Big Maths Ideas · Beta

The Four Colour Theorem

Colour neighbouring regions and explore why four colours are enough for a planar map.

Use the palette to colour this flat map. Neighbours share a border segment; a corner alone does not count.

Neighbours must have different colours\text{Neighbours must have different colours}
Use the palette to colour this flat map. Neighbours share a border segment; a corner alone does not count.012345

No conflicts so far. Finish colouring the map.

Uncoloured regions
6
Conflicting borders
0
Colours used
0
The idea behind the experiment

The Four Colour Theorem is proved: every planar map can be coloured with at most four colours, assuming each region is connected and adjacency means sharing a boundary segment. This small map is an experiment, not a proof of the general theorem.

A proof author’s explanation
Predict, test and explain

Step 1 of 3

Predict

Do two map regions that meet only at a corner need different colours?

InvestigateTry the five-region ring using only three colours. Can the centre also be coloured?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Do two map regions that meet only at a corner need different colours?

corner-only contact\text{corner-only contact}

Hint

Choose a colour before selecting a region.

Reveal answer

Map adjacency means sharing a border segment. Meeting at one point alone does not count.

not adjacent\text{not adjacent}

Build an explanation

Hint 1

Choose a colour before selecting a region.

Hint 2

Every ring region shares a border with the centre and its two ring neighbours.

Hint 3

One successful map is evidence about that map, not a proof for every map.

Worked example

This example uses fixed values, separate from the diagram controls.

Colour a five-region ring and its centre.

  1. An odd cycle needs at least three colours around the ring.
  2. Every ring region borders the centre.
  3. Give the centre a fourth colour.

Answer: Four colours suffice for this map.

Connect this idea

Watch out: Sharing a point is different from sharing a boundary segment.