**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:

- One can label 1-2 side with either of 1 or 2.
- One can label 1-3 side with either of 1 or 3.
- One can label 2-3 side with either of 2 or 3.
- 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 20^{th} 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:

- The person will end up inside a 1-2-3 triangle:

- The person will find him/herself outside of the triangle:

M. Serkan Kalaycıoğlu

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## Published by Serkan

Maths teacher, sports lover and a professional music listener.
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