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## What color is the bear? A new class of solutions

In the 1985 film Young Sherlock Holmes, Holmes tells Watson the following riddle:  “You’re sitting in a room with an all-southern view.  Suddenly, a bear walks by the window. What colour is the bear?”  (He says “colour” not “color” because he’s from the UK.)  As I recall it takes Watson most of the movie to give the obvious answer—white—because the bear must be a polar bear.  All southern view…North Pole…polar bear…you get the idea.

Over the years I’ve hear other versions of this riddle.  The most common seems to go something like this: “A hunter travels a mile south, a mile east, shoots a bear, then travels a mile north to her starting point.  What color was the bear?”  People assume that this riddle is isomorphic to the previous one, because (supposedly) there is only one place on Earth you can travel the same distance south, then east, then north, and return to the beginning.  But this is wrong.  There are an infinite number of places on Earth you can travel in a loop that is 1 mile south, then 1 mile east, then 1 mile north.

Of course, starting at the North Pole is one solution.  But there are also many, many more solutions close to the South Pole.  Imagine, for example, a latitude roughly 1/(2π) miles north of the South Pole; in such a case the parallel along that latitude is the circumference of a circle, given by

C = 2πr = 2π [1/(2π)] = 1 mile.

(I said roughly because we’re on the surface of a sphere, so the circumference of a parallel is not exactlyr—because r is an arc length, not a straight line—but 1 mile is so much smaller than the radius of the Earth that we can assume a locally flat geometry.)  If a hunter started his journey 1 mile north of this latitude, then of course he could go 1 mile south, then 1 mile east (circumnavigating the South Pole!) and then 1 mile north, and return to his starting point; I presume there would be no bears.

There are actually infinitely many solutions that work.  In each case, after going a mile south, the hunter would have to reach a latitude in which the circumference C was an integer fraction of 1 mile.  That is, it must be true that

C = 1/n miles,

where n is an integer.  This means that the parallel would have to be

r = C/(2π) = 1/(2πn)

miles from the South Pole (again, this is a flat approximation to a spherical problem).  So the most general South Pole solution is that the hunter should begin 1 + 1/(2πn) miles north of the South Pole.  For example, take n = 5.  If a hunter starts 1.0318 miles from the South Pole, she can go south 1 mile, east 1 mile (circumnavigating the South Pole exactly 5 times) then north 1 mile, and relax in her hot tub.  No bears will be harmed, unless some evil genius has released them in Antarctica.

The original version of the riddle, as given by the young Sherlock Holmes, is superior, since it has only one solution.  We can conclude that Sherlock Holmes was good at math.

I mean, maths.

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