Col game

Col is a pencil and paper 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 in On Numbers and Games.

Example game

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, she may colour any one of these.

Two moves in:

At this point, the requirement that the graph be proper comes in to 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.

The game continues, players moving alternately, until a player cannot make a move when it is their turn. This player is the loser. A possible continuation of the game is as follows (with each move numbered for clarity):

Game over:

In this outcome, the blue player has lost.