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Problem 6 (Six-Point Problem)

We start with six points in the plane such that no three of them are on the same line. Two players play a game where they take turns connecting any two points with a line. Once two points are connected, they cannot be joined a second time. The players continue to play until one player is forced to connect points resulting in the formation of a triangle. This player is the loser. Show that the person who makes the opening move will always win if he/she plays skillfully.

Solution
Let's suppose you are the player who makes the opening move. We show you how you can always win. The first thing you do is redraw the game as a hexagon as shown in the diagram below. (You can always redraw the points as a hexagon no matter how they were originally drawn -- in fact if you secretly do this so the other person doesn't see the hexagon, you can disguise your strategy.)

You now draw an imaginary line L that separates any three points from the others. In the drawing above, we have drawn the line L so it separates A, B, and C from D,E, and F. From this given imaginary line L, you begin by drawing the mirror line . For example, if your opponent draws the line AF, you draw CD; if your opponent draws AB, you draw DE; if your opponent draws AD, you draw CF, and so on. You can convince yourself after playing a few games that you will never be the one to draw the first triangle. In fact, you could enumerate all the possible games in a tree-type diagram and show that in every case you will win. You can always extend this game for any even number of points in the game, like 8, 10, 12, .... .

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Last modified on Tuesday, January 12, 1999