The 249-Sided Polygon Puzzle

Challenge yourself with this pencil-and-paper game


Here's an awesome puzzle from the desk of Mac Oglesby, a retired math teacher. In fact, it's multiple puzzles packed into a single download. The first thing you need to do is download the pdf and print it out multiple times. If you're a teacher, get ready to hit the photocopier!

The download is a single page, with some instructions, and multiple grids of dots. There's a huge number of ways to use this resource. Here's what Mac Oglesby suggests:

In the 3x3 grid at the top left, draw a seven-sided polygon.

This is a bit of a warmup. You don't need to use all the dots, but you can if you want. Joining the three dots along an edge doesn't count as two edges of the polygon though, and the edges can't touch or cross. Sadly, though there are 9 points in the grid, it's not possible to draw an octagon or nonagon in this way.

In the 4x4 grid at the top middle, draw a sixteen-sided polygon.

All the grid points can be used, and the answer is a cute little spike polygon. Once you've got that, try the next challenge:

Using the 5x5 grid with the centre missing, draw a 24-sided polygon

Or, can you get a 25-sided polygon when you add the centre dot? I suspect you can't, but I'm not 100% sure!

Draw polygons in the 6x6, 8x8 and 10x10 grids, with 36, 64 and 100 points each.

To the mathematicians out there: is there a "pattern" to these polygons? Can you describe a method for building a polygon in any even-sized grid? What about odd-sized grids?

Once you've cracked these individual puzzles, there are other challenges using the whole sheet as a single "grid" of 249 points:

  • Can you draw a single 249-sided polygon using every single grid point?
  • Can you draw 83 individual triangles, none of them overlapping?

  • Can you make sure none of the triangles is right-angled?

Here's a few more challenges that should be possible, but I can't guarantee it:

  • Remove one point from the 3x3 grid. Can you draw 62 non-overlapping quadrilaterals? Or 31 octagons?
  • Add the point in the centre of the 5x5 grid. Can you draw 50 non-overlapping pentagons? Or 25 decagons?

Have fun!

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