Abstract
Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to a new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.
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