Euler's 1751 question to a friend quietly invented the Catalan numbers
In 1751, Leonhard Euler wrote to his friend Christian Goldbach with a geometry question: cutting a convex polygon into triangles using only non-crossing diagonals, how many different ways can it be done? A triangle obviously allows only one way, and a square can be split two ways. How many ways can a seven-sided heptagon be triangulated?
Reveal the answer
Exactly 42 ways. Euler computed the sequence 1, 2, 5, 14, 42... by hand for small polygons and guessed the general formula from the pattern, though he couldn't prove it himself; his friend Goldbach, and later Johann Segner, supplied the missing pieces, with a complete proof by 1759. The sequence was named the Catalan numbers nearly a century later, after Eugène Catalan connected them to bracketings of algebraic expressions in 1838 — Euler had already found them first.