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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