The above clock
face indicates a little before 42
minutes past 4.
The hands will again point at exactly the same spots a little after 23
minutes past 8.
In fact, the hands will have changed places.
How many times do
the hands of a clock change
places between three o'clock p.m. and midnight?
And out of all the pairs of times indicated by these changes, what is
the exact time when the minute hand will be nearest to the point IX?