There are 11 ways to chop a pentagon into pieces by drawing lines between corners - lines that don't intersect. For a square there are just 3 ways, and for a triangle there's just 1, using no lines at all.
We can count the ways for a polygon with any number of corners, and we get these numbers:
1, 3, 11, 45 ,197, 903, 4279, 20793, 103049, …
These are called the Schröder–Hipparchus numbers
, for an interesting reason. Hipparchus was one of the best of the ancient Greek astronomers. He lived around 100 BC, and he discovered the precession of the equinoxes, and invented the 'stereographic projection' - an important way to map a sphere on a plane.
But Plutarch, a philosopher who lived two centuries later, said Hipparchus also showed that the number of “affirmative compound propositions” that can be made from ten “simple propositions” is 103049.
See that number up there?
Nobody had a clue what Plutarch was talking about until 1994, when a grad student put the pieces together. The story is in my blog article here:http://golem.ph.utexas.edu/category/2013/04/permutations_polynomials_and_p.html
and the story moves forwards even further in the comments. I hope this is fun regardless whether you like simple easy math (chopping polygons into pieces), or heavy-duty sophisticated math (operads).