Once upon a time I scored a 42 on the Putnam. Two decades later I placed 23rd at the World Puzzle Championships. I’d be happy to help if I can.
But honestly? This website holds many mathematicians far better than I. Really I’m replying more from a desire to assuage my own curiosity, than from a strong belief that there exists a problem that uniquely I can solve. All I can promise you is that if I don’t know the answer, I’ll say so.
ETA: If anything, folks, this comment is worth downvoting for committing terrible math while bragging of being good at it. Not that anybody here could have caught the error, but however much eleven years between events might feel like two decades, it really, really isn’t.
Oh, that and for commenting on Luke’s post when he specifically asked for emails.
The day-to-day cognitive skills I’ve mastered most completely (I will not say “rationalist skills,” because this is true of my countless irrational skills too) are the ones which I learned during a moment of strong emotion — any emotion, excitement or curiosity or joy or surprise or depression or fear or betrayal.
In the case of this particular skill, it was betrayal. I brought it on myself — the details aren’t important; suffice it that I spent two weeks living in the “should-universe” (I like this term) before a rude reminder of reality — but the emotion, the physical neurendocrine experience of betrayal, was quite real. And I’ve been able to return to it ever since, and if I’m ever in a situation where I might be working from a cached plan, I can relive a hint of it and ask myself, “Now, you don’t want to feel that again, do you?”
Unfortunately, this experience strongly ties the five-second skill of “check consequentialism” to the emotion of betrayal in my mind. It is very easy for me to construct social experiments in which the teacher radically betrays her students, and then turns around and says, “Don’t let anyone do that to you again!” But that is horrible teaching. It’s a lot more difficult for me to imagine what “check consequentialism” would feel like if it carried a strictly positive emotional association, and then extrapolate outward to what kind of social situation would provide that emotional/cognitive link.
Students must abandon a cached plan, and evaluate the real-world consequences of their actions instead, at precisely the moment they get a strong positive emotional charge. Preferably “fun.” Preferably in the sense of a party game, not a strategy game: both because people who have learned to win without disrupting social bonds (or who care more about winning than about socialization) have often already learned this skill, and because the moment I construct “winning” as a state which disrupts social bonds, I’ve set up a false dilemma which misleads my students about what rational thought actually is.
But what’s the chain of causation? A dispassionate experimenter times the payoff to correlate with the decision? That seems awfully Pavlovian. Leaving the plan causes a reward which provides an emotional payoff? Maybe, but if a student only leaves the plan in expectation of reward, they haven’t actually learned anything beyond the latest professorial password. The excitement of getting the right answer to a puzzle inspires leaving the plan? I suspect this is the way to go. But then what sort of puzzle?
I’m going to press the “Comment” button now, even though I don’t think I’ve contributed much beyond a restatement of your original dilemma. Perhaps having done so, I’ll think of some specific scenarios overnight.