Parity
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.
Pascal's triangle, in two colours
Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.
The primes on a spiral, and a pattern nobody ordered
Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.
Numbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
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.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
Named alongside it
The objects these essays reach for when they reach for this one.
PrimesGraphAbstractionBinomial coefficientBinomial distributionComplete graphConjectureConvergenceCounting argumentCramér's modelCyclic groupDegree