[Math Lair] Don't Make a Triangle

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"Don't Make a Triangle" is a simple game for two players. You'll need two different-coloured pens, markers, or the like. First, draw six dots in the shape of a hexagon:

[six dots in the shape of a hexagon]

Players take turns drawing a straight line from one dot to another dot in his or her own colour. Players may not draw a line between two points that are already connected by a line. The first player to make a triangle where all three sides are his or her colour loses. For example, in the diagram below, blue has lost, because of the triangle in the bottom left-hand corner:

[blue has lost]

You may want to play the game for a while with a friend. Then, try the following questions:

  1. If the game lasted until every possible line was drawn, how many turns would the game take? Hint: If you can't find the answer right away, ask yourself how many turns it would take if the game were played on a triangle, square, pentagon, etc.
  2. How many possible triangles could be drawn?
  3. Will one player always win, or can the game end in a tie (i.e. is it possible for all possible lines to be drawn without someone winning)?
  4. Prove your answer to the following question. Hint: There is a very beautiful proof.
  5. Use your answer to the previous two questions to answer the following question: Show that, in any group of six people, there will either be three people who are mutually acquainted, or three people who are mutual strangers.
  6. Does one player or another have a winning strategy? Hint: Read the page on combinatorial games.
  7. How would the game be different if there were other rules? For example, what if there were seven dots (arranged in the form of a heptagon) instead of six, or some other number, or if you weren't allowed to draw lines that crossed other lines, or if both players used the same colour and the first player to complete any triangle lost, or if three players played? Find some rule variants that produce an interesting game.

The answers are on the answers page.

Sources used (see bibliography page for titles corresponding to numbers): 32.