Problem #WSP-5339
Problem
We want to prove Monsky’s theorem as a corollary of Sperner’s lemma:
it is not possible to dissect a square into an odd number of triangles
of equal area. After scaling we can consider the square with
coordinates: \((0,0),(0,1),(1,0),(1,1)\), which we want to
cut into \(n\) triangles with area
\(\frac{1}{n}\) each for an odd \(n\). Consider the following coloring of all
the points with rational coordinates \((\frac{p}{q},\frac{r}{s})\) inside the
square:
We look at the powers of \(2\) in the
fractions \((\frac{p}{q},\frac{r}{s})\), first of all
the numbers \(p,q\) are coprime, and
thus only one of them is divisible by \(2\), same with \(r,s\). Then the following possibilities
might occur:
Neither \(q\) nor \(s\) is divisible by \(2\). In this case we color the point red.
\(\frac{r}{s}\) is divisible by a larger or equal power of \(2\) than \(\frac{p}{q}\) and \(p\) is not divisible by \(2\). In this case we color the point blue.
\(\frac{p}{q}\) is divisible by a strictly larger power of \(2\) than \(\frac{r}{s}\) and \(r\) is not divisible by \(2\). In this case we color the point green.
Under an assumption (which you do not have to prove) that the area of any rainbow triangle is at least \(\frac{1}{2}\) prove the Monsky’s theorem.