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Given $m \times n$ rectangle with area $A$, and $m,n \in \mathbb{N}$. Let $S_k(m,n)$ be the number of way to dissect this rectangle into $k$ non-overlapping triangles whose area is $\frac{A}{k}$.

It is known that when $k$ is odd $S_k(n,n)=0$ (For example see Paul Monsky, "On Dividing A Square Into Triangles", American Mathematical Monthly, Vol 77 no 2, 1970)

Is it true that $S_k(m,n) < \infty$?
Is there any known bound for $S_k(m,n)$ (even when $m=n$ and $k$ even)?

Edit: There are several possible interpretations of the problem, one may assume that the vertices of the triangles are lattice points. Or perhaps (harder?), when we are given an arbitrary rectangle with a fixed dimension (not necessarily has an integer dimension).

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    $\begingroup$ Just to check, are there any conditions on the triangles? Do the vertices need to be lattice points? Then again, judging from your $S_k(m,n) < \infty$, I guess that the answer is no? $\endgroup$
    – Calvin Lin
    Jan 3, 2013 at 7:49
  • $\begingroup$ it's entirely an open question, so it can be lattice point or not. $\endgroup$ Jan 3, 2013 at 8:14
  • $\begingroup$ @AjatAdriansyah, open question for you means unknown answer or no more conditions needed on the triangles? $\endgroup$
    – Sigur
    Jan 3, 2013 at 10:40
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    $\begingroup$ It seems when the triangle’s vertices are nor necessarily the lattice points then $S_k(n,m)$ does not depends on $n$ and $m$, because an affine transformation can transform the dissected rectangle into a square, preserving all triangles and their relative areas. $\endgroup$ May 1, 2013 at 0:22
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    $\begingroup$ For the case where $m=n$, do you happen to know the values of $S_k(n,n)$ for small even $k$? $\endgroup$ May 6, 2015 at 13:28

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In the case where the vertices of the triangles are not lattice points, there exists some $k$ for which $S_k(m,n)$ is not finite. In particular, $S_8(1,1)$ is infinite.

This result comes from the related problem of subdividing triangles into triangles. The number of ways to divide a triangle into 3 equal parts is finite; the number of ways to divide it into 4 equal parts is infinite. See here for reference: http://www.mathpuzzle.com/triangle.html

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