Sperner's Lemma

Dissect a triangle into smaller triangles, such that all have full edge contact with their neighbors. Label the corners 1, 2, and 3. Label all vertices with 1, 2, or 3, with a restriction that the vertices of the side opposite a number lack that number. Thus, the side opposite 1 contains no vertices labelled 1.

Then Sperner's lemma states that any such labelling must contain an odd number of triangles with vertices labelled 1, 2, 3.

Sperner's Lemma is equivalent to the Brouwer fixed point theorem.