# 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.