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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Medium level:

Sperner's Lemma

Figure 1
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 non-constructive 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>.

References:
    F.E. Su, "Rental harmony: Sperner's lemma in fair division",
    Amer. Math. Monthly, 1999.

Keywords:    triangles, combinatorics
Subjects:    combinatorics
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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