Re: Mike Throll 5

Timofei Shatrov <gru...@mail.ru> wrote:

: We have the black blot on the white plane.
: Every dot is added to the blot if the black
: area of its neighborhood (radius 1) is more
: than white area and removed if the white area
: is larger. Is this possible that blot's area
: will be 1000 times more than initial area?

I've been thinking about this puzzle... is this a correct rephrasing:

Given an initial separation of the plane RxR into a subset A and not-A,
and an iterative process which at each step assigns membership in A to
each point outside A whose circular neighbourhood of radius 1 has a
greater intersection with A than with not-A, and removes each point
previously within A whose circular neighbourhood of radius 1 has a greater
intersection with not-A than with A, is it possible to choose initial A
such that its area will increase by a factor of 1000 or more during the
process?

Tell me, does the blot have to be simply connected in the beginning? Do we
need the final limiting area to be more than 1000 times the initial area,
or can some intermediate area be 1000 times? I imagine it is some 
impossible fractal shape to begin with. We need Bill or another real 
mathematician in this group to figure it out...

nice drawings by the way.

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