```You're on the right track, Pat, especially with your second point.
Computers can't prove anything.  Basically, because they can't do real math
(excuse the pun).  They only deal with finite numbers, very big finite
numbers maybe, but still finite.  And there are two types of infinite sets,
countable (like rationals that can be put into 1-to-1 correspondence to
integers) and uncountable (like irrationals that can't, or reals that are a
set with both rational and rational numbers).

Robert

-----Original Message-----
Sent:	Monday, March 12, 2001 1:33 PM
Subject:	Re: (VERY) OFF TOPIC - Map coloring

In a message dated 3/12/01 12:14:55 PM Eastern Standard Time,

It is the job of the mathematicians is to determine what
is
mathematically
meant by "every case". That's done all the time. For
example, in poker, the
"odds" of getting five cards of the same suit is determined
by calculating
the NUMBER of ways five randomly drawn cards can be of the
same suit, and
then dividing that number by the TOTAL NUMBER OF WAYS any
five cards can be
drawn from a 52 card deck.

Yes, and if n colors are going to be used to color n shapes, the
number of
possibilities is n to the nth power. This really includes all
possibilities,
including using only one of the n colors for all of the n shapes.

Most of these color arrangments are not going to meet the criteria
for what
Saaty calls correct k-coloring--that is, no two contiguous countries
are the
same color. For his correct k-coloring, the limit is n!. That is, n
factorial.

Thus, for four color used on a basic four-shape module, there are
256
possible combinations of which only 24 are "correct."

Now here are my questions.

1) This sounds to me like it's leading into a problem in what I
believe is
called combinatorial algebra. So why does one need a computer?

2) For each n in the number series, the nth power of n is an
infinite series.
So is n factorial.  So the computer, if using brute force, is going
to run
out of steam at some point, as it can't deal with an infinite series
by
testing "every" possibility.

3) What was wrong with de Morgan's proof, which was rejected? It
seems to me
that he was on the right track, much more so than what followed
after.

4) Maybe you should rethink your poker example. The number is finite
because
there's a limit to the number of cards in a pack. There's no limit
to the
number of shapes that can be on a map. That's why one has to do
something
along the line of breaking a large map down into, say, units of four
or fewer
shapes.  How, exactly, did they compute the number of possible
combinations,
and what did the number turn out to be? You keep assuring me it was
"gazillions," which I don't find very clear. I'm actually more
interested in
how it was computed than what it turned out to be.

pat

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