Sperner's lemma

Take a triangle ABC, labeled counterclockwise, and subdivide it into lots of smaller triangles in any arbitrary way. Then label all the new vertices as follows: (1) vertices along AB may be labeled either A or B, but not C; (2) vertices along BC may be labeled either B or C, but not A; (3) vertices along CA may be labeled either C or A, but not B; (4) vertices inside triangle ABC may be labeled A or B or C. Now shade in every small triangle that has three different labels. Use two different shadings to distinguish the triangles that have been labeled counterclockwise (i.e. in the same sense as triangle ABC) from the triangles which have been labeled clockwise (i.e. in the sense opposite to that of as triangle ABC). Then there will be exactly one more counterclockwise triangle than clockwise triangles. In particular, the number of shaded triangles will be odd. This is Sperner's lemma, named after its discoverer, the German mathematician Emanuel Sperner (1905-1980). Sperner's lemma is equivalent to the Brouwer fixed-point theorem; a version of it holds in all dimensions.