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-----Oorspronkelijk bericht-----
Verzonden: maandag 12 maart 2001 22:33
Onderwerp: 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.

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

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

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

Arwin

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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<BLOCKQUOTE=20
style=3D"BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; PADDING-LEFT: =
5px">
face=3DTahoma=20
size=3D2>-----Oorspronkelijk bericht-----<BR><B>Van:</B>=20
(VERY)=20
OFF TOPIC - Map coloring<BR><BR></DIV></FONT><FONT =
face=3Darial,helvetica><FONT=20
face=3D"Arial Narrow" lang=3D0 size=3D3 FAMILY=3D"SANSSERIF"><B>In a =
message dated=20
writes:=20
<BR><BR></FONT><FONT color=3D#000000 lang=3D0 size=3D2 =
FAMILY=3D"SANSSERIF">
<BLOCKQUOTE=20
style=3D"BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; =
TYPE=3D"CITE"></B><BR>&nbsp;&nbsp;It is the job of the mathematicians =
is to=20
determine what is <BR>mathematically <BR>meant by "every case". =
That's done=20
all the time. For example, in poker, the <BR>"odds" of getting five =
cards of=20
the same suit is determined by calculating <BR>the NUMBER of ways =
five=20
randomly drawn cards can be of the same suit, and <BR>then dividing =
that=20
number by the TOTAL NUMBER OF WAYS any five cards can be <BR>drawn =
from a 52=20
card deck.</FONT><FONT color=3D#000000 lang=3D0 size=3D3=20
FAMILY=3D"SANSSERIF"></BLOCKQUOTE>
<DIV><BR><BR></FONT><FONT color=3D#000000 lang=3D0 =
FAMILY=3D"SANSSERIF"><FONT=20
size=3D2><STRONG>Yes, and if n colors are going to be used to color n =
shapes,=20
the number of <BR>possibilities is n to the nth power. This really =
includes=20
<U>all </U>possibilities, <BR>including using only one of the n colors =
for all=20
of the n shapes. <BR><BR>Most of these color arrangments are not going =
to meet=20
the criteria for what <BR>Saaty calls correct k-coloring--that is, no =
two=20
contiguous countries are the <BR>same color. For his correct =
k-coloring, the=20
limit is n!. That is, n <BR>factorial. &nbsp; <BR><BR>Thus, for four =
color=20
used on a basic four-shape module, there are 256 <BR>possible =
combinations of=20
which only 24 are "correct." &nbsp; <BR><BR>Now here are my questions. =

<BR><BR>1) This sounds to me like it's leading into a problem in what =
I=20
believe is <BR>called combinatorial algebra. So why does one need a =
computer?=20
<BR><BR>2) For each n in the number series, the nth power of n is an =
infinite=20
series. <BR>So is n factorial. &nbsp;So the computer, if using brute =
force, is=20
going to run <BR>out of steam at some point, as it can't deal with an =
infinite=20
series by <BR>testing "every" possibility. <BR><BR>3) What was wrong =
with de=20
Morgan's proof, which was rejected? It seems to me <BR>that he was on =
the=20
right track, much more so than what followed after. <BR><BR>4) Maybe =
you=20
should rethink your poker example. The number is finite because =
<BR>there's a=20
limit to the number of cards in a pack. There's no limit to the =
<BR>number of=20
shapes that can be on a map.&nbsp;</STRONG><SPAN=20
class=3D132385021-12032001><FONT=20
color=3D#0000ff>&nbsp;</FONT></SPAN></FONT></FONT></DIV>
<DIV><FONT color=3D#000000 lang=3D0 FAMILY=3D"SANSSERIF"><FONT =
size=3D2><SPAN=20
=
class=3D132385021-12032001></SPAN></FONT></FONT>&nbsp;</DIV></BLOCKQUOTE>=

<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001>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 =
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. </SPAN></FONT></FONT></DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001></SPAN></FONT></FONT>&nbsp;</DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001>But this is only what I remember from quick =
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 =
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. </SPAN></FONT></FONT></DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001></SPAN></FONT></FONT>&nbsp;</DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001>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
.... </SPAN></FONT></FONT></DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001></SPAN></FONT></FONT>&nbsp;</DIV>
<DIV><FONT lang=3D0 FAMILY=3D"SANSSERIF"><FONT color=3D#0000ff =
size=3D2><SPAN=20
class=3D132385021-12032001>Arwin</SPAN></FONT></FONT></DIV>
<BLOCKQUOTE=20
style=3D"BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; PADDING-LEFT: =
5px">
<DIV><FONT color=3D#000000 lang=3D0 FAMILY=3D"SANSSERIF"><FONT =
size=3D2><SPAN=20
class=3D132385021-12032001></SPAN></FONT></FONT>&nbsp;</DIV>
<DIV><FONT color=3D#000000 lang=3D0 FAMILY=3D"SANSSERIF"><FONT =
size=3D2><SPAN=20
class=3D132385021-12032001>&nbsp;</SPAN><STRONG></FONT><FONT =
size=3D2>That's why=20
one has to do something <BR>along the line of breaking a large map =
down into,=20
say, units of four or fewer <BR>shapes. &nbsp;How, exactly, did they =
compute=20
the number of possible combinations, <BR>and what did the number turn =
out to=20
be? You keep assuring me it was <BR>"gazillions," which I don't find =
very=20
clear. I'm actually more interested in <BR>how it was computed than =
what it=20
turned out to be. <BR><BR>pat</FONT></STRONG><FONT size=3D2>=20
<BR><BR></FONT></FONT><FONT color=3D#000000 lang=3D0 size=3D3=20
FAMILY=3D"SANSSERIF"><BR></DIV></BLOCKQUOTE></FONT><FONT =
color=3D#000000 lang=3D0=20
size=3D2 FAMILY=3D"SANSSERIF"></FONT></FONT></BODY></HTML>

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