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

From: Date: Mon, 12 Mar 2001 23:15:21 +0100

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-----Oorspronkelijk bericht-----
Verzonden: maandag 12 maart 2001=20   22:33
Onderwerp: Re: = (VERY)=20   OFF TOPIC - Map coloring

In a = message dated=20   3/12/01 12:14:55 PM Eastern Standard Time,

It is the job of the mathematicians = is to=20     determine what is
mathematically
meant by "every case". = That's done=20     all the time. For example, in poker, the
"odds" of getting five = cards of=20     the same suit is determined by calculating
the NUMBER of ways = five=20     randomly drawn cards can be of the same suit, and
then dividing = that=20     number by the TOTAL NUMBER OF WAYS any five cards can be
drawn = from a 52=20     card deck.

Yes, and if n colors are going to be used to color n = shapes,=20   the number of
possibilities is n to the nth power. This really = includes=20   all possibilities,
including using only one of the n colors = for all=20   of the n shapes.

Most of these color arrangments are not going = to meet=20   the criteria for what
Saaty calls correct k-coloring--that is, no = two=20   contiguous countries are the
same color. For his correct = k-coloring, the=20   limit is n!. That is, n
factorial.

Thus, for four = color=20   used on a basic four-shape module, there are 256
possible = combinations of=20   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=20   believe is
called combinatorial algebra. So why does one need a = computer?=20

2) For each n in the number series, the nth power of n is an = infinite=20   series.
So is n factorial.  So the computer, if using brute = force, is=20   going to run
out of steam at some point, as it can't deal with an = infinite=20   series by
testing "every" possibility.

3) What was wrong = with de=20   Morgan's proof, which was rejected? It seems to me
that he was on = the=20   right track, much more so than what followed after.

4) Maybe = you=20   should rethink your poker example. The number is finite because =
there's a=20   limit to the number of cards in a pack. There's no limit to the =
number of=20   shapes that can be on a map.

=
The idea was that although there isn't a = fixed number=20 of shapes, there was a fixed number of shape-combinations. That is, one = country=20 can be limited by one country (say, the Vatican for example), by two = countries,=20 three, etc. Each type of combination and situation can then be used in=20 combination with other combinations, which altogether apparently lead to = about=20 1200 possible configurations of countries you can theoretically have. = For these,=20 no further simplification has been found and thus brute force proof is = required=20 to complete the proof.

But this is only what I remember from quick = reading in=20 a bookstore (was the only book I looked at though in that bookstore) and = most of=20 what I read was about the difficulty to get the proof widely accepted = (gosh, can=20 be pretty tricky to type with a cat resting his head on your hand, = especially a=20 big muscular male 'great white'). You read that book and you'll be = perfectly=20 happy (or not, but I doubt many on this list can detail everything = nearly as=20 well as that book, which was also very pleasantly written). There was = another=20 book that I browsed in Schotland on dreaming - the biggest point the = book made=20 was that there was still so incredibly little known about it, = particularly on=20 why it is even useful. I think it probably has to do with recovery from=20 thinking, some kind of low-power save-mode stuff, but hopefully with the = new big=20 power-magnet and computer analysis they'll be able to unravel those = mysteries of=20 the brain a bit further.

Another mystery is that the cat isn't in the = least bit=20 bothered by having it's head bounce slightly up and down by my typing. = Amazing=20 ....

Arwin

That's why=20   one has to do something
along the line of breaking a large map = down into,=20   say, units of four or fewer
shapes.  How, exactly, did they = compute=20   the number of possible combinations,
and what did the number turn = out to=20   be? You keep assuring me it was
"gazillions," which I don't find = very=20   clear. I'm actually more interested in
how it was computed than = what it=20   turned out to be.

pat
=20

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