It's a cliché that 'even' a stopped clock is right twice a
day,
but one which is fast or slow is never right.

This is intuitively true, but does it have to be the case?
There are some common changes made to timepieces
such
that when stopped they show the correct time less than
twice a day
on average - a 24-hour clock (either with the
hour hand going round once a day, or a separate am/pm
indicator) would be right once a day, while a face
showing
the day of the month is right only twice a month at
most. I
consider these degenerate cases and won't consider them
further.

But it occurs to me that a very fast clock could be right
three -or more- times a day.
The precision broken clock is a post-manufacture
modification, made solely by adjusting its existing parts.
For
the
"three times/day" model this causes the clock to run
exactly two and a half times too fast. One can choose
one
of the three times a day that the clock is correct simply
by
setting it at that time (the other two times are at one-
third day intervals).
Suppose one sets the correct time at midnight. The
clock
will then increment by 2œ hours per hour, so that by 8
am
it will have proceeded through 20 hours; the face will
show
8 o'clock. At 4 pm it will overtake the true time again,
and
the cycle will repeat at midnight.