What is Map Colouring?
Map Colouring is played on a map divided into regions, like countries on a continent or counties on a road atlas. Every region has to be filled with one of four colours, and two regions that share a border may never be the same colour. Some regions arrive already coloured. They cannot be changed, and they are chosen so that the rest of the map can be completed in exactly one way.
The puzzle grows out of the four colour theorem. In 1852 Francis Guthrie, colouring a map of the counties of England, noticed he never seemed to need more than four colours and wondered whether that was true of every possible map. It took until 1976 to prove it: Kenneth Appel and Wolfgang Haken settled the question with a proof that leaned on hours of computer checking, the first major theorem to be proved that way. Four colours are always enough — which is exactly why four is the right number for a puzzle. With three, most maps cannot be coloured at all; with five, there is too much freedom for the logic to bite.
As a logic puzzle — a part-coloured map with a single solution — the idea is best known from Simon Tatham's Portable Puzzle Collection, where it is simply called Map. This page plays the same game on maps of its own: every one is drawn from scratch when you press New game, so no two are alike.
- Fill every region with one of the four colours.
- Two regions that share a border must be different colours.
- The regions coloured at the start are fixed and cannot be changed.
- Each map on this page has exactly one solution.
How to solve a Map Colouring puzzle
Almost every move comes from one observation: a region cannot be any colour that one of its neighbours already is. So look at a blank region and count the different colours around it. If its neighbours already show three different colours, the region must be the fourth. Fill it in, and that new colour immediately narrows the choices for every region it touches, which often hands you the next move.
When no region is down to a single colour, use notes. Switch Notes on and tap a region to pencil in the colours it could still be. Two candidates is the useful number: a region that can only be pink or blue is one neighbour away from being solved, and pairs of them unlock the next level of deduction.
That next level is the locked pair. If two regions touch each other and each can only be pink or blue, then one of them is pink and the other is blue — you do not yet know which way round, but both colours are spoken for. Any third region that borders both of them can be neither pink nor blue. The same idea stretches to three regions that all touch one another and share three colours between them: a region touching all three has to take the fourth.
On the Expert maps there is occasionally nothing left but to test a colour. Pick a region with two candidates, suppose it takes the first, and follow the forced moves in your head or in notes. If the chain ends with some region having no colour left, the supposition was wrong and the region takes its other colour — a deduction, not a guess.
- Start with the blank regions that have the most coloured neighbours.
- Three different colours around a region force the fourth.
- Pencil in candidates when a region is down to two colours.
- Two touching regions with the same two candidates rule those colours out for any region that borders both.
- After every colour you place, re-check each neighbour of that region.
Difficulty levels
The four levels change two things: how big the map is, and how deep the logic has to go. Easy and Medium can be solved from start to finish with the basic rule alone — find a region whose neighbours show three colours, fill in the fourth, repeat. Easy uses a small map and leaves a few extra regions coloured in; Medium uses a larger map and takes away every starting colour it can.
Hard maps are bigger again and are built so that the basic rule runs out: at some point you will need a locked pair or triple to make progress. Expert maps are the largest, with around forty regions, and each one needs at least one what-if chain. Every map at every level is checked by a solver that only uses these same techniques, so the difficulty label describes the logic the map really requires.
- Easy: about 15 regions, a generous start, basic rule only.
- Medium: about 20 regions, fewest possible starting colours, basic rule only.
- Hard: about 30 regions, needs locked pairs or triples.
- Expert: about 40 regions, needs a what-if chain.
Why play Map Colouring online?
On paper, this puzzle needs four coloured pencils and a good eraser. On screen the bookkeeping disappears: pick a colour and tap a region to fill it, tap again to clear it, and undo as far back as you like. Notes let you pencil up to four candidate colours into a region, and when you colour a region its colour is rubbed out of the notes in every neighbouring region automatically. If two bordering regions end up the same colour, both are hatched so the clash is impossible to miss.
The Numbers button prints a number from 1 to 4 on every coloured region and on the palette, so the puzzle can be played without relying on telling the colours apart. On a keyboard, the keys 1 to 4 choose a colour, N toggles notes and right-click adds a note without leaving colouring mode. A hint fills in a region you could have deduced, your progress is saved in your browser, and every map is guaranteed to have one solution.
- A new map every game, at four difficulty levels.
- Colour notes that tidy themselves up as you solve.
- Clashing neighbours highlighted the moment they happen.
- A numbers mode for colour-blind players.
- Hint, undo, clear and a full solution when you want it.
- One unique solution per map, checked by a solver.