latin squares jerzy wojdyło february 17, 2006. jerzy wojdylo, latin squares2 definition and...

Latin Squares

Jerzy WojdyJerzy Wojdyłłoo

February 17, 2006February 17, 2006

February 17, 2006 Jerzy Wojdylo, Latin Squares 2

Definition and Examples

A A Latin squareLatin square is a square array in which is a square array in which each row and each column consists of the each row and each column consists of the same set of entries without repetition.same set of entries without repetition.

aa aa bb

bb aa

aa bb cc

bb cc aa

cc aa bb

aa bb cc dd

bb cc dd aa

cc dd aa bb

dd aa bb cc


Existence

Do Latin squares exist for every Do Latin squares exist for every nnZZ++?? Yes. Yes.

Consider the addition table (the Cayley Consider the addition table (the Cayley table) of the group table) of the group ZZnn..

Or, more generally, consider the Or, more generally, consider the multiplication table of an multiplication table of an nn-element -element quasigroup.quasigroup.


Latin Squares and Quasigroups

A A quasigroupquasigroup is is a nonempty set is is a nonempty set Q Q with with operation operation · : · : Q Q QQ (multiplication) such (multiplication) such that in the equation that in the equation

rr · · cc = s = s the values of any two variables determine the values of any two variables determine the third one uniquely. the third one uniquely.

It is like a group, but associativity and the It is like a group, but associativity and the unit element are optional.unit element are optional.


Latin Squares and Quasigroups

The uniqueness guarantees no repetitions of The uniqueness guarantees no repetitions of symbols symbols ss in each row in each row rr and each column and each column cc..

·· 00 11 22 33

00 00 22 33 11

11 33 11 00 22

22 22 33 11 00

33 11 00 22 33


Operations on Latin Squares IsotopismIsotopism of a Latin square of a Latin square LL is a is a

permutation of its rows, permutation of its rows, permutation of its columns,permutation of its columns, permutation of its symbols. permutation of its symbols.

(These permutations do not have to be the same.)(These permutations do not have to be the same.) LL is is reduced reduced iff its first row is [1, 2, iff its first row is [1, 2, ……, , nn] ]

and its first column is [1, 2, …, and its first column is [1, 2, …, nn]]TT.. LL is is normal normal iff its first row is [1, 2, …, iff its first row is [1, 2, …, nn].].


Enumeration

How many Latin squares (Latin rectangles) are How many Latin squares (Latin rectangles) are there?there?

If order If order 11 11Brendan D. McKay, Ian M. Wanless, “Brendan D. McKay, Ian M. Wanless, “The The number of Latin squares of order elevennumber of Latin squares of order eleven” ” 2004(?) (show the table on page 5)2004(?) (show the table on page 5)http://en.wikipedia.org/wiki/Latin_square#The_number_of_Latin_squareshttp://en.wikipedia.org/wiki/Latin_square#The_number_of_Latin_squares

Order 12, 13, … open problem.Order 12, 13, … open problem.


Orthogonal Latin Squares

Two Two nnnn Latin squares Latin squares LL=[=[llijij] and ] and M M =[=[mmijij] ]

are are orthogonalorthogonal iff the iff the nn22 pairs ( pairs (llijij, , mmijij) are all ) are all

different.different.

aa bb cc

bb cc aa

cc aa bb

aa bb cc

cc aa bb

bb cc aa


Orthogonal LS - Useful Property

TheoremTheoremTwo Latin squares are orthogonal iff their Two Latin squares are orthogonal iff their normal forms are orthogonal. normal forms are orthogonal. (You can symbols so both LS have the first row (You can symbols so both LS have the first row [1, 2, …, [1, 2, …, nn])])

No two 2No two 22 Latin squares are orthogonal.2 Latin squares are orthogonal.

11 22

22 11


Orthogonal Latin Squares

This 4This 44 Latin square does not have an 4 Latin square does not have an orthogonal mate.orthogonal mate.

11 22 33 44

22 33 44 11

33 44 11 22

44 11 22 33

11 22 33 44


Orthogonal LS - History

1782 Leonhard Euler1782 Leonhard Euler The problem of 36 officers, 6 ranks, 6 The problem of 36 officers, 6 ranks, 6

regiments. regiments. His conclusion: No two 6His conclusion: No two 66 LS are 6 LS are orthogonal.orthogonal.

Additional conjecture: no two Additional conjecture: no two nnnn LS are LS are orthogonal, where orthogonal, where nn ZZ++, , n n 2 (mod 4). 2 (mod 4).

1900 G. Tarry verified the case 1900 G. Tarry verified the case n n = 6.= 6.


Orthogonal LS – History (cd)

1960 R.C. Bose, S.S. Shrikhande, E.T. 1960 R.C. Bose, S.S. Shrikhande, E.T. Parker, Parker, Further Results on the Construction Further Results on the Construction of Mutually Orthogonal Latin Squares and of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecturethe Falsity of Euler's Conjecture, Canadian , Canadian Journal of Mathematics, vol. 12 (1960), pp. Journal of Mathematics, vol. 12 (1960), pp. 189-203. 189-203.

There exists a pair of orthogonal LS for all There exists a pair of orthogonal LS for all nnZZ++, with exception of , with exception of n n = 2 and = 2 and n n = 6.= 6.


Mutually Orthogonal LS (MOLS)

A set of LS that are pairwise orthogonal is A set of LS that are pairwise orthogonal is called a set of called a set of mutually orthogonal Latin mutually orthogonal Latin squaressquares ( (MOLSMOLS).).

TheoremTheoremThe largest number of The largest number of nnnn MOLS is MOLS is nn1.1.


Mutually Orthogonal LS (MOLS)

Proof (by contradiction)Proof (by contradiction)

Suppose we have Suppose we have nn MOLS: MOLS:

… … … … … …

LL11 LLii LLjj LLnn

11 22 …… nn 11 22 …… nn 11 22 …… nn 11 22 …… nn


MOLS TheoremTheorem

If If n n = = pp, prime, then there are , prime, then there are nn1 1 nnnn-MOLS.-MOLS. ProofProof

Construction of Construction of LLkk=[=[aakkijij],], k k =1, 2, …, =1, 2, …, nn1: 1:

aakkijij = = ki ki + + jj (mod (mod nn). ).

CorollaryCorollaryIf If nn==pptt, , p p prime, then there are prime, then there are nn1 1 nnnn-MOLS.-MOLS.

Open problemOpen problemIf there are If there are nn1 1 nnnn-MOLS, then -MOLS, then n n = = pptt, , p p prime.prime.


Latin Rectangle

A A ppqq Latin rectangle with entries in Latin rectangle with entries in {1, 2, …, {1, 2, …, nn} } is a is a ppqq matrix with entries in matrix with entries in {1, 2, …, {1, 2, …, nn} with no repeated entry in a row } with no repeated entry in a row or column.or column.

(3,4,5) Latin rectangle(3,4,5) Latin rectangle 11 33 44 55

33 55 11 22

55 11 33 44


Completion Problems

When can a When can a ppqq Latin rectangle with Latin rectangle with entries in {1, 2, …, entries in {1, 2, …, nn} be completed to a } be completed to a nnnn Latin square? Latin square?

11 33 44 55

33 55 11 22

55 11 33 44

11 22 33 44

44 33 11 22


Completion Problems

The good:The good:

11 22 33 44

44 33 11 22

22 11 44 33

33 44 22 11


Completion Problems

The bad:The bad:

Where to put “2” in the last column?Where to put “2” in the last column?

11 33 44 55

33 55 11 22

55 11 33 44


Completion Theorems

TheoremTheoremLet Let p p < < n.n. Any Any ppnn Latin rectangle with Latin rectangle with entries in {1, 2, …, entries in {1, 2, …, nn} can be completed to } can be completed to a a nnnn Latin square. Latin square.

The proof uses Hall’s marriage theorem or The proof uses Hall’s marriage theorem or transversals to complete the bottom transversals to complete the bottom n n pp rows. The construction fills one row at a rows. The construction fills one row at a time. time.


Completion Theorems

TheoremTheoremLet Let pp,, q q < < n.n. A A ppqq Latin rectangle Latin rectangle RR with with entries in {1, 2, …, entries in {1, 2, …, nn} can be completed to } can be completed to a a nnnn Latin square iff Latin square iff RR((tt), the number of ), the number of occurrences of occurrences of tt in in RR, satisfies, satisfies

RR((tt) ) pp + + qq nn

for eachfor each t t withwith 1 1 t t nn..


Completion Theorems

From last slide: From last slide: RR((tt) ) pp + + qq n.n.

Let Let t t = 5. = 5.

Then Then RR(5) = 1 (5) = 1

and and pp++qqn = n = 4+44+46 = 2.6 = 2.

But 1 But 1 2, so R 2, so R cannot cannot be completed to be completed to a Latin a Latin square.square.

66 11 22 33

55 66 33 11

11 33 66 22

33 22 44 66


Completion Problems

The ugly (?)The ugly (?)a. k. a. sudokua. k. a. sudoku

99 77 11

77 44 22 66 55

11 88 99 44

22 88 55 99

11 22 33 66

44 55 11 22

77 44 11 22

33 22 99 88 55

66 55 77


Completion Problems

The ugly (?)The ugly (?)a. k. a. sudokua. k. a. sudoku

99 66 33 44 66 77 22 11 88

88 77 44 11 33 22 99 66 55

66 11 22 88 99 55 44 77 33

22 33 66 77 88 99 55 99 11

11 99 88 22 55 33 77 44 66

77 44 55 66 11 44 33 88 22

55 88 77 33 44 11 66 22 99

33 22 11 99 77 66 88 55 44

44 66 99 55 22 88 11 33 77


Sudoku

History:History: http://http://en.wikipedia.org/wiki/Sudokuen.wikipedia.org/wiki/Sudoku Robin WilsonRobin Wilson, The Sudoku Epidemic, The Sudoku Epidemic, ,

MAA Focus, January 2006.MAA Focus, January 2006. http://sudoku.com/http://sudoku.com/ Google (2/15/2006) Google (2/15/2006)

about 20,300,000 results for sudoku. about 20,300,000 results for sudoku.


Mathematics of Sudoku

Bertram Felgenhauer and Frazer Jarvis: Bertram Felgenhauer and Frazer Jarvis: There are 6,670,903,752,021,072,936,960 There are 6,670,903,752,021,072,936,960

Sudoku grids.Sudoku grids. Ed Russell and Frazer Jarvis:Ed Russell and Frazer Jarvis:

There are 5,472,730,538 essentially There are 5,472,730,538 essentially different Sudoku grids.different Sudoku grids.

http://www.afjarvis.staff.shef.ac.uk/sudoku/http://www.afjarvis.staff.shef.ac.uk/sudoku/


Uniqueness of Sudoku Completion

Maximal Maximal number number of givens of givens while solution while solution

is not unique: is not unique:

81 81 4 = 77. 4 = 77.

?? ??

?? ??


Uniqueness of Sudoku Completion

Minimal number of givens which force a Minimal number of givens which force a unique solution – open problem.unique solution – open problem.

So far: So far: the lowest number yet found for the the lowest number yet found for the

standard variation without a symmetry standard variation without a symmetry constraint is 17,constraint is 17,

and 18 with the givens in rotationally and 18 with the givens in rotationally symmetric cells. symmetric cells.


Example of Small Sudoku11

44

22

55 44 77

88 33

11 99

33 44 22

55 11

88 66


Example of Small Sudoku66 99 33 77 88 44 55 11 22

44 88 77 55 11 22 99 33 66

11 22 55 99 66 33 88 77 44

99 33 22 66 55 11 44 88 77

55 66 88 22 44 77 33 99 11

77 44 11 33 99 88 66 22 55

33 11 99 44 77 55 22 66 88

88 55 66 11 22 99 77 44 33

22 77 44 88 33 66 11 55 99


More Small Sudoku Grids

Sudoku grids with 17 givens Sudoku grids with 17 givens http://www.csse.uwa.edu.au/~gordon/sudokhttp://www.csse.uwa.edu.au/~gordon/sudokumin.phpumin.php

Need help solving sudoku? Try:Need help solving sudoku? Try:http://www.sudokusolver.co.uk/http://www.sudokusolver.co.uk/

The End

latin squares jerzy wojdyło february 17, 2006. jerzy wojdylo, latin squares2 definition and...

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