# Four Color Theorem

My wife is making a quilt. She has a whole bunch of colors and is making a very simple pattern. I enjoy watching the whole process because it's very mathematical, but it has made me question the four color theorem because I've come up with a counterexample that indicates that I either:

1. Do not understand the problem.
2. Am missing a solution for this example.

The "counterexample" is this:

++++
+wx+
+yz+
++++


Unless w andz or x and y are allowed to be the same color you need 5 colors to make this work.

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That is correct: it does not include single-point adjacency. Otherwise the four-corners states would provide a familiar core for a potential counterexample. –  Brian M. Scott Jan 2 '13 at 19:08
Right, that's what made me curious... If you did decide that single-point adjacency (correct term?) was adjacent how many colors would you need then? –  Christopher Pfohl Jan 2 '13 at 19:10
If single-point adjacency were admitted for consideration, then no number of colors would suffice, and the problem wouldn't be very interesting. –  Austin Mohr Jan 2 '13 at 19:11
If you allow single-point adjacencies then you can create maps that require arbitrarily many colors by cutting up a circle with all the pieces meeting at the center (like a pie). –  Jonathan Christensen Jan 2 '13 at 19:12
Also not considered are multiple-component regions, eg Michigan. –  alancalvitti Jan 2 '13 at 19:12