The Mathematics of Map Coloring – The Four-Color Theorem
The Problem
Counties of England
The Counties, Colored
Why Counties of England?
1852 October 23
How to begin?
1 country – how many colors?
1 country – 1 color
2 countries – how many colors?
2 countries – 2 colors
3 countries – how many colors?
3 countries – 3 colors
3 countries – how many colors?
3 countries – 2 colors
4 countries – how many colors?
4 countries – 2 colors
4 countries – how many colors?
4 countries – 3 colors
4 countries – how many colors?
4 countries – 4 colors
Important fact
Do the math
Leonhard Euler (1707 -83)
F+V–E=?
Euler Characteristic
F+V–E=2
5 or fewer neighbors
5 or fewer neighbors
5 or fewer neighbors
5 or fewer neighbors
Six Colors Suffice
Six Colors Suffice
Six Colors Suffice
Six Colors Suffice
Six Colors Suffice
Pick a number: 4, 5, or 6?
Break time
No activity for several decades
Solved, at last
How did he do it?
Is this approach familiar?
Minimum map for which 4 colors are not enough
Get rid of one country
Now we can 4 -color the map
Restore the removed country
Change gray to what?
Red-yellow chain
Swap B-G inside R-Y chain
Red-green chain
Swap B-Y inside R-G chain
Color the gray country blue
Here's a 4 -coloring of the map
Four Colors Suffice!
Hold it a second
Trouble in paradise
Ouch!
The Five Color Theorem
Maybe 4 colors aren't enough?
Gardner: 5 colors are needed
80+ years had gone by. . .
Garner's map: 4 colors work
At last!
Some reactions
What's the problem?
Eventually: Acceptance
In conclusion. . .