It is amusing to muse about what gives this puzzle its
zest.
We note that the solution-set is non-compact. That is,
the southern solutions are spectacularly different from
the northern solution.
It seems that as a general rule, knowing one solution
forms some sort of psychological impediment to finding
other solutions. The mind is attracted to the first
solution like a moth to candle.
How can we understand this? I don't know.
One aggravating factor might be a lack of experience dealing
with such situations. In turn this may be because:
-- Students are rarely (if ever) exposed to problems of
this sort. The typical end-of-chapter homework problem
has a unique cut-and-dried answer.
-- In many real-life problem-solving situations, it
suffices to find _some_ solution, and it is not
necessary to exhaustively describe the solution-set.
For example, I know how to get to the grocery store.
It is useful to know more than one way, but it is
not useful to know all the ways (of which there are
an astronomical number).
On the other hand, there are lots of real-life situations
where it is quite useful to find multiple solutions to a
given problem. In particular, when doing a problem where
the answer really matters, I like to work the problem at
least two different ways, and check that they agree.
===========
I note that nobody has really addressed part (2) of the
question. Are you sure that the solution-set that has
been exhibited is complete? How do you know?
Just because you've been snookered once doesn't mean you
can't be snookered again. :-)
(I think it *is* complete ... but that's not the same as
a proof.)