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How a Doodler's Problem Sparked a Controversy in Math | Scientific American
(www.scientificamerican.com)
It's hard to prove mathematically, but it's not very hard to wrap your brain around. In 2 dimensional space, it is impossible to draw five shapes that all touch each other with a contiguous border. Try it. Once you have four or more shapes, making the four touch requires that one shape be fully enveloped. You could have one shape touch as many shapes as you like, but those shapes cannot be touching more than three of the other shapes.
