In the following discussion pre colored hexagons are referred to as bound ary

In the following discussion pre colored hexagons are

This preview shows page 205 - 207 out of 241 pages.

In the following discussion, pre-colored hexagons are referred to as bound- ary . Uncolored hexagons are called interior . Without loss of generality, we may assume that the edges of the board are made up of pre-colored hexagons (see figure). Thus, the interior hexagons are surrounded by hexagons on all sides. Theorem 10.2.6. For a completed standard Hex board with non-empty in- terior and with the boundary divided into two disjoint yellow and two disjoint blue segments, there is always at least one crossing between a pair of seg- ments of like color. Proof. Along every edge separating a blue hexagon and a yellow one, insert an arrow so that the blue hexagon is to the arrow’s left and the yellow one to its right. There will be four paths of such arrows, two directed toward the interior of the board (call these entry arrows) and two directed away from the interior (call these exit arrows), see Fig. 10.12. Fig. 10.12. On an empty board the entry and exit arrows are marked. On a completed board, a blue chain lies on the left side of the directed path. Now, suppose the board has been arbitrarily filled with blue and yellow hexagons. Starting with one of the entry arrows, we will show that it is possible to construct a continuous path by adding arrows tail-to-head always keeping a blue hexagon on the left and a yellow on the right. In the interior of the board, when two hexagons share an edge with an arrow, there is always a third hexagon which meets them at the vertex toward which the arrow is pointing. If that third hexagon is blue, the next arrow will turn to the right. If the third hexagon is yellow, the arrow will turn to the left. See (a,b) of Fig. 10.13. Loops are not possible, as you can see from (c) of Fig. 10.13. A loop cir- cling to the left, for instance, would circle an isolated group of blue hexagons
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198 Combinatorial games b a c Fig. 10.13. In (a) the third hexagon is blue and the next arrow turns to the right; in (b) — next arrow turns to the left; in (c) we see that in order to close the loop an arrow would have to pass between two hexagons of the same color. surrounded by yellow ones. Because we started our path at the boundary, where yellow and blue meet, our path will never contain a loop. Because there are finitely many available edges on the board and our path has no loops, it eventually must exit the board using via of the exit arrows. All the hexagons on the left of such a path are blue, while those on the right are yellow. If the exit arrow touches the same yellow segment of the boundary as the entry arrow, there is a blue crossing (see Fig. 10.12). If it touches the same blue segment, there is a yellow crossing. 10.2.3 Hex and Y That there cannot be more than one crossing in the game of Hex seems obvious until you actually try to prove it carefully. To do this directly, we would need a discrete analog of the Jordan curve theorem, which says that a continuous closed curve in the plane divides the plane into two connected components. The discrete version of the theorem is slightly easier than the continuous one, but it is still quite challenging to prove.
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