Clock

How many times a day hands on a clock overlap?

Result

n =  22

Solution:

 alpha = beta ; ;
 fraction{ 360° }{ 60} m + 360° k = fraction{ 360° }{ 12} fraction{ m}{ 60} ; ;
6° m + 360° k = 0.5° m ; ;
6m -0 = m/2; m = 2/11*0 = 0 min ==> 0:00:00
6m -360 = m/2; m = 2/11*360 = 65.45 min ==> 1:05:27
6m -720 = m/2; m = 2/11*720 = 130.91 min ==> 2:10:54
6m -1080 = m/2; m = 2/11*1080 = 196.36 min ==> 3:16:21
6m -1440 = m/2; m = 2/11*1440 = 261.82 min ==> 4:21:49
6m -1800 = m/2; m = 2/11*1800 = 327.27 min ==> 5:27:16
6m -2160 = m/2; m = 2/11*2160 = 392.73 min ==> 6:32:43
6m -2520 = m/2; m = 2/11*2520 = 458.18 min ==> 7:38:10
6m -2880 = m/2; m = 2/11*2880 = 523.64 min ==> 8:43:38
6m -3240 = m/2; m = 2/11*3240 = 589.09 min ==> 9:49:05
6m -3600 = m/2; m = 2/11*3600 = 654.55 min ==> 10:54:32
6m -3960 = m/2; m = 2/11*3960 = 720 min ==> 12:00:00
6m -4320 = m/2; m = 2/11*4320 = 785.45 min ==> 13:05:27
6m -4680 = m/2; m = 2/11*4680 = 850.91 min ==> 14:10:54
6m -5040 = m/2; m = 2/11*5040 = 916.36 min ==> 15:16:21
6m -5400 = m/2; m = 2/11*5400 = 981.82 min ==> 16:21:49
6m -5760 = m/2; m = 2/11*5760 = 1047.27 min ==> 17:27:16
6m -6120 = m/2; m = 2/11*6120 = 1112.73 min ==> 18:32:43
6m -6480 = m/2; m = 2/11*6480 = 1178.18 min ==> 19:38:10
6m -6840 = m/2; m = 2/11*6840 = 1243.64 min ==> 20:43:38
6m -7200 = m/2; m = 2/11*7200 = 1309.09 min ==> 21:49:05
6m -7560 = m/2; m = 2/11*7560 = 1374.55 min ==> 22:54:32
6m -7920 = m/2; m = 2/11*7920 = 1440 min ==> 24:00:00 !!! h>23

6m -8280 = m/2; m = 2/11*8280 = 1505.45 min ==> 25:05:27 !!! h>23



n=22








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