Discussion  - 
I introduced my kids to the four color map theorem this evening. I asked "How many colors would you need to color a map of the US so that no two states that touch are the same color?" Their responses illustrate two important problem solving strategies.

One said "50 just to be sure." Not a bad answer actually. It's undeniably true. Sometimes the key to cracking a problem is to first come up with an upper bound, even if it's nowhere near the smallest possible. Sometimes it's better to say "I know 50 will do" than "I think 6 should be enough."

Another tried to prove me wrong when I told her 4 colors would be enough. This is a great way to understand a theorem: try to come up with counterexamples, even though you know you'll fail. If you can find a pattern to all your failed attempts at counterexamples, maybe you can turn that into a proof. And if not, at least you have more intuition for why the theorem might be true.
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If you need to color the map, then there probably are border lines, so technically you would only need one color (that isnt identical to the border color) since the border would separate them so they wouldn't touch.
i like technicalities
I love the Four Colour Theorem.  So easy to state that children understand it.  So hard to solve.
Another thing you have to specify for general maps is that just touching at one point doesn't count; the states have to share an edge to be required to have different colors.

(Otherwise you could require any number of colors by making an arbitrary number of regions touch at a point. For the actual US map, though, I don't think this is a serious problem; there's only one point where this stipulation makes a difference, and only four states meet there.)
That's such a cool thing for a dad to do.  Super cool!  And aren't kids responses to tough questions fantastic?  GOOD JOB :D

This is why upper level math needs to start early so kids can have a long time to think about concepts.  After all, some took centuries to solve.  Young people often get only a day or two understand fairly complicated ideas.
Assuming one state doesn't tunnel to another state, "undermining" a third middle state.
I love when students try to disprove this theorem--causes great math talk. One of the best pd courses I've taken as a teacher was the Discrete Math Institute--awesome!
Please ask them the Monty Hall problem and post the response. 
Well, actually 50 is kind of a bad answer. They are saying that the upper bound is related to the number of regions (i.e. the coloring is trivial). There is nothing particulary good in that. They only recognize that they need as many colors as the number of regions. What if there are 800 regions? How can they find 800 different colors? Let me tell you this little "joke".

Umberto Eco, an Italian semiotician and essayst (not certainly a mathematician!) once proposed the "Theorem of the 800 colors": 800 colors are sufficient to color any map. There are two main problems. First you have to find a map with 800 regions, second you have to be able to describe (hence distinguish between!) 800 colors. This is more a joke, but it's relevant (I don't really know where you can find this, but I guess there is only the Italian version, somewhere)
John Cook is explaining the thought process of some one who didn't take any formal mathematical course and relating it to the standard techniques in mathematics. In that sense 50 is a good answer. It is an upper bound and there is a logic in arriving at the number 50.
From that point of view, it's an interesting answer. But there is no hint about how to improve it (as I pointed out, If you ask the same for a 800 region map, would you accept as an answer that you can only do that with 64 colors because I have that many different crayons? Or is 800 still a good answer, not dealing with the actual coloring?

This could be interesting, what if you ask them about a 800 region coloring. Too bad you already gave them the answer! I would be curious about which of the two bounds is the most appealing.
+Robert King ... which the original state of Massachusetts did, effectively (what is now Maine was part of Massachusetts).
+Stefano Pascolutti "_Well, actually 50 is kind of a bad answer._"  You do realize that you're picking on children, don't you?  

That kind of thinking is what discourages young people from mathematics when what they need is all the encouragement they can get.
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