By: Megan Duke
Jason Holman
Eli Morris
Aaron Wagner
 Color
your map
 http://www.gamedesign.jp/flash/fourcolor/f
ourcolor.html
 The
Four Color Problem came about when
Francis Guthrie was working with coloring a
map of England and discovered that you
could color the map using only 4 colors .
 Francis was trying to show that any map can
be colored using only four colors so that no
regions sharing a common boundary (other
than a point) share the same color.
 Francis’s brother was a student of De
Morgan, and Francis asked his brother to
show the maps to him to prove if this is
always true.

De Morgan studied the problem in 1852 and could not
figure out a solution. He wrote to another
mathematician (Hamilton) about the problem, saying

“A student of mine asked me today to give him a
reason for a fact which I did not know was a fact and do not yet. He says that if a figure be anyhow
divided and the compartments differently colored so
that figures with any portion of common boundary
line are differently colored - four colors may be
wanted, but not more - the following is the case in
which four colors are wanted. Query cannot a
necessity for five or more be invented. ...... If you
retort with some very simple case which makes me
out a stupid animal, I think I must do as the Sphynx
did....”
Many
people continued to
work on it throughout the
years.
In
1879 Alfred Bray
Kempe announced that he
had solved the Four-Color
Problem.2
 In
1890 Percy John Heawood proved that
Kempe’s proof was incorrect, and then
proved that every map can be colored using
5 colors.
 Heawood also proved that if the number of
edges around each region is divisible by 3
then the regions are 4-colorable.1
 George Birkhoff then worked on the problem
for many years. His work led Phillip Franklin
to prove that the four color problem is true
for maps with at least 25 regions. 1
 In
1976 Kenneth Appel and Wolfgang Haken
then used a computer to prove the Four
Color Problem.
 They went about their proof by using a proof
by contradiction. They assumed a map that
needed more than 4 colors and then
developed a formula to look at maps using
numbers and algorithms to express
neighboring countries.3
 Their
program looked at almost 2,000 maps
and proved that it must be true since no
same colored region touched. It took the
program almost 1,200 hours to receive the
results.
 While many believe this is not true, no one
has been able to disprove what they found.3
 Many do not accept this proof because it uses
a computer and the math involved is too
tedious and difficult to prove by hand.
 Edge-
the line between two boundaries
 Vertex- a point where two or more lines
meet
 Graph- a visual representation of edges and
vertices
Vertices
 http://www-groups.dcs.st-
and.ac.uk/~history/HistTopics/The_four_colo
ur_theorem.html
 http://mathworld.wolfram.com/FourColorTheorem.html
 http://unhmagazine.unh.edu/sp02/mathpion
eers.html
 http://www.mathsisfun.com/basic-mathdefinitions.html