In the proof of Monsky's Theorem, which states that it is not possible to dissect a square into an odd number of triangles of equal area, it is common to use a triangulation of a unit square. The demonstration uses 2-adic valuation and Sperner's lemma.
The original article is https://www.jstor.org/stable/2317329?origin=crossref
My question is: Does the use of Sperner's lemma restrict the demonstration? Could it be divided into triangles of the same area, not overlapping, but which do not satisfy the conditions of Sperner's triangulation?
Thanks a lot!
The demonstration in "Proofs from The Book" (Aigner, Ziegler), which follows Monsky's proof (with a simplification due to Lenstra), does not directly use Sperner's lemma, but "an idea from Emmanuel Sperner that will appear also later when talking about Sperner's lemma". The drawing that illustrates the proof is not a valid Sperner triangulation: some triangle vertices are in the midst of another triangle's edge.
It works because it has been previously proven, using p-adic valuations, that any line contains points with at most $2$ different colors. This is a stronger condition than in Sperner's lemma, where this condition is only met on the $3$ edges of the initial triangle: here it is also valid for internal edges.
So in conclusion, the proof is valid for any partitioning into triangles, not only for correct triangulations where any vertex and any edge are vertices and edges for all triangles which they intersect.