TITLE: After swapping the positions of the hour and the minute hand, when will a clock still give a valid time?
QUESTION [21 upvotes]: At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and the minute hand position does a clock still give a valid time?

REPLY [3 votes]: Define a third imaginary hand at the position the minutes hand would be if the actual minutes hand was the hours hand. This imaginary third hand travels 12 times faster than the actual minutes hand, thus completes 144 cycles in a 12 hour period.
Note that the actual minute and hour hands are swap-able iff the third imaginary hand coincides with the actual hour hand. The actual hour hand is traveling at a speed of 1 cycle per 12 hours, and the third imaginary hand is traveling at a speed of 144 cycles per 12 hours, so their relative speed is 143 cycles per 12 hours. Thus the hands of a clock are swap-able 143 times in a 12 hour period.