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My method for solving this includes use of
geometry (a very under-rated tool) and therefore diagrams, but I
will do my best to describe what I did in words!
Firstly, I have assumed that when you say
"Arthur glanced at ", he is not bothering to look at
the number of seconds, but when you say "he checked his watch and
found that it told exactly the same time as the station clock",
the seconds are significant. I think this is reasonable,
and the rest of my theory is based on it.
Arthur's Watch is running very slow, although
at the start of the story it is set a little fast (ahead).
There are two points in the story where there
is an obvious way of getting some information about the exact equation
for the discrepancy, although, as we shall see, nothing in the story
actually gives away the precise figures.
The two points are:
(A) "He checked his digital watch, it read
10:00. He glanced up at the station clock, it read 09:59"
This means that when the time was 09:59:[s1],
Arthur's watch showed "10:00:[s2]", where [s1] and [s2] are two
unknown seconds figures.
(B) "he looked up at the clock again, it
read 10:00, he checked his watch and found that it said 10:01"
This means that when the time was 10:00:[s3],
Arthur's watch showed "10:01:[s4]", where [s3] and [s4] are two
further unknown seconds figures.
However, there is a third point in
the story where you know something about the equation:
(C) "When he arrived he found the train had
just left" -- this implies that it was exactly 11:12:00 when he
arrived (the train kept perfect time - HAH! A likely story). So
as he took five minutes to walk there, we know that it was actually
11:07:00 when he left the tea shop. But the story says "When he
finished the tea he checked his watch and found it was 11:00, so
he set off back for the station" -- so it was actually 11:07:00
and his watch read 11:00:[s5], where [s5] is another unknown quantity
of seconds.
We cannot know the values of [s1] - [s5],
but we can use geometry to define the possible regions in which
they might lie. This is where my explanation falls short of articulate,
because words are not an ideal tool for explaining the method -
so I'm going to try to describe the picture I drew (sorry, I can't
be bothered to make a digital one at the moment).
Draw a grid, with "Real (Clock) time" on
the horizontal axis, ranging from 09:59:00 to 11:07:00, and "Arthur's
Watch" on the vertical axis, ranging from 10:00:00 to 11:01:00.
The idea is to draw a line that represents the "Arthur's Watch"
function, i.e. maps real time to that showing on the watch.
Fact (A) means that the line must
pass through at least one point in the square defined by (real time
between 09:59:00 and 10:00:00) x (watch time between 10:00:00 and
10:01:00)
Fact (B) means that the line must
pass through at least one point in the square defined by (real time
between 10:00:00 and 10:01:00) x (watch time between 10:01:00 and
10:02:00)
Fact (C) means that the line must
pass through at least one point on the line segment defined by (real
time = 11:07:00) x (watch time between 11:00:00 and 11:01:00)
All three of these regions should be shaded
on your diagram - although you will not be able to draw it to scale
because 11:07 is so far off to the top-right!!
Each range given above is inclusive at the
minimum and exclusive at the maximum (i.e. "09:59:00 to 10:00:00"
includes 09:59:00 but does not include 10:00:00 -- but this is a
little picky, so let's ignore it).
There are two obvious bounding lines that
pass through the extremities of the two squares and through the
11:07 line segment. These have to be chosen carefully so that all
lines passing somewhere between them intersect both squares, but
this is not difficult once you realise that Arthur's watch is running
slow, so the gradient is less than 45 degrees. You have to draw
the diagram to understand that comment, I think!!
One line passes through (09:59:00,10:01:00)
and (11:07:00,11:01:00) and has a gradient of 60/68. The other passes
through (10:01:00,10:01:00) and (11:07:00,11:00:00) and has a gradient
of 59/66 (which is steeper than the other line, so the whole thing
is a fan shape with the apex somewhere off to the right).
On the diagram, draw the line of a "perfect
watch", i.e. y=x. The story tells us that there is a particular
moment at which Arthur's watch is correct, so at that moment the
line should intersect the "Arthur's Watch" line, and the task is
to discover where this happens.
The intersection cannot happen before 10:01:00
(real time), because then y=x would lie outside our fan-shaped region.So
we have a minimum time for the event.
It also cannot happen after the intersection
of y=x and the top line, because, again, y=x would lie outside the
fan region.
A little linear algebra yields the intersection
of the top line with y=x as a point with a real time co-ordinate
of 10:07:37.5.
There are "Arthur's Watch" solutions at both
10:01:00 and 10:07:37.5 - in the first case, it follows the bottom
line, and in the second case, it follows the top line. So it is
clear that any value in between the two will also have a solution.
THEREFORE: The time at which Arthur's
watch agreed with the station clock could have been anywhere between
10:01:00 and 10:07:37.5 - a possible range of six minutes and thirty-seven
and a half seconds.
Do I get the prize for the most verbose answer?
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