Figure 1

Divide a triangle T into lots of baby triangles,
so that baby triangles only meet at a common edge or a
common vertex. Label each main vertex of the whole
triangle by 1, 2, or 3; then label vertices on the
(12) side by either 1 or 2, on the (23) side by either
2 or 3, and the (13) side by either 1 or 3. Label the
points in the interior by any of 1, 2, or 3. For
instance, see Figure 1.
Fun Fact: any such labelling must contain an
baby (123) triangle!
(In fact, there must be an odd number!)
Actually, a version of Sperner's Lemma
holds in all dimensions. Can you figure out how it
generalizes?
Sperner's Lemma is equivalent to the
Brouwer fixed point theorem.
Presentation Suggestions:
Have everyone make their own labelled triangle
and see how many (123) triangles they have in their
picture.
The Math Behind the Fact:
There are many proofs of this fact. Some short
nonconstructive proofs rely on parity arguments.
Constructive proofs are the key to many fixed point
algorithms as well as fair division procedures.
See the reference.
How to Cite this Page:
Su, Francis E., et al. "Sperner's Lemma."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
