The post I was planning on putting up today got delayed by completely foreseeable events, so here’s a cool nerd snipe I ran across recently.
Start with a shape that looks like a right triangle except that the hypotenuse has been replaced with part of the arc of a circle. (Convex inward, and intersects the triangle at only those two points.)
Prove that the curve is shorter than the sum of the two sides.
I like this question because it’s immediately obvious just from looking at it, but you’ll probably have to think to get a proof. It seems like there should be a completely obvious proof that you could explain to a six-year-old, and there probably is, but if so I don’t know what. Even the best proof I’ve seen involved a limit and wasn’t all that satisfying. Can you find a good one?
(As incentive for trying, a proof here will also work as an explanation of the troll pi phenomenon. Because there’s only a countably infinite number of points where the shape with all the right angles intersects the circles, and there’s an uncountably infinite number of points on the circle, there must be some points where no matter how many iterations you go, that point will never be on one of the corners. So zoom in on one of those points, and you get this shape. Proving this will prove that the shape with the right angles has a larger perimeter than the circle, and Archimedes will praise you as the savior of mathematics.)
Anyway, the post that was meant for today will probably go up in a day or two.