How did an ancient physician find 63 ways to mix six tastes?

By taking the six tastes one at a time, two at a time, three at a time and so on up to all six together, and counting every mix. The Indian physician Sushruta asserts in the Sushruta Samhita, a medical text that reached its present form by about 500 CE, that six different tastes make 63 combinations. That is exactly 2 to the 6th power minus 1: each taste is either in a mix or left out, which gives 64 choices, minus the single empty 'mix' with no taste at all. Without any modern notation, the text had counted every possible selection.

Why would a doctor care? Taste was part of medicine. The Sushruta Samhita and other ancient Indian texts describe more than 700 medicinal herbs, and their descriptions include each herb's taste alongside its appearance, digestive effects and dosage. Counting mixes of tastes was a physician's question that turned out to be a mathematician's question too. That kind of question has a name today: combinatorics, the area of mathematics concerned with counting and the properties of finite structures.

Similar counting puzzles appear across the ancient and medieval world. The earliest recorded use of combinatorial techniques comes from the Rhind papyrus in Egypt, dating to the 16th century BC. The Greek historian Plutarch described an argument between Chrysippus and Hipparchus over a delicate counting problem. In India, the mathematician Mahavira gave formulas for permutations and combinations around the year 850, and around 1140 Rabbi Abraham ibn Ezra established a symmetry in the numbers now called binomial coefficients, the numbers laid out in what became known as Pascal's triangle.

Combinatorics only became an independent branch of mathematics in the later twentieth century. Pascal, Newton, Bernoulli and Euler laid foundations during the Renaissance, and in the second half of the twentieth century the field grew fast, spurred by connections to algebra, probability and number theory. Today it is used frequently in computer science to analyze algorithms.