When does the mirror agree with the clock?

Question

A clock with no numbers, only tick marks, hangs opposite a mirror. Usually the reflection shows a different time from the real clock: a real 2:00 looks like 10:00 in the mirror.

At which times does the reflection show exactly the same time as the clock itself? List them all, considering the 12-hour dial.

Accepted Answer

Only at 12:00 and 6:00. How the mirror acts. The mirror reflects the dial across its vertical axis, the line through 12 and 6. A hand pointing at 12 or at 6 lies on that line and is unchanged. Any other hand is moved to its mirror position: 3 becomes 9, 1 becomes 11, and so on. In terms of time, a mirror image of H:M reads as (12:00 minus H:M). Both hands must be fixed. For the reflection to show the same time, both the hour hand and the minute hand must be on the vertical line. The minute hand is on that line only at :00 (pointing up) or :30 (pointing down). The hour hand is on the line only…

When does the mirror agree with the clock?

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