The Mathematics of and How to Play to SUDOKU

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The Mathematics of and How to Play to SUDOKU
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• What is SUDOKU?
• A LOGIC BASED placement puzzle
• Played on a 9x9 grid, sub-divided into 9 smaller
  3x3 boxes
• The RULE

To solve the puzzle, each row, column and box
must contain each of the numbers 1 9
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• History and Origins
• First published in USA in 1979 under the name
  Number Place
• Gained popularity in JAPAN as an alternative to
  Crosswords
• Puzzle designed by Howard Garns
• Inspired by the Latin Square invented by
  Leonhard Euler
• 1989 DigitHunt Commodore 64
• First SUDOKU World Championship scheduled for
  March 2006
• The Rubiks Cube of the 21st Century

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So How do you DO Sudoku?

• Elementary Techniques
• Eliminations
• Finding Lone Numbers

• Matching Pairs/Triples
• Twins/Triplets

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• Harder Strategies
• Ariadnes Thread

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But Where is the MATHS?

• Sudoku Squares are special cases of LATIN
  SQUARES with one less
• constraint
• Latin Squares No known formula for computation
  of number of
• order n squares

Lower Bounds
Upper Bounds
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Interesting Mathematical things about Sudoku

• Sudokus are a special case of Latin Squares
  with a regional constraint
• Solving Sudoku puzzles is known to be an NP
  Complete Process
• Sudoku puzzles can be expressed as a graph
  colouring problem with
• the aim to construct a proper 9-colouring,
  given a partial 9-colouring
• The Graph has 81 vertices
• Vertices are joined when i) x x
• 
  ii) y y
• 
  iii) or
• The maximum number of givens not rendering a
  solution unique is 4
• short of a full grid
• The fewest givens rendering a solution unique is
  unsolved

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How many Sudokus are there?
6670903752021072936960 Bertram Felgenhauer
(2005) 5472730538 Essentially
Different Russell Jarvis (2005)
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Computer Solution

• Programmers dont resort to guessing!
• Brute Force
• Use Ariadne find a way through the maze
• Clever Brute Force Donald Knuths Dancing
  Links Algorithm
• Difficulty of solution NOT based on number of
  givens

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Sudoku Variants
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Any Questions?