Graph
Seven bridges, and the invention of throwing things away
Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.
Four colours, and a proof nobody can read
Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.
Every corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
Six people at a party
Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicParityAbstractionChromatic numberComplete graphComputer assisted proofCounting argumentDegreeExistence proofGenusGraph colouringHierholzer's algorithm