Wm.C.Corwin, Ph.D., P.E. billc issi1.com www.ConcurrentInverse.com www.byeless.com 2006 June 10 This letter is maintained at http://www.issi1.com/corwin/byeless/priorart.txt Title: Generation of Symmetric Latin Squares Many sports leagues try to have a number of teams equal to a power of two since the schedules are easy. It is generally believed by many league schedulers that it is impossible for a byeless schedule for other than a power of two; they instead use byes instead of having a full schedule. However, in some sports events the byes are relished for a rest. It seems that not everyone is aware of the application of Latin squares to byeless league schedules, or at least there is not much on the web, and there is a lot of more of more complicated things. However, it is possible to have byeless schedules with twice an odd number of teams as I have demonstrated at www.byeless.com . There may be four systematic methods of generating the appropriate Latin square that I know of, in addition to the method of err, cut, and retry (which is easy for 6(3) and which I have done with great difficulty for as great as for order 18). One(1) does not require a great leap as with the cubic equation. It would be interesting to determine if Latin squares generated with different methods were isomorphic(2), different in any way that could be categorized, or have characteristic invariants. The appropriate Latin square is subject to the additional constraint of symmetry, S(S(T,G),G) = T , where the schedule is S, T is the number of the team, and G is the number of the game. i.e. The team that you are playing is playing you. Thus byeless robin schedules for tournaments are possible for all even number of teams, not just powers of two. i.e. Bowling alleys with 6, 10, 14, 18, 20, 22, 26, ... lanes can have schedules where each team plays each week and all lanes are in use. Parhaps the fact that a power of two is not necessary should be pointed out more clearly. Is this already published anywhere or should it be published in more detail or referenced somewhere in addition to http://www.byeless.com ? I would think that the number of solutions of a particular order that are not isomorphic or distinguishable in whatever way would have applicability in the statistical factors in atomic and nuclear transitions or some other thing in nature. It would be worthwhile finding something in nature to verify and inchorage the theory. It also may be worthwhile to see if a square can be both symmetric and orthoganal. ------------------------------------------------------------------------- ------------------------------------------------------------------------- cc: somewhat in order of relevance and some of less relevance which I may have found before more relevant ones. alexb cut-the-knot.com (1)http://www.cut-the-knot.org/arithmetic/latin2.shtml method: make symmetric matrix, then remove diagonal where a team play itself (2)http://www.cut-the-knot.org/arithmetic/latin.shtml isomorphic alexb cut-the-knot.com (3)http://www.maa.org/editorial/knot/quasi.html #tournament Prof. William A. McWorter, Jr. Ohio State University 6 teams http://www.issi1.com/corwin/byeless/byeless.html billc issi1.com Symmetric Latin Squares to Order 26 for Byeless Robin Schedules [schmidt symcom.math.uiuc.edu] http://www.math.uiuic.edu/~schmidt/cgi-bin/pairing.html Karl Schmidt, Thomas Boutell, Scott Coon, Darrin Doud 19 Aug 96 rusin math.niu.edu exeter rjc maths.ex.ac.uk http://www.math.niu.edu/~rusin/known-math/98/graeco_latin Newsgroups: sci.math hoekstra nlr.nl hoekstra.nospam nlr.nl van Lint and Wilson A Course in Combinatorics, Cambridge UP, 1992 [kemoauc de.ibm.com] iain stt.win-uk.net Constructions and Combinational Problems in Design of Experiments Damaraju Raghavarao, Wiley Series in Probability and Mathematical Statistics John Wiley and Sons Inc. (1971) Bethe, Jungnickel, Lenz Design Theorie B.I. greig sfu.ca [cpu01 my-dejanews.com] rusin math.niu.edu http://lib.stat.cmu.edu/designs/ http://www.research.att.com/~njas/oadir/index.html jgamble ripco.com Martin Gardner Euler's Spoilers www.sciam ! [galvin math.ukans.edu] Latin Squares and Their Applications, J.Denes and A.D. Keedwell http://www.math.niu.edu/~rusin/known-math/index/05-XX.html #sites with this focus http://www.combinatorics.org http://www.combinatorics.net http://www.cs.rit.edu/~spr/ http://www.nada.kth.se/~viggo/wwwcompendium/wwwcompendium.html http://www.cs.cf.ac.uk:8088/User/S.U.Thiel/ra/section3_7.html http://front.math.ucdavis.edu/math.CO http://www.ms.uky.edu/~pagano/Matridx.htm http://www.ams.org/mathweb/mi-mathbyclass.html#MR05 http://www.sub.uni-goettingen.de/ssgfi/math/subject/math_msc05_on_en.html http://archives.math.utk.edu/topics/discreteMath.html Tony Phillips [(webmaster ams.org delayed)] http://www.math.sunysb.edu/~tony/whatsnew/column/latin-squaresI-0701/latinI4.html http://buzzard.ups.edu/squares.html http://www.cut-the-knot.com/arithmetic/latin.html http://www.tfrec.wsu.edu/ANOVA/Latin.html http://www.uky.edu/Ag/Agronomy/Extension/ssnv/ssvl212.pdf ritter ciphersbyritter.com http://www.ciphersbyritter.com/RES/LATSQ.HTM literature survey ! daviddarling daviddarling.info http://www.daviddarling.info/encyclopedia/L/Latin_square.html Heisenberg /C/Cayley.html amulets ca.1200 R.A.Fisher design of statistical experiments /contact/ http://mathworld.wolfram.com/LatinSquare.html Bammel,S.E. and Rothstein,J. "The Number of 9x9 Latin Squares." Disc. Math. 11,93-95,1975. Cayley,A. "On Latin Squares." Oxford Cambridge Dublin Messenger Math. 19,135-137,1890. Colbourn,C.J. and Dinitz,J.H.(Eds.), CRC Handbook of Combinotorial Designs, BocaRatonFL:CRCPress,1996. ... maya math.ucdavis.edu http://www.mathucdavis.edu Discovered by Euler in 1783 Latin Squares: New Development in the Theory and Applications. Denes,J. and A.D. Keedwell Amsterdam, Elsevier 1991 webmaster ams.org Tony Phillips 2001 July/August http://www.ams.org/featurecolumn/archive/latinI1.html Euler conjectured that there was no graeco-latin square of size 2 plus a multiple of 4; he was right for 2 and 6 but wrong otherwise. Box, Hunter, Hunter Fisher Fisher, Yates Pearson doughertys1 uofs.edu http://academic.uofs.edu/faculty/DOUGHERTYS1/square.htm 2 mod 4 2 6 36 officer problem Bose, Shrikhande and Parker in 1960 /euler.htm 1901 1988 D. Stinson /euler.tex Designs, Codes and Cryptography,4,123-128,1994. /latin.htm I.M.Wanless A generalization of transversals for Latin squares Elc.J.Combin,9,2002,R12 www.combinatorics.org/People/index.html Jeanette Janssen /new.htm http://en.wikipedia.org/wiki/Design_of_experiments joc st-andrews.ac.uk efr st-andrews.ac.uk http://www-history.mcs.st-andrews.as.uk/Biographies/Knuth.html http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Fisher.html http://www.umetrics.com/default.asp/pagename/methods_DOE_intro/c/1 http://www.itl.nist.gov/div898/handbook/pri/section1/pri1.htm http://en.wikipedia.org/widki/Latin_square http://www.cut-the-knot.org/Curriculum/Algebra/Latin.shtml http://www.cut-the-knot.org/Curriculum/Combinatorics/InfiniteLatinSquare.shtml http://www.muljadi.org/MagicSudoku.htm Scientific American 2006 June p81 The Science behind SUDOKU Jean-Paul Delahaye Leonhard Euler 1707-1783 http://www.britannica.com/eb/article-21897 orthogonal -------------------------------------------------------------------------- less so: william.cherowitzo cudenver.edu http://www-math.cudenver.edu/~wcherowi/courses/m6406/csln3.html orthogonal mate ! cyclical http://www-math.cudenver.edu/~wcherowi/mathlinks.html lam vax2.concordia.ca http://www.cecm.sfu.ca/organics.papers/lam/paper/html/node2-an.shtml history ci430fa01 pingry.ed.uiuc.edu Shannon Dolan http://www.mste.uiuc.edu/courses/ci430fa01/students/sbwalke1/THE%20ESSENCE%20OF%20MATHEMATICS.doc bmekdeci engmail.uwaterloo.ca http://www.geocities.com/mekdeci/magicsquares.htm parallel computing, graph theory, cryptography Edythe Parker Woodruff, Ph.D. eaddy ra.msstate.edu http://www2.msstate.edu/~eaddy/html/etparker.htm http://www.cut-the-knot.com/htdocs/dcforum/DCForumID4/683.shtml Soduku qualifies as mathematics -------------------------------------------------------------------------- http://www.combinatorics.org/Software/index.html ------------------------------------------------------------------------- 6 plays (1+n) mod 5 (5+n) mod 5 plays (2+n) mod 5 (4+n) mod 5 plays (3+n) mod 5 e.g. 1 4 5 6 3 2 2 5 3 4 6 1 3 6 2 5 1 4 4 1 6 2 5 3 5 2 1 3 4 6 6 3 4 1 2 5 ------------------------------------------------------------------------- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html http://www.grogono.com/magic/history.php