Re: navigation riddle

["Carl E. Mungan" <mungan@USNA.EDU>]

> 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.)

Taking a crack at a proof, it seems to me that one should proceed as
follows. Specify the initial position on the Earth's surface in terms
of a colatitude (polar angle) and longitude (azimuthal angle). Use
radians for convenience and let R be Earth's radius in miles.

One can then proceed by simply applying the 3 required operations and
seeing for what starting positions you get a solution. It is easy to
formally derive Herb's wonderful solution this way. However, it is
easy to miss some solutions, such as the North pole!

In particular one needs to be careful about limiting values:

1. For example, what exactly does it mean to walk "one mile south" if
you start half a mile north of the south pole?

2. You need to be careful about when two sets of angular positions
(theta1,phi1) and (theta2,phi2) are equal. For example, if
theta1=theta2=0 (the North pole), then the two positions are equal
for *any* values of phi.

Both of these arise because of "weirdnesses" in the definitions of
the angles. This needs to be brought explicitly.

So here are the conclusions I draw from John's nice "moth attraction" simile:

a. A formal mathematical proof using angles is important. If we
weren't so familiar with this riddle, probably more of us would have
started with a math proof of this sort.

b. Examination of an actual globe and/or awareness of common "tricks"
helps us to catch limiting values. It seems to me however that a very
careful math proof should have caught these. Perhaps this is an
example of where mathematician's insistence on rigor is to be
commended. We physicists tend to skip over it too quickly sometimes
and rely overly on "intuition."
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu       http://usna.edu/Users/physics/mungan/