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Subject: Re: (VERY) OFF TOPIC - Map coloring

From: Date: Mon, 12 Mar 2001 16:33:19 EST

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 --part1_23.8a48b04.27de9a9f_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit In a message dated 3/12/01 12:14:55 PM Eastern Standard Time, [log in to unmask] writes: > > 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 --part1_23.8a48b04.27de9a9f_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit 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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