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.
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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
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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
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http://www.combinatorics.org/Software/index.html
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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
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http://www-history.mcs.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html
http://www.grogono.com/magic/history.php