I have been co-leading math circles at GVSU since 2017. I find/make math circles by:
Spending a little time on YouTube
Asking students about cool math stuff they have seen.
Below are some specific math circles I've co-led with colleague Feryal Alayont and Lora Bailey.
3 Cups Trick: Inspired by this YouTube video and this paper. You flip cups 2 at a time, trying to get them all up right. The trick is that you change the starting configuration when you let your friend try. But why does it work one way and not the other? Leads to parity arguments, and when generalized, modular arithmetic. Notes.
Eleusis Express: Inspired by our colleague Dr. John Golden who was inspired by Dr. Robert Abbot. In this card game the "dealer" picks a secret rule and other members try to guess the rule. To aid in guessing, they can choose a card from their hand to play, and the dealer discloses if their choice follows the rule or not. Harder than you think! Flyer. Notes.
Even Quads: Inspired by the Even Quads game by Dr. Lauren Rose and Jeffrey Pereira. The game is like SET with a twist. In this math circle, we play a few rounds, and then participants ask mathematical questions like "Can we figure out how many cards there are without counting?", "How many possible quads are there?" Flyer. Slides/notes.
Shoddy Splitting: Inspired by this video of how to fold a tie into thirds by Dr. James Tanton, we demonstrate how to fold a piece of string into thirds, and then ask why does it work? For undergraduate participants it invoked ideas of limits, integration by substitution, and numerical analysis. We then ask, what other fractions can you make folding the string? Notes. Flyer.
Spy Games: How can you use math to do some detective work? Inspired by this vanishing point math circle by Anna Davis and Dino Seketa and the art gallery problem. Flyer.
Family Feud: Flyer.
Penny Problems: This is a collection of "magic tricks" all involving pennies. The first is one of those "magic number tricks" where you start with any number, do some operations, and everyone ends up with the same number. The second is inspired by a bar trick with flipping cups. The final trick has to do with what glass door claims is an Apple interview question. Flyer. Notes.
Voronoi Diagrams: Flyer.
When Tricks Become Methods: Inspired by the paper "Three Tricks or One Method" by Dr. James Propp. Start with the classic mutilated chess board problem. Flyer.