A Fractal Inducing Problem about Polygons

Here’s a question I like: take an n by n grid (start small), and draw straight lines connecting dots. If you end where you started and don’t cross lines, you get a polygon. What’s the most number of sides the polygon can have?

I received those words in an e-mail from Dan Finkel in April of 2014. I read the e-mail on the way from my front door to my car, and I visualized the 3×3 case for most of my commute to Tacoma.

Convinced that the problem would be perfect for my younger of two groups of elementary age homeschoolers that day, I launched into it–still with no clear ideas about the solutions. By the end of the hour, I still hadn’t had a chance to pick up a pencil, and I was skeptical that the this group of children had managed to produce the best solution–it’s hard to be satisfied by an asymmetrical solution to a problem like this. I pitched the problem to two groups that day and spent multiple hours after class exploring the different sizes–the 4×4, 8×8, 16×16 are particularly interesting to me for a particularly elegant solution. This simple problem has captivated me more than any other I’ve considered since, and I’m looking forward to hearing what you think.

Post your findings in the comments!

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