Real MATHEMATICS – Game #12

Triangle Invasion

Invasion is a multiplayer game which needs only a pencil and a pen. To start the game one should draw a big triangle on a paper whilst assigning the corners with 1-2-3:


Then the big triangle should be triangulated (without any rules):


Once the construction is over, we can start labeling the triangle corners with the direction of the following rules:

  1. One can label 1-2 side with either of 1 or 2.
  2. One can label 1-3 side with either of 1 or 3.
  3. One can label 2-3 side with either of 2 or 3.
  4. One can label the inside of the original triangle with any of 1, 2 or 3.

Progress of the Game

  • Assign the corners according to the rules.
  • In order to invade a triangle, player must assign the final unattached corner of that specific triangle.
  • Goal is to avoid invading a triangle that has corners 1-2-3. Winner is the player who invaded least number of such triangles.

Sperner’s Triangle

Idea of the invasion game comes from the Sperner’s triangle which is discovered by Emanuel Sperner in the 20th century. Triangulated shape in the invasion game is an example of Sperner’s triangle.

After labeling the corners a Sperner triangle will always give a small triangle that has corners 1-2-3:


Actually, Sperner’s triangle always has odd-numbered 1-2-3 triangles.

This is why invasion game never ends with draw.

Dead End

Sperner’s triangle can be used to construct many games. For example, let’s say in a Sperner’s triangle all the sides of the little triangles that are connected between 1 and 2 are doors. And all the other sides are walls:

If one tries to walk through these doors, that person will end up in two situations:

  1. The person will end up inside a 1-2-3 triangle:
  2. The person will find him/herself outside of the triangle:

    M. Serkan Kalaycıoğlu

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