Date: Jun 14, 1996 4:03 AM
Author: Joel Neely
Subject: Re: "Impossible Problem"



Brian Raiter wrote:
> A similar problem
...
> Fill in the blanks so as to make this whole page logically
> self-consistent.
>
> On this page, the digit zero appears times.

[snip]
> " nine " times.
...
> Clifford Pickover describes this problem, and mentions that Martin
> Gardner has shown that there is only one solution.


And an interesting approach (described, I think, in a Sci Am article
by MG) is to treat the set of sentences as a function mapping from
10-tuples to 10-tuples. I.e., fill in the blanks with the values of
the argument 10-tuple, then count the digits to get the resultant
10-tuple.

Now iterate.

Depending on the initial value, the iteration will either cycle
through a set of tuples (length > 1) or converge to a solution
tuple (i.e., a cycle of length 1).

Someone with better discrete dynamical systems skills than I have
might address the question of which properties of an initial value
could be used to predict which kind of basic that initial tuple
lies in.