Map colour
Colour every region so no two touching regions match.
The maths Graph colouring and the four-colour theorem
6 regions to colour with 3 colours.
This map needs 3 colours.
Pick a colour, then choose a region. Regions are numbered 1 to 6, and the status line above names any pair that clashes.
How to play
Pick a colour from the swatches, then choose a region to paint it. Two regions that share a border must not have the same colour. Meeting at a single corner does not count as sharing a border, so those two may match.
You are given exactly as many colours as this map needs, and not one more. Four are always enough for a flat map; whether three will do depends on the map, and working that out is the puzzle. A region sharing a colour with a neighbour is outlined in red and carries a warning mark, and the pair is named in the line above the map.
Every colour also has its own pattern — solid, dots, diagonal stripes, cross-hatch — so the map can be read without relying on hue.
Keyboard: Tab steps through the regions in order, and the arrow keys jump to the nearest region in that direction. Home and End go to the first and last region. Number keys 1 to 4 paint the focused region, 0 or Backspace clears it, and Enter or Space paints it with the selected swatch.
Why this is mathematics
Any flat map, however tangled, can be coloured with four colours. It was conjectured in 1852 and stayed unproved for 124 years, then became the first major theorem proved with a computer. Four is always enough. Three often is not, and finding out which is the puzzle.