When can the hands swap places and still show a real time?

Question

Take an analogue clock at some moment and imagine swapping its two hands: the hour hand takes the minute hand's position and vice versa. Usually the result is nonsense: no real time has the hands there. But occasionally the swapped picture is itself a genuine time.

How many moments in a 12-hour period have this property? Count moments when the hands coincide too, since swapping then changes nothing.

Accepted Answer

143 moments (132 if you exclude the 11 moments where the hands coincide). Positions as fractions of a turn. Let the time be x hours after 12, with x between 0 and 12. The hour hand is at position x/12. The minute hand is at position x minus (whole hours), that is, x/12 times 12 with the whole turns removed; write it as x minus a, where a is the whole number of hours elapsed, so 0 at most x minus a less than 1. The swap condition. The swapped picture is a valid time y if the minute hand of y sits where the hour hand of x is, and the hour hand of y sits where the minute hand of x is: 1. Minute …

When can the hands swap places and still show a real time?

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