50	03:20			05:30
03:51	03:21			15:30
03:52	03:25
03:53	03:23
03:55	03:22
03:58	03:28
04:00				00:40
04:01				10:40
04:02				20:40
04:10				01:40
04:11				11:40
04:12				21:40
04:20				02:40
04:21				12:40
04:22				22:40
04:30				03:40
04:31				13:40
04:32				23:40
04:40				04:40
04:41				14:40
04:50				05:40
04:51				15:40
05:00	02:00	00:20	00:20	00:50
05:01	02:01	10:20	10:20	10:50
05:02	02:05			20:50
05:03	02:03
05:05	02:02	20:20	20:20
05:08	02:08
05:10	02:10	01:20	01:20	01:50
05:11	02:11	11:20	11:20	11:50
05:12	02:15			21:50
05:15	02:12	21:20	21:20
05:18	02:18
05:20	02:50	05:20	05:20	02:50
05:21	02:51	15:20	15:20	12:50
05:22	02:55			22:50
05:23	02:53
05:25	02:52
05:28	02:58
05:30	02:30			03:50
05:31	02:31			13:50
05:32	02:35			23:50
05:33	02:33
05:35	02:32
05:38	02:38
05:40				04:50
05:41				14:50
05:50	02:20	02:20	02:20	05:50
05:51	02:21	12:20	12:20	15:50
05:52	02:25
05:53	02:23
05:55	02:22	22:20	22:20
05:58	02:28
08:00	08:00
08:01	08:01
08:02	08:05
08:03	08:03
08:05	08:02
08:08	08:08
08:10	08:10
08:11	08:11
08:12	08:15
08:15	08:12
08:18	08:18
08:20	08:50
08:21	08:51
08:22	08:55
08:23	08:53
08:25	08:52
08:28	08:58
08:30	08:30
08:31	08:31
08:32	08:35
08:33	08:33
08:35	08:32
08:38	08:38
08:50	08:20
08:51	08:21
08:52	08:25
08:53	08:23
08:55	08:22
08:58	08:28
10:00	10:00	00:01	00:01	00:01
10:01	10:01	10:01	10:01	10:01
10:02	10:05			20:01
10:03	10:03
10:05	10:02	20:01	20:01
10:08	10:08
10:10	10:10	01:01	01:01	01:01
10:11	10:11	11:01	11:01	11:01
10:12	10:15			21:01
10:15	10:12	21:01	21:01
10:18	10:18
10:20	10:50	05:01	05:01	02:01
10:21	10:51	15:01	15:01	12:01
10:22	10:55			22:01
10:23	10:53
10:25	10:52
10:28	10:58
10:30	10:30			03:01
10:31	10:31			13:01
10:32	10:35			23:01
10:33	10:33
10:35	10:32
10:38	10:38
10:40				04:01
10:41				14:01
10:50	10:20	02:01	02:01	05:01
10:51	10:21	12:01	12:01	15:01
10:52	10:25
10:53	10:23
10:55	10:22	22:01	22:01
10:58	10:28
11:00	11:00	00:11	00:11	00:11
11:01	11:01	10:11	10:11	10:11
11:02	11:05			20:11
11:03	11:03
11:05	11:02	20:11	20:11
11:08	11:08
11:10	11:10	01:11	01:11	01:11
11:11	11:11	11:11	11:11	11:11
11:12	11:15			21:11
11:15	11:12	21:11	21:11
11:18	11:18
11:20	11:50	05:11	05:11	02:11
11:21	11:51	15:11	15:11	12:11
11:22	11:55			22:11
11:23	11:53
11:25	11:52
11:28	11:58
11:30	11:30			03:11
11:31	11:31			13:11
11:32	11:35			23:11
11:33	11:33
11:35	11:32
11:38	11:38
11:40				04:11
11:41				14:11
11:50	11:20	02:11	02:11	05:11
11:51	11:21	12:11	12:11	15:11
11:52	11:25
11:53	11:23
11:55	11:22	22:11	22:11
11:58	11:28
12:00	15:00	00:51		00:21
12:01	15:01	10:51		10:21
12:02	15:05			20:21
12:03	15:03
12:05	15:02	20:51
12:08	15:08
12:10	15:10	01:51		01:21
12:11	15:11	11:51		11:21
12:12	15:15			21:21
12:15	15:12	21:51
12:18	15:18
12:20	15:50	05:51		02:21
12:21	15:51	15:51		12:21
12:22	15:55			22:21
12:23	15:53
12:25	15:52
12:28	15:58
12:30	15:30			03:21
12:31	15:31			13:21
12:32	15:35			23:21
12:33	15:33
12:35	15:32
12:38	15:38
12:40				04:21
12:41				14:21
12:50	15:20	02:51		05:21
12:51	15:21	12:51		15:21
12:52	15:25
12:53	15:23
12:55	15:22	22:51
12:58	15:28
13:00				00:31
13:01				10:31
13:02				20:31
13:10				01:31
13:11				11:31
13:12				21:31
13:20				02:31
13:21				12:31
13:22				22:31
13:30				03:31
13:31				13:31
13:32				23:31
13:40				04:31
13:41				14:31
13:50				05:31
13:51				15:31
14:00				00:41
14:01				10:41
14:02				20:41
14:10				01:41
14:11				11:41
14:12				21:41
14:20				02:41
14:21				12:41
14:22				22:41
14:30				03:41
14:31				13:41
14:32				23:41
14:40				04:41
14:41				14:41
14:50				05:41
14:51				15:41
15:00	12:00	00:21	00:21	00:51
15:01	12:01	10:21	10:21	10:51
15:02	12:05			20:51
15:03	12:03
15:05	12:02	20:21	20:21
15:08	12:08
15:10	12:10	01:21	01:21	01:51
15:11	12:11	11:21	11:21	11:51
15:12	12:15			21:51
15:15	12:12	21:21	21:21
15:18	12:18
15:20	12:50	05:21	05:21	02:51
15:21	12:51	15:21	15:21	12:51
15:22	12:55			22:51
15:23	12:53
15:25	12:52
15:28	12:58
15:30	12:30			03:51
15:31	12:31			13:51
15:32	12:35			23:51
15:33	12:33
15:35	12:32
15:38	12:38
15:40				04:51
15:41				14:51
15:50	12:20	02:21	02:21	05:51
15:51	12:21	12:21	12:21	15:51
15:52	12:25
15:53	12:23
15:55	12:22	22:21	22:21
15:58	12:28
18:00	18:00
18:01	18:01
18:02	18:05
18:03	18:03
18:05	18:02
18:08	18:08
18:10	18:10
18:11	18:11
18:12	18:15
18:15	18:12
18:18	18:18
18:20	18:50
18:21	18:51
18:22	18:55
18:23	18:53
18:25	18:52
18:28	18:58
18:30	18:30
18:31	18:31
18:32	18:35
18:33	18:33
18:35	18:32
18:38	18:38
18:50	18:20
18:51	18:21
18:52	18:25
18:53	18:23
18:55	18:22
18:58	18:28
20:00		00:05	00:05	00:02
20:01		10:05	10:05	10:02
20:02				20:02
20:05		20:05	20:05
20:10		01:05	01:05	01:02
20:11		11:05	11:05	11:02
20:12				21:02
20:15		21:05	21:05
20:20		05:05	05:05	02:02
20:21		15:05	15:05	12:02
20:22				22:02
20:30				03:02
20:31				13:02
20:32				23:02
20:40				04:02
20:41				14:02
20:50		02:05	02:05	05:02
20:51		12:05	12:05	15:02
20:55		22:05	22:05
21:00		00:15	00:15	00:12
21:01		10:15	10:15	10:12
21:02				20:12
21:05		20:15	20:15
21:10		01:15	01:15	01:12
21:11		11:15	11:15	11:12
21:12				21:12
21:15		21:15	21:15
21:20		05:15	05:15	02:12
21:21		15:15	15:15	12:12
21:22				22:12
21:30				03:12
21:31				13:12
21:32				23:12
21:40				04:12
21:41				14:12
21:50		02:15	02:15	05:12
21:51		12:15	12:15	15:12
21:55		22:15	22:15
22:00		00:55		00:22
22:01		10:55		10:22
22:02				20:22
22:05		20:55
22:10		01:55		01:22
22:11		11:55		11:22
22:12				21:22
22:15		21:55
22:20		05:55		02:22
22:21		15:55		12:22
22:22				22:22
22:30				03:22
22:31				13:22
22:32				23:22
22:40				04:22
22:41				14:22
22:50		02:55		05:22
22:51		12:55		15:22
22:55		22:55
23:00				00:32
23:01				10:32
23:02				20:32
23:10				01:32
23:11				11:32
23:12				21:32
23:20				02:32
23:21				12:32
23:22				22:32
23:30				03:32
23:31				13:32
23:32				23:32
23:40				04:32
23:41				14:32
23:50				05:32
23:51				15:32

12 Hour Clock

Time	Vertical	Horizontal	Rotated	Reversed
01:00	01:00
01:01	01:01	10:10	10:10	10:10
01:02	01:05
01:03	01:03
01:05	01:02
01:08	01:08
01:10	01:10	01:10	01:10	01:10
01:11	01:11	11:10	11:10	11:10
01:12	01:15
01:15	01:12
01:18	01:18
01:20	01:50	05:10	05:10	02:10
01:21	01:51			12:10
01:22	01:55
01:23	01:53
01:25	01:52
01:28	01:58
01:30	01:30			03:10
01:31	01:31
01:32	01:35
01:33	01:33
01:35	01:32
01:38	01:38
01:40				04:10
01:50	01:20	02:10	02:10	05:10
01:51	01:21	12:10	12:10
01:52	01:25
01:53	01:23
01:55	01:22
01:58	01:28
02:00	05:00
02:01	05:01	10:50	10:50	10:20
02:02	05:05
02:03	05:03
02:05	05:02
02:08	05:08
02:10	05:10	01:50	01:50	01:20
02:11	05:11	11:50	11:50	11:20
02:12	05:15
02:15	05:12
02:18	05:18
02:20	05:50	05:50	05:50	02:20
02:21	05:51			12:20
02:22	05:55
02:23	05:53
02:25	05:52
02:28	05:58
02:30	05:30			03:20
02:31	05:31
02:32	05:35
02:33	05:33
02:35	05:32
02:38	05:38
02:40				04:20
02:50	05:20	02:50	02:50	05:20
02:51	05:21	12:50	12:50
02:52	05:25
02:53	05:23
02:55	05:22
02:58	05:28
03:00	03:00
03:01	03:01			10:30
03:02	03:05
03:03	03:03
03:05	03:02
03:08	03:08
03:10	03:10			01:30
03:11	03:11			11:30
03:12	03:15
03:15	03:12
03:18	03:18
03:20	03:50			02:30
03:21	03:51			12:30
03:22	03:55
03:23	03:53
03:25	03:52
03:28	03:58
03:30	03:30			03:30
03:31	03:31
03:32	03:35
03:33	03:33
03:35	03:32
03:38	03:38
03:40				04:30
03:50	03:20			05:30
03:51	03:21
03:52	03:25
03:53	03:23
03:55	03:22
03:58	03:28
04:01				10:40
04:10				01:40
04:11				11:40
04:20				02:40
04:21				12:40
04:30				03:40
04:40				04:40
04:50				05:40
05:00	02:00
05:01	02:01	10:20	10:20	10:50
05:02	02:05
05:03	02:03
05:05	02:02
05:08	02:08
05:10	02:10	01:20	01:20	01:50
05:11	02:11	11:20	11:20	11:50
05:12	02:15
05:15	02:12
05:18	02:18
05:20	02:50	05:20	05:20	02:50
05:21	02:51			12:50
05:22	02:55
05:23	02:53
05:25	02:52
05:28	02:58
05:30	02:30			03:50
05:31	02:31
05:32	02:35
05:33	02:33
05:35	02:32
05:38	02:38
05:40				04:50
05:50	02:20	02:20	02:20	05:50
05:51	02:21	12:20	12:20
05:52	02:25
05:53	02:23
05:55	02:22
05:58	02:28
08:00	08:00
08:01	08:01
08:02	08:05
08:03	08:03
08:05	08:02
08:08	08:08
08:10	08:10
08:11	08:11
08:12	08:15
08:15	08:12
08:18	08:18
08:20	08:50
08:21	08:51
08:22	08:55
08:23	08:53
08:25	08:52
08:28	08:58
08:30	08:30
08:31	08:31
08:32	08:35
08:33	08:33
08:35	08:32
08:38	08:38
08:50	08:20
08:51	08:21
08:52	08:25
08:53	08:23
08:55	08:22
08:58	08:28
10:00	10:00
10:01	10:01	10:01	10:01	10:01
10:02	10:05
10:03	10:03
10:05	10:02
10:08	10:08
10:10	10:10	01:01	01:01	01:01
10:11	10:11	11:01	11:01	11:01
10:12	10:15
10:15	10:12
10:18	10:18
10:20	10:50	05:01	05:01	02:01
10:21	10:51			12:01
10:22	10:55
10:23	10:53
10:25	10:52
10:28	10:58
10:30	10:30			03:01
10:31	10:31
10:32	10:35
10:33	10:33
10:35	10:32
10:38	10:38
10:40				04:01
10:50	10:20	02:01	02:01	05:01
10:51	10:21	12:01	12:01
10:52	10:25
10:53	10:23
10:55	10:22
10:58	10:28
11:00	11:00
11:01	11:01	10:11	10:11	10:11
11:02	11:05
11:03	11:03
11:05	11:02
11:08	11:08
11:10	11:10	01:11	01:11	01:11
11:11	11:11	11:11	11:11	11:11
11:12	11:15
11:15	11:12
11:18	11:18
11:20	11:50	05:11	05:11	02:11
11:21	11:51			12:11
11:22	11:55
11:23	11:53
11:25	11:52
11:28	11:58
11:30	11:30			03:11
11:31	11:31
11:32	11:35
11:33	11:33
11:35	11:32
11:38	11:38
11:40				04:11
11:50	11:20	02:11	02:11	05:11
11:51	11:21	12:11	12:11
11:52	11:25
11:53	11:23
11:55	11:22
11:58	11:28
12:01		10:51		10:21
12:10		01:51		01:21
12:11		11:51		11:21
12:20		05:51		02:21
12:21				12:21
12:30				03:21
12:40				04:21
12:50		02:51		05:21
12:51		12:51

Code

The JavaScript code that created this data is:

var fs = require('fs');

function saveTime(output, type, time, newtime) {
	if (!(time in output)) output[time] = {};
	output[time][type] = newtime;
}

function createMarkdownTable(times) {
	var table = '| Time       | Vertical   | Horizontal | Rotated    | Reversed   |\n| ---------- | ---------- | ---------- | ---------- | ---------- |\n';
	times.forEach((time) => {
		if (time && 'time' in time) {
			table += '| ' + time.time + '      | ' + (time.vertical || '     ') + '      | ' + (time.horizontal || '     ') + '      | ' + (time.rotated || '     ') + '      | ' + (time.reversed || '     ') + '      |\n';
		}
	});
	return table;
}

function createHTMLTable(times) {
	var table = '<table><thead><tr><td>Time</td><td>Vertical</td><td>Horizontal</td><td>Rotated</td><td>Reversed</td></tr></thead><tbody>\n';
	times.forEach((time) => {
		if (time && 'time' in time) {
			table += '<tr><td>' + time.time + '</td><td>' + (time.vertical || '') + '</td><td>' + (time.horizontal || '') + '</td><td>' + (time.rotated || '') + '</td><td>' + (time.reversed || '') + '</td></tr>\n';
		}
	});
	return table + '</tbody></table>';
}

function istwelvehour(timestring) {
	if (timestring) {
		var hour = parseInt(timestring.slice(0, 2));
		var minute = parseInt(timestring.slice(3, 5));
		if (hour >= 1 && hour <= 12 && minute >= 0 && minute <= 59) return true;
	}
}

function buildTransposedTimes(output, type, times) {
	Object.entries(times[type].hours).forEach(([hour, newminute]) => {
		Object.entries(times[type].minutes).forEach(([minute, newhour]) => {
			saveTime(output, type, [hour, minute].join(":"), [newhour, newminute].join(":"));
		});
	});
}

function buildTimes(output, type, times) {
	Object.entries(times[type].hours).forEach(([hour, newhour]) => {
		Object.entries(times[type].minutes).forEach(([minute, newminute]) => {
			saveTime(output, type, [hour, minute].join(":"), [newhour, newminute].join(":"));
		});
	});
}

module.exports = (clock = 24, format = 'js') => {
	const times = require('./times.json');
	var output = {};
	
	buildTimes(output, 'vertical', times);
	['horizontal', 'rotated', 'reversed'].forEach((type) => {
		buildTransposedTimes(output, type, times);
	});
	
	var ordered = [];
	Object.keys(output).sort().forEach(function(key) {
		ordered.push(Object.assign({
			time: key
		}, output[key]));
	});
	
	if (clock == 12) {
		ordered = ordered.map(time => {
			if (!istwelvehour(time.time)) return;
			['vertical', 'horizontal', 'rotated', 'reversed'].forEach((type) => {
				if (!istwelvehour(time[type])) delete time[type];
			});
			if (Object.keys(time).length > 1) return time;
		});
	}
	var response;
	switch(format) {
		case 'js':
			response = ordered;
			break;
		case 'json':
			response = JSON.stringify(ordered);
			break;
		case 'table':
			response = createHTMLTable(ordered);
			break;
		case 'markdown':
			response = createMarkdownTable(ordered);
			break;
	}
	return response;
};

This is the data file (times.json) that is used by the code. It is a list of times and their reflection in the different directions:

{
	"vertical": {
		"hours": {
			"00": "00",
			"01": "01",
			"02": "05",
			"03": "03",
			"05": "02",
			"08": "08",
			"10": "10",
			"11": "11",
			"12": "15",
			"15": "12",
			"18": "18"
		},
		"minutes": {
			"00": "00",
			"01": "01",
			"02": "05",
			"03": "03",
			"05": "02",
			"08": "08",
			"10": "10",
			"11": "11",
			"12": "15",
			"15": "12",
			"18": "18",
			"20": "50",
			"21": "51",
			"22": "55",
			"23": "53",
			"25": "52",
			"28": "58",
			"30": "30",
			"31": "31",
			"32": "35",
			"33": "33",
			"35": "32",
			"38": "38",
			"50": "20",
			"51": "21",
			"52": "25",
			"53": "23",
			"55": "22",
			"58": "28"
		}
	},
	"horizontal": {
		"hours": {
			"00": "00",
			"01": "10",
			"02": "50",
			"05": "20",
			"10": "01",
			"11": "11",
			"12": "51",
			"15": "21",
			"20": "05",
			"21": "15",
			"22": "55"
		},
		"minutes": {
			"00": "00",
			"01": "10",
			"05": "20",
			"10": "01",
			"11": "11",
			"15": "21",
			"20": "05",
			"21": "15",
			"50": "02",
			"51": "12",
			"55": "22"
		}
	},
	"rotated": {
		"hours": {
			"00": "00",
			"01": "10",
			"02": "50",
			"05": "20",
			"10": "01",
			"11": "11",
			"15": "21",
			"20": "05",
			"21": "15"
		},
		"minutes": {
			"00": "00",
			"01": "10",
			"05": "20",
			"10": "01",
			"11": "11",
			"15": "21",
			"20": "05",
			"21": "15",
			"50": "02",
			"51": "12",
			"55": "22"
		}
	},
	"reversed": {
		"hours": {
			"00": "00",
			"01": "10",
			"02": "20",
			"03": "30",
			"04": "40",
			"05": "50",
			"10": "01",
			"11": "11",
			"12": "21",
			"13": "31",
			"14": "41",
			"15": "51",
			"20": "02",
			"21": "12",
			"22": "22",
			"23": "32"
		},
		"minutes": {
			"00": "00",
			"01": "10",
			"02": "20",
			"10": "01",
			"11": "11",
			"12": "21",
			"20": "02",
			"21": "12",
			"22": "22",
			"30": "03",
			"31": "13",
			"32": "23",
			"40": "04",
			"41": "14",
			"50": "05",
			"51": "15"
		}
	}
}