1. 2009/11/14GPW20091 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

2. 2009/11/14GPW20092 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

3. Why Sudoku? Our goal is to implement Marvin Minsky’s Society of Mind theory 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 2009/11/14GPW20093

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 2009/11/14GPW20094 182463 4935 53624 4678 893547 296 745619 348 617942

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 2009/11/14GPW20095

6. Recording human behavior Five subjects in their early twenties Two Sudoku puzzles For each step P in a Sudoku puzzle, a number N P has to be penciled on square S P Explain the reason for selecting N P and S P 2009/11/14GPW20096

7. Rule Extraction 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 2009/11/14GPW20097 Too specific, so we used more general rules

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 2009/11/14GPW20098 7316 6128 8249 698 31 254 5874 3681 1469 7316 6128 8249 698 31 254 5874 3681 1469

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 2009/11/14GPW20099

10. Simulation vs. Human Behavior Output ratio Single position C P is the total number of candidate squares, B P is the total number of blank squares Whole puzzle 2009/11/14GPW200910

11. Simulation vs. Human Behavior Cover Ratio here R c 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 2009/11/14GPW200911

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 2009/11/14GPW200912

13. Rule Set and Comparison Rule 1 Generate all squares with corresponding subgrid, row or column having less than T 1 empty squares Rule 2 Generate all squares in a subgrid with less than T 2 empty squares Rule 3 Generate all squares in a row or column with less than T 3 empty squares 2009/11/14GPW200913

14. Results for Sudoku Puzzle 1 Individual rule results Ordered rule results 1.Rule 1 with T 1 = 2 2.Rule 2 with T 2 = 1 3.Rule 3 with T 3 = 3 2009/11/14GPW200914 182463 4935 53624 4678 893547 296 745619 348 617942 RuleThresholdOutput ratioCover ratio 1954.6%96.8% 2454.0%94.7% 3352.8%95.8% Output ratio: 40.0% Cover ratio: 92.1%

15. Results for Sudoku Puzzle 2 Individual rule results Ordered rule results 1.Rule 2 with T 2 = 1 2.Rule 1 with T 1 = 2 2009/11/14GPW200915 RuleThresholdOutput ratioCover ratio 11266.8%97.1% 2846.7%91.8% 3599.9%98.0% Output ratio: 20.8% Cover ratio: 98.4% 7316 6128 8249 698 31 254 5874 3681 1469

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 2009/11/14GPW200916