Sperner's lemma
= Sperner's lemma
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Triangulate a triangle and label every vertex by $1$, $2$, or $3$, forbidding label $i$ on the side opposite vertex $i$. Then the number of small triangles carrying all three labels is odd, and in particular at least one such triangle exists. Counting edges whose endpoint labels are $1$ and $2$ proves the parity statement: boundary incidences are odd, while a small triangle has odd incidence exactly when it has all three labels.