Newsgroups: rec.puzzles
From: Hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 1997/03/26
Subject: Re: You're all jealous!!

dk...@raspberry.bbn.com (David Karr) writes:

> Duncan McKenzie <dun...@cable.com> wrote:

> >[...] "imagine four intersecting spheres -- what is the
> >maximum number of regions they could create?" [...]
> >there's a certain level of problem where my mind just gets
> >confused, and hers obviously doesn't.

> Right.  The problem is you think too much.
> The obvious answer is that each sphere added in turn can at most
> divide all the existing regions in two.  So the answer is 16....

Perhaps he was also thinking too rigorously.  An added sphere can
divide a region into more than two parts.  For instance, if three
spheres intersect in three-dimensional lune, the fourth sphere could
cut the lune into three parts.  If so, the fourth sphere won't cut the
region exterior to the first three spheres, so perhaps 16 is still an
upper bound.  But more cases need to be considered if you want a proof.

I join in the suspicion that part of what passes for MvS's ability to
avoid confusion is a tendency to make plausible guesses instead of
addressing the hard parts of a problem.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil