Katana VentraIP

Mutually orthogonal Latin squares

In combinatorial mathematics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.

An outdated term for pair of orthogonal Latin squares is Graeco-Latin square, found in older literature.

Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 978-1-58488-506-1

Handbook of Combinatorial Designs

Dénes, J.; Keedwell, A. D. (1974), Latin squares and their applications, New York-London: Academic Press, p. 547,  0-12-209350-X, MR 0351850

ISBN

(1966), Martin Gardner's New Mathematical Diversions from Scientific American, Fireside, ISBN 0-671-20913-2

Gardner, Martin

; Meynert, Alison; Myrvold, Wendy (2007), "Small Latin Squares, Quasigroups and Loops" (PDF), Journal of Combinatorial Designs, 15 (2): 98–119, doi:10.1002/jcd.20105, S2CID 82321|doi-broken-date=2020-10-03| zbl=1112.05018 | citeseerx=10.1.1.151.3043}}

McKay, Brendan D.

(1988), Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.), New York: Dover

Raghavarao, Damaraju

& Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.

Raghavarao, Damaraju

Stinson, Douglas R. (2004), Combinatorial Designs / Constructions and Analysis, Springer,  978-0-387-95487-5

ISBN

; Street, Deborah J. (1987), Combinatorics of Experimental Design, Oxford U. P. [Clarendon], ISBN 0-19-853256-3

Street, Anne Penfold

van Lint, J.H.; Wilson, R.M. (1993), A Course in Combinatorics, Cambridge University Press,  978-0-521-42260-4

ISBN

AMS featured column archive (Latin Squares in Practice and Theory II)

Leonhard Euler's Puzzle of the 36 Officiers

"36 Officer Problem". MathWorld.

Weisstein, Eric W.

at Convergence

Euler's work on Latin Squares and Euler Squares

at cut-the-knot

Java Tool which assists in constructing Graeco-Latin squares (it does not construct them by itself)

Anything but square: from magic squares to Sudoku

Historical facts and correlation with Magic Squares, and related source code (Javascript in Firefox browser and HTML5 mobile devices)

Javascript Application to solve Graeco-Latin Squares from size 1x1 to 10x10

Grime, James. (video). YouTube. Brady Haran. Archived from the original on 2021-12-12. Retrieved 9 May 2020.

"Euler Squares"