Proofs (not) from The Book

We all know about The Book in which, according to Erdős, God keeps the most elegant proof of each mathematical result. It is interesting to speculate whether every theorem has a “Book proof ” (say what about the Four Color Theorem or the Collatz 3n+1 Conjecture?) or whether a “Book proof ” of a theorem is unique. I believe many mathematicians have their own private collections of proofs from The Book. In fact, such a collection, and a highly success-ful one, was published by M. Aigner and G. M. Ziegler [1] (see [2] for a complement). This article contains several proofs, not included in [1], that, in my opinion, are serious contenders for inclusion in The Book. Needless to say, the selection reflects my taste and mathematical interests. Most of the theorems discussed here are 100–150 years old, but these proofs are considerably newer. This is another issue to mull over: how long does it take for a Book proof of a theorem to emerge? Pick’s Formula Pick’s Formula gives the area of a plane polygon whose vertices are points of the standard lattice Z2. Theorem 1 (G. Pick, 1899 [21]) Let I be the number of lattice points inside a simple polygon, B the number of lattice points on its boundary, including the vertices, and A its area. The