As OEIS Foundation, Inc (2020) and GeeksforGeeks (2018) notes, the applications of Catalannumbers pop up frequently in many different problems in number theory. In a presentationgiven on Catalan Numbers, Mirando (n.d.) notes that applications of Catalan numbers ap-pear in Computer Science when using stacks and pushing and popping data values.
And this is not even considering the huge impact teaching and working through this problem gives for students. This silly little problem may have an Achilles' Heel with the Algebraic Solution but it still proved to be a formidable challenge to find (much less understand) that solution in the first place. And along the way, this problem can expose students to a diverse spectrum of content from solving quadratic equations, to discrete mathematics, to Calculus and Real Analysis, to bits of Statistics on top of a good about of review in probability.
Included below is a series of images taken over the course of two weeks when we first started to attempt this problem. More than anything else I can comment on, this problem has had immense significance in my life, and these images serve to remind me of some of the many bad ideas and the few good ideas that we tackled in this problem. Even after nearly a year working on this problem on and off, it still stuns me just how much I have learned by working through just this one problem.
