id: 06350833 dt: j an: 2014f.00686 au: Stephenson, Paul ti: From Latin to Graeco-Latin. III. so: SYMmetryplus, No. 53, 17-19 (2014). py: 2014 pu: Mathematical Association (MA), Leicester la: EN cc: H40 G90 F60 N70 ut: orthogonal Latin squares; algorithms; combinatorics; symmetry operations; symmetry group of the rectangle; abstract groups; Klein four-group; isomorphic objects; algebraic structures; isomorphisms; automorphisms; modular arithmetic; Galois fields; isomorphisms ci: ME 2014a.00730; ME 2014b.00740 li: ab: 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)]. rv: