This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
How many times per day can the hour and minute hands on an analog clock switch
roles and still signify a valid time, ignoring the second hand?
Have 12 clocks C1, C2 ... C12 show 1:00, 2:00, ..., 12:00. Have a
clock C0 show 12:00
Now turn C0 around 12 hours, simultaneously turning C1-C12 so their
hour hands always coincide with the minute hand of C0, i.e., as C0
spans 12 hours, C1-C12 will span 1 hour, but for each possible placing
of the hour hand, all 12 possible 'true' placings of the minute hand
will be represented by one of the 12 clocks.
Each time the hour hand of C0 coincides with the minute hand of a
C1-C12 clock we have a reversible valid time. This happens regularly
12 times each C0 hour, but the first and last time is equal (12:00), so
the number of reversible true times is 12*12-1 = 143 spaced regularly
in the 12-hour interval, ie. each 5 min 2.0979+ sec
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