How to play Four Colours

Colour every region with one of four colours so that regions sharing a border never match. Also known as Map colouring puzzle, Four-colour map.

▶ Play Four Colours ⊞ Open the solver

The rules

Every region is coloured red, yellow, green or blue.
Two regions that share a border — even a short one — must have different colours.
Regions that only touch at a corner may have the same colour.
Some regions start coloured, and there is exactly one way to complete the map.

Playing Four Colours on Puzzle Solvers

Choose a map size and a difficulty, then press New Puzzle.
Pick a colour (or press 1–4) and tap a region to paint it. Drag to paint several. Tap again or right-click to clear.
Each colour also has a shape (● ▲ ■ ◆), so the map works for colour-blind players and in black-and-white print.
Hint explains the next deduction — usually a region whose neighbours already use three colours.

Open the Four Colours game →

Using the Four Colours solver

Choose the size, draw the regions and add the given colours (or press Sample).
Press Solve and step through: regions forced by their neighbours, narrowing down, and what-if chains.

Open the solver →

A worked example

The region in the top corner touches three regions coloured red, green and blue — so it is yellow.

A region touches yellow and blue, so it is red or green. Its neighbour touches red, so if the first were green, the neighbour would be squeezed to one colour — follow it through.

Strategies

Three around one. If a region’s neighbours already use three different colours, it must be the fourth.
Narrow it down. Cross off the colours a region’s neighbours already use; keep track of what is left.
Pairs. Two neighbouring regions that can each only be red or blue use up both colours — any region touching both can be neither.
What if. When stuck, try a colour in a region with two options and follow what it forces. If something ends up surrounded by all four colours, the try was wrong.
Big regions first. Regions with many neighbours are the most constrained — they are often decided early.

Common mistakes

Treating a corner touch as a border.
Changing a region that started coloured.
Guessing without following through what the guess forces.

Where it comes from

The Four Colour Theorem was suggested in 1852 by Francis Guthrie, who noticed that four colours were enough for a map of English counties. It was finally proved in 1976 by Kenneth Appel and Wolfgang Haken — one of the first major proofs to rely on a computer.

Frequently asked questions

Why four colours?

Four is always enough for any flat map — that is the Four Colour Theorem. Three is not always enough.

What do the shapes mean?

Each colour has a shape (● red, ▲ yellow, ■ green, ◆ blue) so you can tell them apart without colour.

Is there always one answer?

Yes — the starting colours are chosen so there is exactly one way to finish.

How do I solve one from a book?

Open the Four Colours Solver, draw the regions, add the given colours and press Solve.

Keep going

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