Four Colours Are Always Enough

Step 1 of 5

Mapmakers had noticed it long before anyone could prove it.

The rule is that regions sharing a border must differ in colour — touching at a single corner does not count. In 1852 a student colouring the counties of England noticed he never needed more than four, and asked why. That question stayed open for 124 years.

Step 2 of 5

Three is definitely not enough, and five is easy to prove.

It is simple to draw a map needing four — four countries each touching all the others. Proving five is always enough was managed in 1890. The gap between four and five is where the whole difficulty lived, for eighty-six years.

Step 3 of 5

You cannot check every map, because there are infinitely many.

So the strategy was to prove that any map needing five colours would have to contain one of a specific finite set of arrangements. Then show that every arrangement in that set can in fact be coloured with four. If none survive, no such map exists.

Step 4 of 5

The list came to 1,936 cases, and checking it took a computer.

Kenneth Appel and Wolfgang Haken finished the proof in 1976. The finite list was far too large and fiddly for anyone to verify by hand — the machine ran for over a thousand hours. It was the first time a major mathematical theorem depended on a computation no human could reproduce.

Step 5 of 5

Some mathematicians were genuinely unhappy about it.

A proof is supposed to be something a person can follow and be convinced by. This one asked you to trust a program and the hardware it ran on. The objection was never that the answer was wrong — it was about what counts as understanding. Independent programs have since confirmed it, and a version has been checked by proof-assistant software.

The rule only holds for flat maps or a sphere. On a doughnut-shaped surface you need seven colours, and that result was proved long before the flat case.