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High Quality Content by WIKIPEDIA articles! In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points, in root-finding algorithms, and are applied in fair division algorithms.