Enigma 1694 - The town clock prank

[Chappy]

Enigma 1694 - The town clock prank
New Scientist magazine, 21 April 2012.
By Ian Kay.

Our town clock has a standard 12-hour face,
divided into 60 minute marks. The hands do
not move smoothly: the minute hand moves
sharply on to the next minute mark at the
end of each minute, and the hour hand moves
at the end of each 12 minutes.

The other day, a prankster swapped the hands
(an operation that only took a few seconds
between movements of the mechanism).
Immediately after the swap, the clock showed
the wrong time, but it was a valid time; in
other words, the relative positions of the
hands made sense. Within the next 10 minutes,
the clock had shown the right time for exactly
2 minutes.

What time did the clock show immediately
before the hands were swapped?

Ciao,
Chappy.

[John Jones]

In article <f67dc381-7611-40fd...@googlegroups.com>,
petergreg...@hotmail.com says...

The way the clock moves means that at h:m the minute hand points to
division m and the hour hand to division 5h + [m/12]
where [] means integer part of.

If the time with swapped hands is to be correct, then these two must be
the same, ie
5h + [m/12] = m mod 60

This has 12 solutions, one for each hour:- 12:00 1:05 2:10 3:16 4:21
5:27 6:32 7:38 8:43 9:49 10:54 11:59

To be right for two minutes in ten two of these must be within ten
minutes of each other and the only two solutions within ten minutes of
each other are 11:59 and 12:00. These are also consecutive.

We are told that when swapped, the time was not right, so it wasnt
swapped at 11:59, but at some time upto ten minutes before then.

Not all h:m give valid h':m' when the hands are swapped. The condition
is that

m = [(5h + [m/12]) /12] mod 5.

By inspection, the solutions to that in the region of 11:59 is 11:54 and
11:49. But 11:49 is too early - ten minutes on misses 12:00.

So the clock display at the time of the prank was 11:54 (which made the
apparent time 10:59).

HTH
JJ