. We prove that then T can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T * can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. q

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We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up-and down-oriented triangles as different. We computationally prove W. T. Tutte's conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.

Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:for some distinct c, c ′ , c ′′ and some distinct s, s ′ , s ′′ . In this paper we show that any such P completes to a latin square which is diagonally cyclic.