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Pancakes and Waffles

There are many theorems which are really easy to prove after you develop some technical machinery, tricks and shortcuts. This is analogous to waffles: you need the iron and the mix, but once you have them it's just a matter of pour and wait.

On the other hand many of these same theorems have more difficult "direct" proofs, which is analogous to pancakes: the ingredients and tools required are simple, but you have to have skill to apply them properly.

A waffle iron

A great example of this is the theorem that you can always find as many primes as you like. A pancake proof would be like Euclid's, assuming very little and deriving the result after a clever argument. A waffle proof would be Furstenberg's version which is short but uses topology, an entire field of mathematics with its own notions and terminology.

This is similar, but distinct, from a notion my physics teacher Ruby Musgrow had about solving problems. She described two different problem solving methods as the Ulysses S. Grant approach and the Robert E. Lee approach. The Grant approach is throwing all your resources at a problem thus overpowering it and arriving at the solution, while the Lee approach is using your resources cleverly and selectively and still resolving the problem.

While we're at it I might as well mention Paul Erdős, a famous mathematician who once said that there was a book with all the best proofs written down, and our attempts at proofs are simply trying to emulate these proofs. He would call a proof "from the book" if it were particularly elegant, powerful and at the same time simple. So now given this discussion one could also call a "proof from the book" a Robert E. Lee Pancake proof.

Bunny with a pancake on its head

Ever wondered what New York looks like to me? These pictures are as close as I can get you. Click the pic below to see more. Enjoy.

Looking North on 1st Avenue from 158 1st Ave

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