Chapter 9 Proof techniques IV — Magic
If you can keep your head when all about you are losing theirs, it’s just possible you haven’t grasped the situation.
—Jean Kerr
The famous mathematician Paul Erdös is said to have believed that God has a Book in which all the really elegant proofs are written. The greatest praise that a collaborator72 could receive from Erdös was that they had discovered a “Book proof.” It is not easy or straightforward for a mere mortal to come up with a Book proof but notice that, since the Book is inaccessible to the living, all the Book proofs of which we are aware were constructed by ordinary human beings. In other words, it’s not impossible!
The title of this final chapter is intended to be whimsical — there is no real magic involved in any of the arguments that we’ll look at. Nevertheless, if you reflect a bit on the mental processes that must have gone into the development of these elegant proofs, perhaps you’ll agree that there is something magical there.
At a minimum we hope that you’ll agree that they are beautiful — they are proofs from the Book73.
Acknowledgment: Several of the topics in this section were unknown to the author until he visited the excellent mathematics website maintained by Alexander Bogomolny at http://www.cut-the-knot.org/
The collaborators of Paul Erdös were legion. His collaborators, and their collaborators, and their collaborators, etc. are organized into a tree structure according to their so-called Erdös number.↩
There is a lovely book entitled Proofs from the Book, Aigner, Hofmann, and Ziegler (2014) that has a nice collection of Book proofs.↩