Things I wish I could do to my logic students but I can't, so I won't.

I *heart* BBC cooking competition shows, and watch them whenever they come around. Last spring I was watching too much MasterChef when it occurred to me one day that this could be fun to adapt to logic, so I introduced my students one day to MasterLogician:

• "Today's challenge is the invention test. You can choose from the sweet axioms or the savory axioms from which to prove the best theorems you can. You have one hour and 15 min."
• "These proofs were a little messy, you could do with cleaning up your presentation. But you know, I really like that theorem! This is good!"
• "You've got four hours to produce your best proofs for the finest logicians in the land. They are expecting excellence."
• "For your finals, we are sending you to Poland. You will be taught Polish notation, and expected to prove all your theorems using it."

I had way too much fun with this. I remembering commenting on twitter, "I wonder if there would be as much hugging on MasterLogician as there is on MasterChef."

Yesterday, a student visited office hours, and in the course of trying to prove one of the homework questions, we realized it would be easy if we had the associativity of multiplication. So we started to prove that. Realized that would be easy if we could prove that 0 was idempotent for + and 1 idempotent for *. At one point he commented that we've really proven a lot of number theoretic facts in class so far, and I joked that for the exam, I should simply set them "Prove every number theoretic fact." Now, even I know that that's not reasonable in three hours, but it made me think: Wouldn't it be interesting to sit them down and simply ask "How many number theoretic facts can you prove in PA in the next three hours?" Of course, not all facts are equal, so you couldn't grade it just on number. You'd need some sort of weighting system, so that "1+1=2" and "1+1=1+1" and the like would garner you only minimal points, whereas things like "x+y=x->y=0" would be worth more, and proving there are infinitely many primes even more, and, say, the Chinese remainder theorem would be worth a whole bunch. What would the right strategy be? Would you simply try to prove as many of the basic facts as quickly as you could, and try to garner lots and lots of little amounts of points? Or would you gamble on being able to do one big proof in three hours, facing the possibility that you might completely bomb out? Or would you do a handful of easy ones to start off with, the ensure you had some points, and then head to the big ones? Or simply aim for three-four midrange ones? Each strategy could pay off big, and each could end up bombing.

Now, as someone pointed out on twitter, one needn't be a fast logician to be a good logician, so this type of test could be argued to be inherently unfair -- and even apart from this I am sure that there is no way I could get such an exam past the exam scrutiny board, but...it does sound like fun.