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Take an arbitrary vertex. Since this vertex connects to five others, on the completed board at least three of the lines coming from it are the same colour. Let these be red. Now consider the other endpoints of the three lines, and the three lines connecting these to each other. If one of these is
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Why is there a
Spanish sentence in the English Knowledge? Babel Fish Translator says the sentence says "In May of the 2007 Peruvian mathematician David Palomino it discovered a winning strategy for the second player. The strategy this cradle in the demonstration of the theorem of the four colors."
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page, I think you needn't worry about NOR. I agree that your proof should be in the article. A reference to Ramsey numbers is a simpler proof, for someone who already knows about Ramsey numbers. But I think we should suppose that many of the readers of this page will not find that illuminating,
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Why? Because completing a cycle of four is an excellent move for the like color. There are already two forbidden edges formed by a path of three, and completing the cycle does not create a new forbidden edge with either leg. (If the move is safe for the opponent, taking that option from them is
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All else being equal, the best move forms the fewest number of new forbidden edges for yourself. If there are several which are also available to your opponent, select an edge that your opponent would most want because it creates the fewest number of forbidden edges for him- or herself.
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Why? Because it doesn't create any forbidden edges for either color, hence it is a safe move for either color on its own merits, and therefore also an offensive move in taking it from the opponent.
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Here is an algorithm I'm using for the second player that defeats a computer oppenent fairly consistently. It's probably not fool-proof, but I wouldn't imagine anything more than a few tweeks.
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the article makes it unclear as to whether players take turns coloring lines all the way from one dot to another dot or from one intersection of lines to the next intersection of lines.
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Why go into all this business of Ramsey numbers to prove that the game can't be a draw? I think it would be nice to have here a simple proof. How about this?
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A considerable exception to this rule are the two opening moves, which should be targeted at the most populated verticies of the opposite color.
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Since I am one of the authors of the applet, I provided a hopefully more permanent link which I will try to maintain for the years to come.
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red, it forms a red triangle with two of the original three lines. So colour them blue. However, these lines then form a blue triangle.
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The Java applet linked by the first reference is no longer available at that link. Can someone update or remove the link, as applicable?
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Just tried the strategy as a second player against the applet playing as the first player, and immediately lost the first game.
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Otherwise, if there is a path of three of either color, close it unless your opponent cannot.
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Among the edges that are not forbidden to either player, select from these very good moves:
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Avoid taking an edge that is forbidden to your opponent unless there are no other options.
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First, if there is a path of two of mixed color, close it unless your opponent cannot.
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Consider edges that would complete a triangle of your color to be absolutely forbidden.
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Not sure why, but perhaps ignoring the other player essentially leads to a set-up.
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Of course the point would be to choose rules that scale.
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I added something similar to above proof to the text.
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This seems to have previously already been deleted.
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