Col (game)

Col is a pencil and paper game, specifically a map-coloring game, involving the shading of areas in a line drawing according to the rules of Graph coloring. With each move, the graph must remain proper (no two areas of the same colour may touch), and a player who cannot make a legal move loses. The game was described and analysed by John Conway, who attributed it to Colin Vout, in On Numbers and Games.

In the following game, the first of the two players is using red, and the second is using blue. The last move in each image is shown brighter than the other areas.

The starting graph:

The first player may colour any of the areas to begin. However, the region around the outside of the graph is not included as an area for this game.

After the first move:

The second player now colours a white cell. As no areas are currently blue, any white cell is allowed.

Two moves in:

At this point, the requirement that the graph be proper comes into effect, as a red area must be made which does not touch the existing one:

Once the third region is coloured:

Note that areas only count as touching if they share edges, not if they only share vertices, so this move is legal.

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