Conjecture vs. Proof .1
Unit LAUNCH:The FOUR Color Problem
Conjecture vs. Proof Learning Targets
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 7. Look for and make use of structure.
Language
conjecture
counterexample
proof
theorem
An educated guess.
An example which disproves a proposition. For example, the prime
number 2 is a counterexample to the statement "All prime numbers
are odd."
A rigorous mathematical argument which unequivocally
demonstrates the truth of a given proposition. A mathematical
statement that has been proven is called a theorem.
An assertion that can be proved true using the rules of logic. A
theorem is proven from axioms, postulates, or other theorems
already known to be true.
The Four Color Conjecture
Here is a colorful map of the United States of America.
Can you find a map of the United States of America with fewer colors?
What is the least number of colors you can find on a map of the United States of America?
What is the least number of colors you can find on a map of the world?
What is the least number of colors you can find of counties in our state?
Conjecture vs. Proof .1
Unit LAUNCH:The FOUR Color Problem
The Four Color Problem – Team Task
Given any separation of a plane into contiguous regions, called a map, the regions can be
colored using at most four colors so that no two adjacent regions have the same color.
Work in your teams to make up your own “maps” to test this conjecture. Try to find a
counterexample.
The Four Color Game
The Four Color Problem1. It's a turn-based strategy game between you and a computer
opponent.
A grid of hexagons is created for you, and the hexagons are divided into large chunks
(similar to the layout for Dice Wars). Your goal is to color in as much space as you can,
while the computer opponent tries to do the same. The catch the size of Maryland is that
like the map puzzle, no two adjacent areas can be filled with the same color. Can you
dominate the majority of the map with your color?
You play as the black and grey colorer, while the computer plays with green and orange.
You alternate turns with your opponent, and alternate between colors (black, green, grey,
orange, black, green...). A tally of how many hexagons you've claimed appears at the bottom
of the screen. While the immediate instinct is to color in the largest region you can, further
play might reveal strategies to help you beat the computer. Each round is won by gaining
over half of the hexagons, or if there is a draw (as in, a region can't be filled by either player
because all four colors already border it), then the majority takes the cake.
How long can you last against the computer? My record is eight rounds, can you beat it?
1
http://jayisgames.com/archives/2007/07/the_four_color_problem.php
Conjecture vs. Proof .1
Unit LAUNCH:The FOUR Color Problem
Four-Color Theorem2
The four-color theorem states that any map in a plane can be colored using four-colors in
such a way that regions sharing a common boundary (other than a single point) do not
share the same color. This problem is sometimes also called Guthrie's problem after
F. Guthrie, who first conjectured the theorem in 1852. The conjecture was then
communicated to de Morgan and thence into the general community. In 1878, Cayley wrote
the first paper on the conjecture.
Fallacious proofs were given independently by Kempe (1879) and Tait (1880). Kempe's
proof was accepted for a decade until Heawood showed an error using a map with 18 faces
(although a map with nine faces suffices to show the fallacy). The Heawood
conjecture provided a very general assertion for map coloring, showing that in
a genus 0 space(including the sphere or plane), four colors suffice. Ringel and Youngs
(1968) proved that for genus
, the upper bound provided by the Heawood
conjecture also give thenecessary number of colors, with the exception of the Klein
bottle (for which the Heawood formula gives seven, but the correct bound is six).
Six colors can be proven to suffice for the
case, and this number can easily be
reduced to five, but reducing the number of colors all the way to four proved very difficult.
This result was finally obtained by Appel and Haken (1977), who constructed a computerassisted proof that four colors were sufficient. However, because part of the proof
consisted of an exhaustive analysis of many discrete cases by a computer, some
mathematicians do not accept it. However, no flaws have yet been found, so the proof
appears valid. A shorter, independent proof was constructed by Robertson et al. (1996;
Thomas 1998).
In December 2004, G. Gonthier of Microsoft Research in Cambridge, England (working with
B. Werner of INRIA in France) announced that they had verified the Robertson et al.proof
by formulating the problem in the equational logic program Coq and confirming the
validity of each of its steps (Devlin 2005, Knight 2005).
J. Ferro (pers. comm., Nov. 8, 2005) has debunked a number of purported "short" proofs of
the four-color theorem.
2
http://mathworld.wolfram.com/Four-ColorTheorem.html
Conjecture vs. Proof .1
Unit LAUNCH:The FOUR Color Problem
Martin Gardner (1975) played an April Fool's joke by (incorrectly) claiming that the map of
110 regions illustrated above requires five colors and constitutes a counterexample to the
four-color theorem. However, the coloring of Wagon (1998; 1999, pp. 535-536), obtained
algorithmically using Mathematica, clearly shows that this map is, in fact, four-colorable.
Video