An ancient Indian physician counted 63 ways to mix six tastes
The Sushruta Samhita, a classic of Indian medicine, notes that six basic tastes can be combined in 63 different ways, singly, in pairs, in threes and so on. That tally, every possible mix except the empty one, is an early worked example of combinatorics, the mathematics of counting arrangements that now shapes lotteries, tournament schedules and cryptography.
Counting puzzles surface all over the ancient record. Problem 79 of Egypt's Rhind papyrus, from the sixteenth century BC, involves a geometric series. Plutarch describes a dispute between the Stoic Chrysippus and the astronomer Hipparchus over a delicate counting problem whose answer was later tied to what are now called Schröder–Hipparchus numbers, and Archimedes may have counted the ways to assemble a tiling puzzle called the Ostomachion.
Medieval progress happened mostly outside Europe. Around 850 the Indian mathematician Mahavira gave formulas for permutations and combinations. Abraham ibn Ezra showed around 1140 that the counts are symmetrical, and Gersonides found a closed formula in 1321. The triangle of numbers later named after Pascal appeared in treatises as early as the tenth century. English church bell ringers, working through every order of their bells without repeats, were unknowingly tracing what mathematicians now call Hamiltonian cycles.
Pascal, Newton, Jacob Bernoulli and Euler laid foundations in the early modern era, and J. J. Sylvester and Percy MacMahon built on them around 1900, while the four colour problem stirred interest in graphs. The subject only became a unified branch in the later twentieth century, when general methods replaced one-off tricks and dozens of new journals appeared. Defining it remains awkward; H. J. Ryser noted it cuts across too many areas to pin down.
Design theory shows the practical side. Kirkman's schoolgirl problem of 1850 asked how to arrange girls walking in rows of three so that no two share a row twice, and its solution is a special case of a structure that later helped classify finite simple groups. Ronald Fisher used related designs to plan biological experiments.
Source: Combinatorics
