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Suppose you have a clock with two identical hands (there is no second hand). What are the exact times when this clock and its vertically mirrored image are identical?

image of the clock and its mirrored image

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  • $\begingroup$ Penguino's answer to "Mirrored Clocks" lists these times. $\endgroup$
    – Bass
    Mar 6, 2022 at 8:50
  • $\begingroup$ @Bass That's correct, but it looks like those are only approximate values. This question asks for exact times. $\endgroup$ Mar 6, 2022 at 8:53
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    $\begingroup$ Does this answer your question? Mirrored clocks $\endgroup$
    – Glorfindel
    Mar 7, 2022 at 19:36

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Ignoring AM and PM the clocks will be the same at 12:00 and every 55 5/13 minutes thereafter (i.e. 12:55 and 5/13 minutes, 1:50 and 10:13 minutes etc.) and also at 6:00.

To see this, consider another clock where the hour hand moves anticlockwise. The mirrored clocks will be equal only at 6:00 or at times when the new clock's hand overlap. We note that the minute hand passes the hour hand on our new clock 13 times in a twelve hour period (once when both hands are vertical and once in each of the hour long sectors between numbers). Now we need only show that the time between coincident hands on this clock are evenly distributed.

To see this consider another clock, like the previous but now with the whole clock regularly rotating clockwise every 12 hours. To an observer, the hour hand of this clock appears stationary and the minute hand rotates clockwise 13 times every 12 hours. The coincident times are the same as the previous clock, but because the hour hand is now (relativistically) stationary, we see that every coincident moment returns the clock to the same indistinguishable state, with both hands vertical. We conclude that the time between coincident moments is constant and because there are 13 coincident moments every 12 hours they must occur every 12/13 hours which is every 55 5/13 minutes or every 55 minutes 23 1/13 seconds.

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  • $\begingroup$ Can you please explain the last part further? I'm trying to visualize it and not sure if I understand correctly. $\endgroup$ Mar 7, 2022 at 6:27
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    $\begingroup$ I’ve added more words; I hope that they help. $\endgroup$
    – Daniel S
    Mar 8, 2022 at 21:20
  • $\begingroup$ Thank you, it's clearer now! $\endgroup$ Mar 9, 2022 at 8:19

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