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.
It's restful sitting in Tom's cosy den, talking quietly and sipping
a glass of his Madeira.
I was there one Sunday and we had the usual business of his clock.
When the radio gave the time at the hour, the Ormolu antique was
exactly 3 minutes slow.
"It loses 7 minutes every hour", my old friend told me, as he had done
so many times before. "No more and no less, but I've gotten used to
it that way."
When I spent a second evening with him later that same month, I remarked
on the fact that the clock was dead right by radio time at the hour.
It was rather late in the evening, but Tom assured me that his treasure
had not been adjusted nor fixed since my last visit.
What day of the week was the second visit?
From "Mathematical Diversions" by Hunter + Madachy
The answer is 17 days and 3 hours later, which would have been a Wednesday.
This is the only other time in the same month when the two would agree at all.
In 17 days the slow clock loses 17*24*7 minutes = 2856 minutes,
or 47 hours and 36 minutes. In 3 hours more it loses 21 minutes, so
it has lost a total of 47 hours and 57 minutes. Modulo 12 hours, it
has *gained* 3 minutes so as to make up the 3 minutes it was slow on
Sunday. It is now (fortnight plus 3 days) exactly accurate.
Since the clock was not adjusted since the last visit, it's also
possible that the radio time shifted by one hour due to a change to or
from summer daylight saving time. However, it turns out that the only
additional possibilities that need to be considered are those of 4 days
15 hours later, when the clock would have lost 12 hours 57 minutes, and
29 days 15 hours later, when the clock would have lost 3 days 10 hours
57 minutes. Without even considering the rules for when in the month the
clock is changed, these possible solutions are ruled out because we know
that both visits were in the evening ("I spent a second evening with him").
and they involve times in a different part of the day.