From the introduction: A Graeco-Latin square is a pair of Latin squares in which each pairing of corresponding elements is distinct. Thus in an order $k$ Graeco-Latin square all $k^2$ possible pairings occur. Leonhard Euler was the first to study Graeco-Latin squares seriously. The name comes from his practice of using $a,b,c, \dots$ for one set of elements and $?, ?, ?, \dots$ for the other. Finding these so-called orthogonal pairs makes a challenging puzzle when $k=4$, (the first composite number). I had the chance over several years to observe how students tackle the problem. It has so many interesting aspects that I want to spend the next 7 sections by going back to the Latin squares themselves (Sections 1 to 4) before assembling the orthogonal pairs (Sections 5 to 7). The third part of the article series contains the sections 5 to 7 on “Detecting orthogonal Latin squares from their “ patterns’"; “Creating orthogonal Latin squares by arranging ready-made cell pairs"; and “Algorithms to generate Graeco-Latin squares". For Part I see [the author, ibid., No. 51, 10‒13 (2013; ME 2014a.00730)] and for Part II [the author, ibid., No. 52, 14‒16 (2013; ME 2014b.00740)].