When you find your students working during break time on a problem that was the topic of a lesson three weeks prior, you know that something has gone right and your heart is filled with joy and excitement.
What they are trying to do is to create a map that cannot be properly colored using just 4 colors. What does this mean? Roughly, The Four Color Theorem states that given a map in the plane (think map of USA or map of Africa for example), one can always color the regions (think countries/states) using just 4 colors so that regions that share a border get different colors. These students cannot believe that such a thing could possibly be true and are therefore trying hard to find a counterexample.
This theorem (which only became a theorem 40 years ago, and before that was just a problem) has fascinated and stumped mathematicians young and old for several centuries. How can a statement so easy to state that it can be explained to a first grader be so difficult to prove? Even today, the only known proofs require the aid of a computer.
But the proof is not what we dealt with at our lesson with second and third graders. Instead, the students had fun coloring maps with 2, 3 and 4 colors.
Coloring the following map with 2 colors was an easy task:
Finding a 3-coloring of this one was slightly harder:
And coloring this map with 4 colors was a very big challenge:
As I already mentioned, some students refuse to believe that 4 colors will suffice for any map and are convinced that they will find a counterexample soon. One student claims that she will grow up and find a flaw in the proof! She has my full support and encouragement for doing so!
And here are some more pictures of the students coloring away:
Musings of a Mathematical Mom 
Can you share your worksheets?
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Thank you for this idea! It became a Saturday morning coloring project for my 8-year-old. He completed the 2-color and 3-color patterns, then seemed to sail through the 4-color pattern…until it didn’t work. He had no interest in trying again at the time, but I’ve left a collection of empty pattern circles and markers lying on the kitchen table to see if he’ll come back it at some point on his own…
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If you google “four color theorem coloring” then you will find many more examples of maps of various difficulty that can be printed out. That 4-color one is particularly tricky (much more so than it looks!). But I think that you’re totally doing the right thing by just leaving it out as a passive reminder. I do that sometimes with my 8 year old as well.
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