Monsky's Theorem states:
One cannot dissect a square into an uneven number of triangles with equal area.
I was wondering how close one could get to a equidissection, i.e.
For $n$ an uneven number,
let $T_n$ be the set of triangulations of the unit square into $n$ triangles.
For all $t \in T_n$ let $s(t)$ be the area of the smallest, $b(t)$ that of the biggest triangle in $t$.
Question: What is $g(n) :=min_{t \in T_n} (b(t)-s(t))$
One can easily split a square into $\frac{n-1}{2}$ triangles of area $\frac{1}{n-1}$ and $\frac{n+1}{2}$ triangles of area $\frac{1}{n+1}$, which implies $g(n) \le \frac{1}{n-1} - \frac{1}{n+1}$
This gives the correct value for n=3. Can you think of any better bounds for higher arguments? Is there something known?
