← UC-328: Five Colours Are Not Enough
The Hadwiger–Nelson problem · asked in 1950

Colouring the Plane

Give every point of a flat surface a colour. One rule: two points exactly one unit apart must be different colours. How few colours do you need?

1. The rule is about one distance

Pick any point. Every point on the circle one unit away from it must be a different colour. Points inside the circle, or beyond it, are free to match.

1 unit on the circle: must differ closer: may match farther: may match

2. Why three colours are not enough

Three points in a triangle with one-unit sides all need different colours. Join two such triangles and the far tip is forced back to the first colour. Do that twice from the same starting point, with the two tips exactly one unit apart, and the two tips clash.

The Moser spindle (1961): seven points, eleven one-unit links. Every solid line is exactly one unit long. A fourth colour fixes this figure — which is why the minimum was known to be at least four.

3. Why seven colours always work

Tile the plane with hexagons just under one unit across, coloured in a repeating pattern of seven. Inside one hexagon, no two points are a full unit apart. Two hexagons of the same colour are always more than a unit apart. So a one-unit ruler can never land on the same colour at both ends.

Try it: drag the ruler anywhere and turn it with the slider.

Each hexagon is 0.8 units corner to corner; the ruler is exactly 1 unit. Numbers mark the seven colours so the pattern reads without relying on colour alone.

4. Where the answer stands

Seven is enough. The open question is how far below seven you can go.

1not enough
2not enough
3ruled out 1950
4ruled out 2018
5ruled out? claimed 2026
6possible
7possible; always works
2018: Aubrey de Grey, with a computer-checked pattern of 1,581 points. 2026: a paper written by an OpenAI model claims five are not enough; no expert has verified it yet. The hard part is that a colouring need not be tidy regions like these hexagons — it can be any assignment of colours to points, and a proof has to rule out all of them.
Next: Inside the Trap — how four colours were ruled out →
A companion visual to UC-328 · Five Colours Are Not Enough, a StratIQX case study. It illustrates the problem; it is not part of the case’s scored analysis.
Sources: A. Soifer, history of the problem (2003); L. and W. Moser (1961) for the spindle; A. de Grey, arXiv:1804.02385 (2018); OpenAI, “The Euclidean plane is not five-colorable” (2026, unverified).