Math Is Fun Forum

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Four Colour Theorem : Formula Challenge

Here is an Interesting Challenge!

Is it possible to make a Formula to prove the Four Colour Theorem Problem?

The Proof as it stands was put forward by Haken & Appel ( 1976 ) using a Computer! and then double checked using an independant Computer,taking aproximately 1200 Computer hours.
The Computer found 1482 different possible Map configurations,and is accepted as a Proof! even though some Mathematicians are not happy! because it was not fully checked by Humans! the main reason being they have not found anyone or group of people ( as far as I know ) willing to check all 1482 configurations.
What makes the Challenge of finding a Formula to Prove the Four Colour Theorem,Interesting is! will the Formula agree with the Computer and produce the same total 1482 different possible Maps! if not? then it can again be looked into further.

I am interested in this myself at the moment,and if anyone else would like to put ideas forward on how it can be done! then please do.
I think using Variables would be a good idea,so as to see what is happening in the Calculations,a start area might be as with the Variables below.

R = Regions : B = Common borders : Colours could be Numbers 1,2,3,4 etc..

History:
The original Four Colour Theorem idea came from," Francis Guthrie " England ( 1852 ) and said!

" What is the minimum number of colours required,to colour any conceivable map,such that no two regions having a common border have the same colour. "

A.R.B

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Four Colour Theorem : Formula Challenge

A map on an orientable surface of genus g needs a maximum of γ colors, where γ is given by

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Re: Four Colour Theorem : Formula Challenge

To Zhylliolom

Quote: " A map on an orientable surface of genus g needs a maximum of γ colors, where γ is given by "

Could you possibly give your example using Simple Variables! ie R,B.1,2,3,4 etc. so we can look at the Math! in a way to make it Into a Formula!

Patrick

Real Member

Registered: 2006-02-24

Posts: 1,005

Re: Four Colour Theorem : Formula Challenge

it is a formula already

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kylekatarn

Member

Registered: 2005-07-24

Posts: 445

Re: Four Colour Theorem : Formula Challenge

.

Last edited by kylekatarn (2020-01-03 11:44:04)

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Re: Four Colour Theorem : Formula Challenge

Quote: " it is a formula already "

Quote: " I think it's pretty clear what Zhylliolom said..."

A.R.B

Does the formula? show  whether the 1482 different possible Maps are the Same! etc......

Does the formula show how many Regions : Common borders : Colours etc......there are!

Last edited by Anthony.R.Brown (2007-03-01 00:53:42)

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Four Colour Theorem : Formula Challenge

genus g is, indeed, a new concept to most of us. I hope it is not too long to explain.
But A.R.B., I suggest you to give up, it looks harder than it is thought. (That's why I have given up)

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X'(y-Xβ)=0

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Re: Four Colour Theorem : Formula Challenge

Thanks George,Y!

Unfortunately I never give up! even more now you have said that!
It would help if others explained things in a more clearer way! I think some people are afraid of explaining in a simple way! because they worry others may think they don't know what they are talking about! elaborate Math prevents Ideas Expanding!!

A.R.B

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Re: Four Colour Theorem : Formula Challenge

---------------------------------------------------------------------------------------------------------------
FOUR COLOUR PROBLEM FORMULA : BREAKTHROUGH! 16/03/07 By Anthony.R.Brown
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The way to tackle this problem is based on the 1482 possible maps! Found by the Computer.

I looked at the way the Numbers 1 to 10 divide into the Magic Number 1482

1482 / 1 = 1482 *
1482 / 2 = 741 *
1482 / 3 = 494 *
1482 / 4 = 370.5
1482 / 5 = 296.4
1482 / 6 = 247 *
1482 / 7 = 211.71428
1482 / 8 = 185.25
1482 / 9 = 164.666...
1482 / 10 = 148.2

4 COLOURS:
I noticed from the above Calculations that only four Numbers divide into 1482 marked * with no amount left over!
Could this be the Calculation that says we need Four Colours for any Map!

6 MAIN POINTS:
From the above Calculations 1482 / 6 = 247 on any Map that requires the Four Colours,there are 6 main points.

3 POINTS AT EVERY MAIN POINT:
From the above Calculations 1482 / 3 = 494  on any Map that requires the Four Colours,there are 3 points that meet at every one of the 6 main points.

9 BORDERS:
From the above Calculations 1482  the 6 = 247 main points + the 3 = 494 points at every main point make up the connections for the 9 Borders on any Map that requires the Four Colours.

The above observations are a starting point! In trying to put together a Formula for this marvellous problem!

The more contributions put forward by everyone! The better! It's there for the taking,it could make this Math forum famous!

A.R.B