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: