The clock in the mirror

[bonanova]

A while ago, about a clock with indistinguishable hour and minute hands and asked at what times of day, between the hours of noon and midnight, it was impossible to unambiguously determine the time. The hands moved continuously.

This puzzle asks a related question. At what times of day, between the hours of noon and midnight, is it impossible to distinguish the hands of such a clock from those of its mirror image? Clearly noon is one of these times, but not in general thereafter -- since the clock's hands will move clockwise while the hands of its mirror image will move counterclockwise.

[Rainman]

For the sake of mathematical consistency, if we include noon we should not include midnight. Midnight will be the start of the next 12-hour period rather than the end of this one.

Now, there are two cases:

1) The minute hand and hour hand are the mirror images of themselves.

2) The minute hand and hour hand are the mirror images of each other, but not of themselves. (The but prevents noon from being counted twice)

Case 1 happens iff both hands are at either 12 or 6. The hour hand is only at 12 once and at 6 once, so it can happen at most twice. Fortunately, both these times the minute hand will be at 12, so we have two instances of case 1.

Let a rough hour be the time defined only from the hour number on a digital clock. For example, the rough hour 4 is the time starting at 4:00 and ending just before 5:00. Case 2 happens exactly once every rough hour. To see why, consider the mirror image of the hour hand as it moves counter-clockwise slowly and continuously. The minute hand makes a continuous clockwise sweep around the clock, and so it must meet the mirrored hour hand. The mirrored hour hand has nowhere to run and nowhere to hide.

So we have 2 instances of case 1, and 12 instances of case 2. That makes a total of 14.