Passing Notes in Class

I just got back from a four-day camping trip for the Fourth. While roasting marshmallows and hiking along trails, I managed to fall into a few puzzles. Justin posed a puzzle that we’d formulated a while ago: a multi-person variation on the unreliable postal service puzzle. At some later point, I brought up The League Problem. On the ride back, Amit mentioned the water jug puzzle and tried explaining his prior work on inaccessible cardinals to me.

I came home to good news about some of my prior work. Specifically, a puzzle that I like to call Passing Notes in Class had been posted while I was away: Alice and Bob want to pass notes to each other in class, but they don’t want the teacher to notice (or at the very least, not catch them very often). If Alice and Bob can both see how much the teacher is paying attention to them, and the rustling of notes between the two will make the teacher more likely to look their way in the future, what is the optimal tradeoff between passing as much information as possible between the two of them while limiting the number of times they get caught by the teacher?

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