|In this sequel to The No-sided 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)
counter-example 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
Gardner apparently no longer likes to see this story in print. His recent collection The No-Sided Professor does not include it. He explains that "(1) It was based on a confusion between the four-color map theorem and a simpler theorem, easily proved, which says that five regions on the plane cannot be mutually contiguous, (2) the true four-color 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.
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.