Four Colour Theorem — Math Gallery

Map

Simplified Singapore planning areas — outlines and names are approximate.

Your colours

Cycle mode: tap a region to step through the four colours. Pick a swatch to paint with just that colour.

Score

0regions

0colours used

0uncoloured

0clashes

Tap regions to colour them, then press Done with 4?

Actions

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Where’s the maths? This is graph theory — the branch of mathematics that studies dots joined by lines, and asks what can and cannot be arranged. Forget the shapes for a moment: what actually matters about a map is which region touches which. Shrink every region to a single dot, and join two dots whenever those two regions share a stretch of border, and the whole map becomes a network (a graph). Colouring the map so that neighbours differ is then exactly the puzzle of colouring the dots so that no line joins two dots of the same colour. The astonishing fact is that for any map you can draw on a flat sheet of paper — any country, any doodle, any number of regions — four colours are always enough. Nobody has ever found a map needing five, and since 1976 we know for certain that nobody ever will.

📘 On the Sec 1–4 syllabus

uses: Sec 1 · N1 — counting and logical reasoning

Beyond the syllabus: graph colouring is enrichment; the 1976 computer-assisted proof is a talking point.

Every map drawn on a flat sheet can be coloured with at most 4 colours, so that no two regions sharing a length of border get the same colour. Regions meeting at a single corner point (like the corners of a chessboard) are allowed to match — a corner is not a border.

Three is not enough. Take one region surrounded by a ring of three, all touching each other: that already forces four colours. So the answer is exactly 4, not 3.
Why not five? That is the hard part. Proving 5 is enough is a page of school-level reasoning; proving 4 is enough took 124 years. Appel and Haken finally did it in 1976 by getting a computer to check 1 936 unavoidable configurations — the first famous proof no human could read in full.
Flat only. On a doughnut (a torus) the theorem fails: there you may need 7 colours. Being flat — having no holes — is what keeps the number down to four.
How the app colours it. It repeatedly picks the region with the fewest colours still available (this is the DSatur rule), colours it, and backs up whenever it paints itself into a corner. It has never needed a fifth colour — and by the theorem, it never will.
The counting behind it. Every planar map has a region with at most 5 neighbours. That single counting fact, which follows from Euler’s V − E + F = 2, is the seed of every proof.

Try this. Switch to Draw your own and deliberately try to build a map that needs five colours — ring a region with five neighbours, nest regions, make long snaking borders. Then press Solve it for me. Watch the “colours used” counter: it never reaches 5. Now turn on the Dual graph and see why — to force a fifth colour you would need five regions all touching each other, and on a flat sheet those five lines can never be drawn without a crossing.

Controls: tap or click a region to change its colour · drag on the blank canvas in “Draw your own” to add a region · scroll or pinch to zoom in, drag to pan, double-click to reset the view · arrow keys select a region, Enter recolours it.