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Two earlier related posts:

  1. Cutting the unit square into pieces with rational length sides
  2. On a possible variant of Monsky's theorem

Question: for odd n, how does one cut the unit square into n equal area triangles of maximum possible area - thereby minimizing the portion of the square left over? Does this question have to be answered for each n separately?

Define f(n) as the fraction of the area of the unit square that gets left out by the optimal way of cutting n equal area triangles from it. Then, how does f(n) behave as n increases?

Further question: Does requiring that the areas of the triangles cut be rational have any interesting implications?

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    $\begingroup$ The rational requirement certainly doesn't affect $f(n)$, because for a given triangle one can choose a slightly smaller triangle with rational coordinates whose area is arbitrarily close to the starting area, so a solution which leaves some area $A$ behind can easily be converted into a rational-coordinate solution which leaves $A+\epsilon$ behind for any $\epsilon>0$. $\endgroup$
    – RavenclawPrefect
    Commented May 4, 2023 at 20:55
  • $\begingroup$ Thanks for pointing this out! The question may be nontrivially affected if say, we insist all sides of all triangles to be rational. But that would be another question. $\endgroup$
    – Nandakumar R
    Commented May 5, 2023 at 3:41

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