overtakes.py

mins_per_day = 24 * 60

def identify_overtakes(num_hours):
    num_mins = num_hours * 60
    overtakes = []
    last_diffs = [0.0, 0.0]

    for i in xrange(num_mins):
        hour = (i / 60) % 12
        minute = (i % 60) % mins_per_day

        # work the hand positions out as fractions 0.0 -> 1.0
        hour_frac = ((i / 60.0) % 12) / 12.0
        min_frac = (i % 60) / 60.0

        diff = min_frac - hour_frac

        # we can't be more than half way away, e.g either side of vertical
        if diff > 0.5:
            diff = diff - 1.0
        if diff < -0.5:
            diff = diff + 1.0

        if diff > 0.0:
            if last_diffs[-1] < 0.0 or (last_diffs[-1] == 0.0 and last_diffs[-2] < 0.0):
                overtakes.append((hour, minute))
        last_diffs = last_diffs[1:] + [diff]

    return overtakes

if __name__ == '__main__':
    num_hours = 24 * 5
    overtakes = identify_overtakes(num_hours)
    for i in overtakes:
        print '{:02d}:{:02d}'.format(*i)
    print '{} overtakes in {} hours'.format(len(overtakes), num_hours)


# Here are the results:
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 00:01
# 01:06
# 02:11
# 03:17
# 04:22
# 05:28
# 06:33
# 07:39
# 08:44
# 09:50
# 10:55
# 109 overtakes in 120 hours

In 5 days we see 10 12-hour periods, each with the same 10 overlap times: 01:06, 02:11, 03:17, 04:22, 05:28, 06:33, 07:39, 08:44, 09:50, 10:55 with each of the 9 joins causing an overlap at 00:01.

On line 12, minute = (i % 60) % mins_per_day, the 2nd % does nothing and might as well be removed.

To clarify, this code steps forwards minutely and records every overtake.