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Left-Handed Times

A "left-handed" clock has its numbers printed counter-clockwise and runs
counter-clockwise to point at them, effectively performing as a mirror image to
a normal clock. Between noon and midnight, at what times are the hands of
a left-handed clock pointing in exactly the same directions as those of a
regular clock (comparing only hour and minute hands, which may be swapped.)

Source: Original. [You may ignore any
effect due
to the electromagnetic weak force.]

Solutions were received from Joseph DeVincentis, Kirk Bresniker, Kevin
Haborak, and Dan Chirica. Everyone noticed that the problem was
basically one of finding clock times with vertical symmetry.

Joseph listed these times as precise fractions:

In order for the hands of left and right handed clocks to be in the same positions at the same time, one
of two things can be true:
(1) the hour hands are in the same place, and the minute hands are in the same place. This happens only at
6:00 and 12:00.
(2) the hour and minute hands are swapped. This requires the hour and minute hands to form a mirror-image
pair through the vertical bisector of each clock face.
For case 2, at a time of A hours and B minutes past noon, the hour hand will be pointing between
the A and A+1, B/60 of the way past the hour. The minute hand will be at B/5. We need A + B/60 + B/5 = 12,
because one hand must be as far past the 12 as the other one needs to reach it. So A + 13B/60 = 12.
For each A from 0 to 11 you can find a solution; A=12 gives the 12:00 solution already found in the first part.
For A = 0, 13B/60 = 12, or B = 720/13 = 55 and 5/13, for a time of 12:55 and 5/13.
For A = 1, 13B/60 = 11, or B = 660/13 = 50 and 10/13, or a time of 1:50 and 10/13.
For A = 2, 13B/60= 10, B = 600/13 = 46 and 2/13, for the time 2:46 and 2/13.
Likewise we get
3:41 and 7/13,
4:36 and 12/13,
5:32 and 4/13,
6:27 and 9/13,
7:23 and 1/13,
8:18 and 6/13,
9:13 and 11/13,
10:09 and 3/13, and
11:04 and 8/13.

Dan Chirica listed these times digitally. In addition to 12:00 and 6:00,
0:55:23
1:50:46
2:46:9
3:41:32
4:36:55
5:32:18
6:27:41
7:23:4
8:18:27
9:13:50
10:9:13
11:4:36

Mail to Ken