Orthogonal_Latin_Squares - Mutually Orthogonal Latin...

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Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions MOLS Mutually Orthogonal Latin Squares Serge Ballif [email protected] The Pennsylvania State University February 26, 2008
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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
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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.)
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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
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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?
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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?
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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
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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 NP-complete.
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