I am a big fan of boardgames. The older I get, the more I value spending time with other people. However, my favorite thing about boardgames is not actually playing them but reading the rules. This might sound wrong to you, but I am dead serious. Reading well-written rules is a bliss. Games are for the most part formal systems. A rulebook is like a math book without theorems: just definitions with a few examples. This is for a good reason: exploring the consequences, proving theorems, going through examples - this is what playing the game is.
Boardgames are hence an excellent way to introduce people to formal systems and mathematics in general. I would love to explore this perspective further and it sounds like an excellent excuse to finally learn TLA or Alloy. Both are tools for formalizing and analyzing systems and specifications. Using these to model some classical boardgames might be a fun way to explore edge-cases of the rules (and if game designers actually did that, we would not end up with so many games that are just helplessly underspecified).
On a somewhat related note, this analogy between boardgames and mathematics goes the other way around as well: I believe that truly great mathematicians do not just prove theorems, they coin definitions. Definitions set the playing field. They set the limits of what is and is not thinkable. They separate the relevant from the irrelevant. Theorems “merely” illuminate definitions.
This is not to say that theorems are not relevant, but even the most important definitions like topology and convergence are not often celebrated as the achievements they are. I guess that part of it might be that definitions need to prove themselves useful once they have been established, so they shift shape until they are useful (and they only appear so ingenious in hindsight - it’s a form of survivorship bias). On the other hand, even an unproven theorem can be understood as a question worth answering: The relevance of a (potential) theorem can be understood without knowing its proof, whereas a definition can only be appreciated in hindsight. Maybe this is also why boardgames do not have their rules on the box ;)