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Unformatted text preview: Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions MOLS Mutually Orthogonal Latin Squares Serge Ballif ballif@math.psu.edu The Pennsylvania State University February 26, 2008 Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions MOLS 3 Challenges Challenge I Challenge II Challenge III Definitions Latin Squares How Many? MOLS MOLS Upper Bound on # of MOLS Constructing MOLS Solution to Challenge II MOLS Produce More MOLS Solution to Challenge III Concluding Facts Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Challenge I Challenge II Challenge III Definitions MOLS Challenge I Consider the 16 aces, kings, queens, and jacks from a regular 52 card deck of playing card. Can the 16 cards be arranged in a 4 4 array so that no suit and no single kind of card occurs twice in any row or column? (The suits are spades, diamonds, hearts, and clubs.) Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Challenge I Challenge II Challenge III Definitions MOLS Challenge I Consider the 16 aces, kings, queens, and jacks from a regular 52 card deck of playing card. Can the 16 cards be arranged in a 4 4 array so that no suit and no single kind of card occurs twice in any row or column? (The suits are spades, diamonds, hearts, and clubs.) Solution A K Q J K A J Q Q J A K J Q K A Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Challenge I Challenge II Challenge III Definitions MOLS Challenge II In addition to the above requirements, is it possible to color the cards 4 different colors (red, orange, blue, green) such that 1 No two cards have the same color and suit, 2 No two cards have the same color and value, 3 no row or column has the same color twice? Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Challenge I Challenge II Challenge III Definitions MOLS Challenge III If there were n 2 cards consisting n suits and n types of cards, is it possible to arrange an n n array such that each suit and each type of card is present in each row and column? Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions Latin Squares How Many? MOLS MOLS Definition A latin square of order n is an n n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. Example Here are latin squares of orders 3 and 4 respectively. 1 2 3 2 3 1 3 1 2 and A K Q J K A J Q Q J A K J Q K A Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions Latin Squares How Many? MOLS MOLS Example 1 Sudoku puzzles are latin squares. 2 Group multiplication tables are latin squares. Facts 1 The problem of determining if a partially filled square can be completed to form a Latin square is NPcomplete....
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 Fall '08
 BROWNAWELL,WOODRO
 Math, Algebra

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