# 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.