In this sequel to The Nosided Professor, our
heroes tackle the Four Color Theorem, which was
unproved at the time. (See here for a brief summary of a recent proof.) Included are some historically accurate details
about the prior attempts to prove the theorem, and a (fictional)
counterexample in the form of an African island divided into five
simply connected districts  each of which borders the other
and the ocean. Unfortunately, we do not get to hear
Prof. Slapenarski's explanation of this phenomenon because he is
pulled into the depths of a Klein bottle by what appears to be a giant
insect.
Gardner apparently no longer likes to see this story in print. His recent collection The NoSided Professor does not include it. He explains that "(1) It was based on a confusion between the fourcolor map theorem and a simpler theorem, easily proved, which says that five regions on the plane cannot be mutually contiguous, (2) the true fourcolor theorem, unproved when I wrote my story, has since been established by computer programs, though not very elegantly. As science fiction, the tale is now as dated as a story about Martians or about the twilight zone of Mercury." I don't know, I still like it!
Reprinted in Fantasia Mathematica.
Contributed by
Paul Kainen
I'm writing since apparently Gardner doesn't know (or forgot)
that there _is_ a way to have such a situation  even with
an arbitrary finite number of such mutually bordering regions.
It's a topological "pathology" refered to as "the lakes of Wada"
(a Japanese name I guess but the pun is nice). The catch is
that the regions cannot have boundaries of finite length.
If one has an island with n distinct lakes, first dig a canal
from the first lake so that it gets within, say 1 km of every
point on the boundary of the island and within 1 km of every
point on the boundary of the other lakes. Now repeat this process
for the second lake, etc. Each lake remains a simply connected
region, albeit with a rather long boundary. Now repeat the whole
process replacing 1 km by, e.g., 1/10 km, and keep going. As I
vaguely recall, one can show that in the limit, the lakes remain
simply connected and all of them share a common boundary.
Probably Gardner did know it (as he picked an island as a setting
for his story).
My ratings are based on having read other stories by Gardner,
though not this one, and on your description above.

