Dan Asimov writes:
Speaking of muffinsectors, here is a problem about chocosectors:
You are given a rectangular chocolate bar that is scored in the usual way into KxL small squares. Repeat this step: Choose one of the pieces (at first, there's only the whole bar) and break it all the way along one of its scorings . . . until all the pieces are the small squares.
There is a fun way to present this problem so that it has the surprising property that mathematically sophisticated undergrads take LONGER to see the answer than less sophisticated undergrads. (It'd be fun to make a whole collection of problems that smart and/or knowledgeable people tend to do poorly on: an anti-intelligence or anti-aptitude test if you will.) Specifically, present the candy bar problem as a problem in combinatorial game theory, with players taking turns, and ask WHO WINS?, without revealing that the parity of the duration of the game, and indeed the duration of the game itself, isn't affected by the strategies the players adopt. (This is in fact the way I invented this problem twenty years ago: I like to call the game "Impartial Cutcake" to disguise the fact that there's no real game-theory going on at all.) Lead the students to prove that an even-by-even or even-by-odd bar is a win for the first player (strategy: the first player cuts the bar into two identical pieces), and a fancier argument to prove that an odd-by-odd bar is a win for the second player (imagine that the pieces are still in place, and copy whatever your opponent just did but rotated 180 degrees). Now challenge the students, working alone or in small groups, to figure out who wins under optimal play when the initial configuration is a pair of chocolate bars, one a-by-b and one c-by-d. It is vital to keep a straight face, and not to hint that there's lurking triviality. My experience is that students who know combinatorial game theory jump in and compute Grundy values and get really distracted from the heart of the matter. Students who don't know the theory of impartial games but who have some mathemtical taste and a fondness for proof take longer to find the slick solution than the others. Why? I think it's partly because they value the work that the group has already done in finding strategies for the players, so they emotionally resist making a discovery that would render the earlier arguments pointless. Also, they're used to the idea that in math, you keep building on what you already know, so they want to use what the group has already proved about single rectangles as a tool for analyzing the two-rectangle case. Also, they're inclined to use a trick that's worked before on new problems to see how far the trick can be pushed. The less sophisticated students have a kind of blankness that on the one hand makes it hard for them to get started on the problem but on the other hand keeps them from getting distracted once they do start. I did meet one sophisticate who put his sophistication to good use by imediately asking "What are the invariants here?", but this kind of reaction was not the norm. Jim Propp