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'The Triangle Game' printed from

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An equilateral triangle ABC is labelled anticlockwise.
Parallel to each side, five equally spaced lines are drawn across triangle ABC, thus dividing it into 36 small equilateral triangles.
This is the board for playing the Triangle Game.

The game is for two players, who take it in turns to label the 25 unlabelled vertices of the figure, in accordance with the following rules:

  • a vertex on AB may be labelled either A or B, but not C
  • a vertex on BC may be labelled either B or C, but not A
  • a vertex on CA may be labelled either C or A, but not B
  • a vertex inside ABC may be labelled A or B or C.

When all the vertices have been labelled, the scores of the two players are calculated.

The score of Player 1 is the number of small triangles which are labelled ABC anticlockwise.

The score of Player 2 scores the number of small triangles which are labelled ABC clockwise.

The winner is the player with the higher score.

Play the Triangle Game with a friend or if you have java enabled, play against the computer below.

Can you work out a winning strategy?
Is it better to play first or second?
Is a draw possible?

These questions appear to be very difficult. However, they turn out to have surprisingly simple answers, which are discussed in the article " Sperner's Lemma" from November, 2000.