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clock angle problems

by peach1227 2024. 12. 27.
clock angle problem graph Graphical solution to the puzzle of what times hour and minute hands of a 12-hour clock are aligned by CMG Lee. The vertical axis, ''T'' denotes time elapsed from 12 o'clock in hours; the horizontal axis, ''P'' denotes the hands' positions and ''θ'' their angles. The thick solid red line denotes the hour hand; the thin solid blue lines denote the minute hand. Their intersections, denoted by red squares, give the times they are aligned. Similarly, the orange circles (dash-dot line) are when they are in opposition, and the pink triangles (dashed line) are when they are perpendicular. In the SVG file, hover over the graph to show the positions of the hands on a clock face. 0 1 2 3 4 5 6 7 8 9 10 11 12 T 00:00:00 1 ~01:05:27 2 ~02:10:55 3 ~03:16:22 4 ~04:21:49 5 ~05:27:16 6 ~06:32:44 7 ~07:38:11 8 ~08:43:38 9 ~09:49:0510 ~10:54:3311 12:00:00 XII I II III IV V VI VII VIII IX X XI XII P 90° 180° 270° 360° θ 00:00 00:00 00:015.5° 00:0211° 00:0316.5° 00:0422° 00:0527.5° 00:0633° 00:0738.5° 00:0844° 00:0949.5° 00:1055° 00:1160.5° 00:1266° 00:1371.5° 00:1477° 00:1582.5° 00:1688° 00:1793.5° 00:1899° 00:19104.5° 00:20110° 00:21115.5° 00:22121° 00:23126.5° 00:24132° 00:25137.5° 00:26143° 00:27148.5° 00:28154° 00:29159.5° 00:30165° 00:31170.5° 00:32176° 00:33178.5° 00:34173° 00:35167.5° 00:36162° 00:37156.5° 00:38151° 00:39145.5° 00:40140° 00:41134.5° 00:42129° 00:43123.5° 00:44118° 00:45112.5° 00:46107° 00:47101.5° 00:4896° 00:4990.5° 00:5085° 00:5179.5° 00:5274° 00:5368.5° 00:5463° 00:5557.5° 00:5652° 00:5746.5° 00:5841° 00:5935.5° 01:0030° 01:0124.5° 01:0219° 01:0313.5° 01:04 01:052.5° 01:06 01:078.5° 01:0814° 01:0919.5° 01:1025° 01:1130.5° 01:1236° 01:1341.5° 01:1447° 01:1552.5° 01:1658° 01:1763.5° 01:1869° 01:1974.5° 01:2080° 01:2185.5° 01:2291° 01:2396.5° 01:24102° 01:25107.5° 01:26113° 01:27118.5° 01:28124° 01:29129.5° 01:30135° 01:31140.5° 01:32146° 01:33151.5° 01:34157° 01:35162.5° 01:36168° 01:37173.5° 01:38179° 01:39175.5° 01:40170° 01:41164.5° 01:42159° 01:43153.5° 01:44148° 01:45142.5° 01:46137° 01:47131.5° 01:48126° 01:49120.5° 01:50115° 01:51109.5° 01:52104° 01:5398.5° 01:5493° 01:5587.5° 01:5682° 01:5776.5° 01:5871° 01:5965.5° 02:0060° 02:0154.5° 02:0249° 02:0343.5° 02:0438° 02:0532.5° 02:0627° 02:0721.5° 02:0816° 02:0910.5° 02:10 02:110.5° 02:12 02:1311.5° 02:1417° 02:1522.5° 02:1628° 02:1733.5° 02:1839° 02:1944.5° 02:2050° 02:2155.5° 02:2261° 02:2366.5° 02:2472° 02:2577.5° 02:2683° 02:2788.5° 02:2894° 02:2999.5° 02:30105° 02:31110.5° 02:32116° 02:33121.5° 02:34127° 02:35132.5° 02:36138° 02:37143.5° 02:38149° 02:39154.5° 02:40160° 02:41165.5° 02:42171° 02:43176.5° 02:44178° 02:45172.5° 02:46167° 02:47161.5° 02:48156° 02:49150.5° 02:50145° 02:51139.5° 02:52134° 02:53128.5° 02:54123° 02:55117.5° 02:56112° 02:57106.5° 02:58101° 02:5995.5° 03:0090° 03:0184.5° 03:0279° 03:0373.5° 03:0468° 03:0562.5° 03:0657° 03:0751.5° 03:0846° 03:0940.5° 03:1035° 03:1129.5° 03:1224° 03:1318.5° 03:1413° 03:157.5° 03:16 03:173.5° 03:18 03:1914.5° 03:2020° 03:2125.5° 03:2231° 03:2336.5° 03:2442° 03:2547.5° 03:2653° 03:2758.5° 03:2864° 03:2969.5° 03:3075° 03:3180.5° 03:3286° 03:3391.5° 03:3497° 03:35102.5° 03:36108° 03:37113.5° 03:38119° 03:39124.5° 03:40130° 03:41135.5° 03:42141° 03:43146.5° 03:44152° 03:45157.5° 03:46163° 03:47168.5° 03:48174° 03:49179.5° 03:50175° 03:51169.5° 03:52164° 03:53158.5° 03:54153° 03:55147.5° 03:56142° 03:57136.5° 03:58131° 03:59125.5° 04:00120° 04:01114.5° 04:02109° 04:03103.5° 04:0498° 04:0592.5° 04:0687° 04:0781.5° 04:0876° 04:0970.5° 04:1065° 04:1159.5° 04:1254° 04:1348.5° 04:1443° 04:1537.5° 04:1632° 04:1726.5° 04:1821° 04:1915.5° 04:2010° 04:214.5° 04:22 04:236.5° 04:2412° 04:2517.5° 04:2623° 04:2728.5° 04:2834° 04:2939.5° 04:3045° 04:3150.5° 04:3256° 04:3361.5° 04:3467° 04:3572.5° 04:3678° 04:3783.5° 04:3889° 04:3994.5° 04:40100° 04:41105.5° 04:42111° 04:43116.5° 04:44122° 04:45127.5° 04:46133° 04:47138.5° 04:48144° 04:49149.5° 04:50155° 04:51160.5° 04:52166° 04:53171.5° 04:54177° 04:55177.5° 04:56172° 04:57166.5° 04:58161° 04:59155.5° 05:00150° 05:01144.5° 05:02139° 05:03133.5° 05:04128° 05:05122.5° 05:06117° 05:07111.5° 05:08106° 05:09100.5° 05:1095° 05:1189.5° 05:1284° 05:1378.5° 05:1473° 05:1567.5° 05:1662° 05:1756.5° 05:1851° 05:1945.5° 05:2040° 05:2134.5° 05:2229° 05:2323.5° 05:2418° 05:2512.5° 05:26 05:271.5° 05:28 05:299.5° 05:3015° 05:3120.5° 05:3226° 05:3331.5° 05:3437° 05:3542.5° 05:3648° 05:3753.5° 05:3859° 05:3964.5° 05:4070° 05:4175.5° 05:4281° 05:4386.5° 05:4492° 05:4597.5° 05:46103° 05:47108.5° 05:48114° 05:49119.5° 05:50125° 05:51130.5° 05:52136° 05:53141.5° 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07:2950.5° 07:3045° 07:3139.5° 07:3234° 07:3328.5° 07:3423° 07:3517.5° 07:3612° 07:376.5° 07:38 07:394.5° 07:4010° 07:4115.5° 07:4221° 07:4326.5° 07:4432° 07:4537.5° 07:4643° 07:4748.5° 07:4854° 07:4959.5° 07:5065° 07:5170.5° 07:5276° 07:5381.5° 07:5487° 07:5592.5° 07:5698° 07:57103.5° 07:58109° 07:59114.5° 08:00120° 08:01125.5° 08:02131° 08:03136.5° 08:04142° 08:05147.5° 08:06153° 08:07158.5° 08:08164° 08:09169.5° 08:10175° 08:11179.5° 08:12174° 08:13168.5° 08:14163° 08:15157.5° 08:16152° 08:17146.5° 08:18141° 08:19135.5° 08:20130° 08:21124.5° 08:22119° 08:23113.5° 08:24108° 08:25102.5° 08:2697° 08:2791.5° 08:2886° 08:2980.5° 08:3075° 08:3169.5° 08:3264° 08:3358.5° 08:3453° 08:3547.5° 08:3642° 08:3736.5° 08:3831° 08:3925.5° 08:4020° 08:4114.5° 08:42 08:433.5° 08:44 08:457.5° 08:4613° 08:4718.5° 08:4824° 08:4929.5° 08:5035° 08:5140.5° 08:5246° 08:5351.5° 08:5457° 08:5562.5° 08:5668° 08:5773.5° 08:5879° 08:5984.5° 09:0090° 09:0195.5° 09:02101° 09:03106.5° 09:04112° 09:05117.5° 09:06123° 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