I find Jim Tanton’s videos so charming that I can’t understand why they’re so rarely viewed. Look at this one, on the relatively well known two pancake problem. It’s clear to anyone, relevant to a student of calculus—it’s a lovely application of the intermediate value theorem, and makes the intermediate value theorem meaningful and obvious (if you slowly move your knife across the pancake, at some point half of it will be on the left and half on the right)—and, most impressively, novel: I’d never seen the pizza problem or the quarter pancake problem before, and they’re great additions to the lesson. And it’s just a fun problem.
There’s a three dimensional analogue called the Ham Sandwich Problem: can you cut, in a single cut, a ham sandwich so that both slices of bread and the ham in between are cut precisely in half? The link gives an abstruse discussion.
Having finished two math books recently, I’m very interested in looking at Jim Tanton’s Solve This! The art of coming up with great problems is quite different from knowing math. From what I’ve seen (I referred to Tanton earlier in this post), Tanton is a master of thinking up clever problems.
Here are a few additional observations and questions for his video:
- Can there be a 3 pancake problem? Can you show with a simple example that it’s impossible to cut 3 pancakes in half with a single cut?
- Notice that when he’s cutting a pancake at different angles of rotation, all the cuts go through a single “center” point. Is he doing that on purpose, and is there some mathematical reason for it? [Hint: think about easy shapes, like squares or equilateral triangles… does every cut that cuts them in half go through some special point?]
- With 2 cuts, could you cut 2 pancakes into thirds?
Other thoughts/questions/observations/answers?
