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If we consider when the two hands are superimposed, there are 11 times that this occurs during the 12 hour period of the clock (see

Enigma 1761). So if the clock is showing 12:00, then by placing it on any of the other sides it should read one of the remaining 10 valid superimposed times. So:Solution:The clock has 11 sides.The clock is a regular hendecagon.

Here’s a program that verifies that

n=11is the solution. It runs in 59ms.