Because any colored Y board contains one and only one winning Y, it followsthat any colored Hex board contains one and only one crossing.
1.2. PARTISAN GAMES27Figure 1.15.An illustration of the four possibilities for how a bluepath going throughTkandTk+1reduces to a blue path in the smallerboard.Remark1.2.10.Why did we introduce Y instead of carrying out the proofdirectly for Hex? Hex corresponds to a subclass of Y boards, but this subclass isnot preserved under the reduction we applied in the proof.1.2.4. More general boards*.The statement that any colored Hex boardcontains exactly one crossing is stronger than the statement that every sequence ofmoves in a Hex game always leads to a crossing, i.e., a terminal position. To seewhy it’s stronger, consider the following variant of Hex.Example1.2.11.Six-sided Hexis similar to ordinary Hex, but the boardis hexagonal, rather than square. Each player is assigned three nonadjacent sidesand the goal for each player is to create a crossing in his color between two of hisassigned sides. Thus, the terminal positions are those that contain one and onlyone monochromatic crossing between two like-colored sides.In Six-sided Hex, there can be crossings of both colors in a completed board,but the game ends when the first crossing is created.Theorem1.2.12.Consider an arbitrarily shaped simply-connected3filled-inHex board with nonempty interior with its boundary partitioned intonblue andnyellow segments, wheren≥2.Then there is at least one crossing between somepair of segments of like color.Exercise1.2.13.Adapt the proof of Theorem 1.2.7 to prove Theorem 1.2.12.(As in Hex, each entry and exit arrow lies on the boundary between a yellow andblue segment.Unlike in Hex, in shapes with with six or more sides, these foursegments can be distinct. In this case there is both a blue and a yellow crossing.See Figure 1.16.)1.2.5. Other partisan games played on graphs.We now discuss severalother partisan games which are played on graphs.For each of our examples, wecan explicitly describe a winning strategy for the first player.Example1.2.14.The Shannon Switching Gameis a variant of Hex playedby two players, Cut and Short, on a connected graph with two distinguished nodes,AandB. Short, in his turn, reinforces an edge of the graph, making it immune to3“Simply-connected” means that the board has no holes.Formally, it requires that everycontinuous closed curve on the board can be continuously shrunk to a point.
281. COMBINATORIAL GAMESFigure 1.16.A filled-in Six-sided Hex board can have both blue andyellow crossings. In a game when players take turns to move, one of thecrossings will occur first, and that player will be the winner.being cut. Cut, in her turn, deletes an edge that has not been reinforced. Cut winsif she manages to disconnectAfromB. Short wins if he manages to linkAtoBwith a reinforced path.