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Reinventing Discovery by Michael Nielsen

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Appendix: The Problem Solved by the Polymath Project

The Polymath Project aimed to prove a mathematical result known as the density Hales-Jewett (DHJ) theorem. Although the proof of DHJ is complex, the basic statement can be understood by anyone. Take a look at the following three-by-three grid:

images

I’ve marked seven of the squares on the grid with a dot; as you can see, it’s possible to draw a line through three of those dots. By contrast, the configuration in the following picture is line-free—you can’t draw a line through any three of the dots:

If you play around a bit, you’ll discover that this configuration is the largest possible line-free ...

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