Colour the map
Pick a colour, then click regions. Two regions that share a border must not match. Try to finish every map with as few colours as you can. Four is always possible. Then press Greedy auto: it colours the regions in numbered order, always picking the first colour that is free. Watch that sensible method sometimes need five.
What's going on
Every map here can be coloured with four colours. That is the Four Colour Theorem, and it took from 1852 to 1976 to prove. The hard part is not finding a colouring for one map. It is showing there is no map, anywhere, that needs a fifth.
Greedy colours the regions in numbered order, always picking the first colour not used by an already-coloured neighbour. It never fails to finish, but it can paint itself into a corner: a region whose neighbours have already used colours 1, 2, 3 and 4 gets a fifth. The theorem says a better order always exists. Finding one by hand is the puzzle.
The odd wheel is the classic trap. A ring with an odd number of regions needs three colours on its own, and the centre touches all of them, so it needs a fourth. An even ring only needs two, and the centre a third.
Sheet: Four colours are always enough.