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        riddles >> medium >> Interchange Analog Watch Hands
        
(Message started by: THUDandBLUNDER on Jan 5^th, 2007, 5:08pm)

Title: Interchange Analog Watch Hands
Post by THUDandBLUNDER on Jan 5^th, 2007, 5:08pm

At what times between midnight and noon is it possible to interchange the minute and hour hands of a watch to get another valid time? For example, if it is 4 o'clock and you interchange the hands you won't get another valid time because the hands will not subtend the right angle.

Title: Re: Interchange Analog Watch Hands
Post by balakrishnan on Jan 5^th, 2007, 7:39pm

Let h and m be the angles of the hour and minute hand respectively(the reference being the position of 12)
So we have
h=k*pi/6+m/12
where k is an integer
Now for the given condition to hold,we must have
m=n*pi/6+h/12
where n is another integer
Solving these 2 equations,we find that
h=pi*(24k+2n)/143
m=pi*(24n+2k)/143

For any k(we are looking from k'o clock to (k+1)'o clock)
So we clearly want m/12 to be less than pi/6
or m<2*pi..as otherwise it will go to the next hour
in other words
24n+2k<246
or
0<=n<(143-k)/12
with k=0,1,2,3..11
we see that for k=0 to 11,we have 11 solutions of n
hence the total number of solutions is
143 times(excluding 1 boundary) 144(including both the boundaries)
Here are those times(of course rounded upto seconds)
0: 0: 0
0: 5: 2
0:10: 4
0:15: 6
0:20: 8
0:25:10
0:30:13
0:35:15
0:40:17
0:45:19
0:50:21
0:55:23
1: 0:25
1: 5:27
1:10:29
1:16:31
1:21:34
1:26:36
1:31:38
1:36:40
1:41:42
1:46:44
1:51:46
1:56:48
2: 1:50
2: 6:52
2:11:55
2:16:57
2:21:59
2:26: 1
2:31: 3
2:36: 5
2:41: 7
2:46: 9
2:51:11
2:56:13
3: 1:16
3: 6:18
3:11:20
3:16:22
3:21:24
3:26:26
3:31:28
3:37:30
3:42:32
3:47:34
3:52:37
3:57:39
4: 2:41
4: 7:43
4:12:45
4:17:47
4:22:49
4:27:51
4:32:53
4:37:55
4:42:57
4:47:60
4:52: 2
4:57: 4
5: 2: 6
5: 7: 8
5:12:10
5:17:12
5:22:14
5:27:16
5:32:18
5:37:21
5:42:23
5:47:25
5:52:27
5:57:29
6: 3:31
6: 8:33
6:13:35
6:18:37
6:23:39
6:28:42
6:33:44
6:38:46
6:43:48
6:48:50
6:53:52
6:58:54
7: 3:56
7: 8:58
7:13: 0
7:18: 3
7:23: 5
7:28: 7
7:33: 9
7:38:11
7:43:13
7:48:15
7:53:17
7:58:19
8: 3:21
8: 8:23
8:13:26
8:18:28
8:23:30
8:29:32
8:34:34
8:39:36
8:44:38
8:49:40
8:54:42
8:59:44
9: 4:47
9: 9:49
9:14:51
9:19:53
9:24:55
9:29:57
9:34:59
9:39: 1
9:44: 3
9:49: 5
9:54: 8
9:59:10
10: 4:12
10: 9:14
10:14:16
10:19:18
10:24:20
10:29:22
10:34:24
10:39:26
10:44:29
10:50:31
10:55:33
10:60:35
11: 5:37
11:10:39
11:15:41
11:20:43
11:25:45
11:30:47
11:35:50
11:40:52
11:45:54
11:50:56
11:55:58