“In mathematics, as in the physical sciences, we may run an experiment or check a few cases to come up with a conjecture for a theorem. However, in mathematics experiments cannot replace proof, no matter how natural and obvious the conjecture is that they support. For example, the maximum number of regions defined by 1, 2, 3, 4, 5, and 6, points on a circle are 1, 2, 4, 8, 16, and…31, not 32!

Or, take the famous Goldbach conjecture which claims that every even number greater than two is the sum of two prime numbers as, for example, 12=5+7 or 30=23+7. Although this conjecture has been checked for millions of cases, unless a proof is found, we cannot be sure that the next case we check won’t show that the conjecture is false.

Proofs should be as short, transparent, elegant, and insightful as possible. Our proof that the number 0.999…, with infinitely many 9s, is equal to 1 is of this kind and its main argument can be easily adapted to convert any decimal number with one of those slightly worrying infinitely repeating tails into a number we are more comfortable with. The proof that the indented chess board cannot be tiled with dominos is another example. Of course, the argument here applies to many other mutilated chess boards.”