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Analysis of the Behavior of People Solving Sudoku Puzzles

Analysis of the Behavior of People Solving Sudoku Puzzles. Reijer Grimbergen School of Computer Science, Tokyo University of Technology Akihiro Nakamura Department of Informatics, Yamagata University. Outline. Why Sudoku? Sudoku rules Sudoku solving behavior Recording human behavior

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Analysis of the Behavior of People Solving Sudoku Puzzles

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  1. Analysis of the Behavior of People Solving Sudoku Puzzles Reijer Grimbergen School of Computer Science, Tokyo University of Technology Akihiro Nakamura Department of Informatics, Yamagata University GPW2009

  2. Outline • Why Sudoku? • Sudoku rules • Sudoku solving behavior • Recording human behavior • Rule extraction • Computer simulation • Simulation vs. Human behavior • Rule set and comparison • Conclusions and future work GPW2009

  3. Why Sudoku? Our goal is to implement Marvin Minsky’sSociety of Mindtheory for two-player games like shogi First analyze the behavior of humans solving puzzles Advantage: No interaction with other players Sudoku is a popular puzzle that has not been used for cognitive modeling before GPW2009

  4. Sudoku rules 99 grid with nine 3 3 subgrids Fill each row, column and subgrid with the numbers from 1 to 9 No row, column or subgrid should have a number more than once GPW2009

  5. Sudoku Solving Behavior • Steps in analyzing the behavior of humans • A number of subjects were asked to solve a Sudoku puzzle and explain their decisions • The output of the subjects was used to extract a number of rules • The rules were implemented in a computer program • The output of the program was compared with the original output of the subjects GPW2009

  6. Recording human behavior Five subjects in their early twenties Two Sudoku puzzles For each step P in a Sudoku puzzle, a number NPhas to be penciled on square SP Explain the reason for selecting NP and SP GPW2009

  7. Rule Extraction Too specific, so we used more general rules • The written protocols were analyzed to find rules guiding problem solving behavior • Example rule • If there is only one empty square on a row or column, fill it with the missing number GPW2009

  8. Computer Simulation • Input: A Sudoku position P • Output: A set of squares {SP1,…, SPn} that were selected based on the our rules • Note: No actual numbers were suggested GPW2009

  9. Simulation vs. Human Behavior • Compare the output of the program to the output of the subjects • Output ratio • A measure of how many unnecessary candidates are produced by the rules • Should be as small as possible (1 is ideal) • Cover ratio • Number of positions where a square selected by the human solver was generated by the program • Should be as close to 100% as possible GPW2009

  10. Simulation vs. Human Behavior • Output ratio • Single position CP is the total number of candidate squares, BP is the total number of blank squares • Whole puzzle GPW2009

  11. Simulation vs. Human Behavior • Cover Ratio here Rc is the number of times the square selected by the human solver was part of the generated candidates,S is the number of positions in the Sudoku puzzle GPW2009

  12. Rule Set and Comparison • Likely strategies of human subjects • Give priority to rows and columns with three empty squares or less • Try to fill subgrids, rows and columns with few empty squares • Explore subgrids, rows and columns with only 2 or 3 empty squares GPW2009

  13. Rule Set and Comparison • Rule 1 • Generate all squares with corresponding subgrid, row or column having less than T1 empty squares • Rule 2 • Generate all squares in a subgrid with less than T2 empty squares • Rule 3 • Generate all squares in a row or column with less than T3 empty squares GPW2009

  14. Results for Sudoku Puzzle 1 Output ratio: 40.0% Cover ratio: 92.1% • Individual rule results • Ordered rule results • Rule 1 with T1 = 2 • Rule 2 with T2 = 1 • Rule 3 with T3 = 3 GPW2009

  15. Results for Sudoku Puzzle 2 Output ratio: 20.8% Cover ratio: 98.4% • Individual rule results • Ordered rule results • Rule 2 with T2 = 1 • Rule 1 with T1 = 2 GPW2009

  16. Conclusions and Future Work • Conclusions • The rules suggest that human Sudoku solvers prefer rows and columns over subgrids • General rules that can be applied to all puzzles are hard to find • Future work • Look at more complex rules • Include search to decide between multiple candidates • Add suggestions for numbers to pencil in GPW2009

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