Why Are Four Colours Enough for Any Planar Map?

Take any planar map and require regions that share a boundary to have different colours. However complicated the map becomes, no more than four colours are needed. This is the four-colour theorem. “No more than” matters: many maps need only two or three colours, while some arrangements genuinely require a fourth.

The conditions must be precise. Regions that meet only at one corner are not adjacent; they must share a boundary segment. Each region is also assumed to be connected. A real country may have an exclave or islands. If separated pieces of the same country are required to share one colour, the problem has changed and the theorem cannot simply be applied unchanged.

The map becomes mathematics through a translation. Put a point in every region and connect the points for each adjacent pair. Those connections can be arranged in the plane without crossings, producing a planar graph. Colouring the map is now equivalent to colouring the points so that connected points differ. The theorem guarantees that every such planar graph can be coloured with four colours. It is not an approximation inferred from trying many maps, but a proved theorem; the first generally accepted proof, in 1976, used computers to check a large collection of configurations.

Its interest is not really about saving paint. It shows how local restrictions create a global limit. A region must avoid only the colours of its direct neighbours, not its neighbours' neighbours, while planarity restricts how densely those relationships can interlock. Four colours suffice because these constraints work together, not because maps are usually simple.

https://mathworld.wolfram.com/Four-ColorTheorem.html

https://distributedmuseum.illinois.edu/exhibit/four-color-theorem/

https://mathshistory.st-andrews.ac.uk/HistTopics/The_four_colour_theorem/