# Because any colored y board contains one and only one

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Because any colored Y board contains one and only one winning Y, it follows that any colored Hex board contains one and only one crossing.
1.2. PARTISAN GAMES 27 Figure 1.15. An illustration of the four possibilities for how a blue path going through T k and T k +1 reduces to a blue path in the smaller board. Remark 1.2.10 . Why did we introduce Y instead of carrying out the proof directly for Hex? Hex corresponds to a subclass of Y boards, but this subclass is not preserved under the reduction we applied in the proof. 1.2.4. More general boards*. The statement that any colored Hex board contains exactly one crossing is stronger than the statement that every sequence of moves in a Hex game always leads to a crossing, i.e., a terminal position. To see why it’s stronger, consider the following variant of Hex. Example 1.2.11 . Six-sided Hex is similar to ordinary Hex, but the board is hexagonal, rather than square. Each player is assigned three nonadjacent sides and the goal for each player is to create a crossing in his color between two of his assigned sides. Thus, the terminal positions are those that contain one and only one 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. Theorem 1.2.12 . Consider an arbitrarily shaped simply-connected 3 filled-in Hex board with nonempty interior with its boundary partitioned into n blue and n yellow segments, where n 2 . Then there is at least one crossing between some pair of segments of like color. Exercise 1.2.13 . 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 and blue segment. Unlike in Hex, in shapes with with six or more sides, these four segments 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 several other partisan games which are played on graphs. For each of our examples, we can explicitly describe a winning strategy for the first player. Example 1.2.14 . The Shannon Switching Game is a variant of Hex played by two players, Cut and Short, on a connected graph with two distinguished nodes, A and B . Short, in his turn, reinforces an edge of the graph, making it immune to 3 “Simply-connected” means that the board has no holes. Formally, it requires that every continuous closed curve on the board can be continuously shrunk to a point.
28 1. COMBINATORIAL GAMES Figure 1.16. A filled-in Six-sided Hex board can have both blue and yellow crossings. In a game when players take turns to move, one of the crossings 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 wins if she manages to disconnect A from B . Short wins if he manages to link A to B with a reinforced path.