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Unformatted text preview: Mutually Orthogonal Latin Squares Serge Ballif, February 25, 2008 The main reference for the talk is Discrete Mathematics Using Latin Squares by Charles F. Laywine and Gary L. Mullen 1 3 Challenges 1.1 Challenge I 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 1.2 Challenge II Challenge II In addition to the above requirements, is it possible to color the cards 4 different colors (red, yellow, 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? 1.3 Challenge III 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? 2 Definitions 2.1 Latin Squares Definition 1. 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 2 . 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 Example 3 . 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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