How four colours were ruled out · Aubrey de Grey, 2018

Inside the Trap

A trap is a set of points arranged so that, however you colour them, two points one unit apart end up the same colour. This one catches every attempt with four colours. It is built in stages from one small hexagon.

1. The seed: one hexagon

Six points around a centre, every link one unit long. With four colours there are only four really different ways to colour it. In two of them, three of the outer points share a colour — a triple. In the other two, none do.

The whole trap turns on that triple. One part of it will force a triple to appear. The other part cannot survive one.

2. Growing the first jaw

Copies of the hexagon are overlapped, then the whole thing is doubled twice, each time turned by a precise angle so that particular points land exactly one unit apart.

3. It always catches something

Here is the finished 121-point jaw, properly coloured with four colours — no linked points match. These colourings were chosen to come as close as possible to escaping: each has a triple in only one or two of the 52 hexagons. Try the others. The triple moves, but it never goes away.

a same-colour triplethe four colours

Why it cannot be avoided: with no triples anywhere, the points marked A and B would be forced to match, and so would A and B′. But B and B′ are exactly one unit apart.

4. The second jaw, and the whole trap

The second jaw is a much larger arrangement around a single hexagon, built so that it cannot be four-coloured if that hexagon has a triple. Put a copy of it on every hexagon of the first jaw and the two demands collide: one jaw forces a triple, the other forbids it. No four-colouring survives.

1,345points in the second jaw

20,425points in the first complete trap

1,581after de Grey trimmed it, 2018

510a later, smaller trap, 2019

The 510-point trap, drawn from its published coordinates: 510 points and 2,504 links, every link exactly one unit long. No way of four-colouring these points keeps every linked pair different — a fact established by computer proof, not something the eye can check. The record today is 509 points.

Back to basics: Colouring the Plane — the problem itself →

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: construction from A. de Grey, arXiv:1804.02385 (2018), rebuilt here and matching its stated counts; 510-point coordinates and links from M. Heule’s published data, CNP-SAT; see also J. Parts, arXiv:2010.12665. That the 510-point graph cannot be four-coloured rests on the published computer proof.