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Analysis of the Behavior of People Solving Sudoku Puzzles

Reijer GrimbergenSchool of Computer Science, Tokyo University of Technology

Akihiro NakamuraDepartment of Informatics, Yamagata University

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OutlineWhy Sudoku?Sudoku rulesSudoku solving behavior

Recording human behaviorRule extractionComputer simulationSimulation vs. Human behavior

Rule set and comparisonConclusions and future work

Why Sudoku?Our goal is to implement Marvin Minsky’s Society of Mind theory for two-player games like shogiFirst analyze the behavior of humans solving puzzlesAdvantage: No interaction with other playersSudoku is a popular puzzle that has not been used for cognitive modeling before

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Sudoku rules

99 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

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1 8 2 4 6 3

4 9 3 5

5 3 6 2 4

4 6 7 8

8 9 3 5 4 7

2 9 6

7 4 5 6 1 9

3 4 8

6 1 7 9 4 2

Sudoku Solving BehaviorSteps in analyzing the behavior of humans

A number of subjects were asked to solve a Sudoku puzzle and explain their decisionsThe output of the subjects was used to extract a number of rulesThe rules were implemented in a computer programThe output of the program was compared with the original output of the subjects

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Recording human behavior

Five subjects in their early twenties

Two Sudoku puzzles

For each step P in a Sudoku puzzle, a number NP has to be penciled on square SP

Explain the reason for selecting NP and SP

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Rule Extraction

The written protocols were analyzed to find rules guiding problem solving behavior

Example ruleIf there is only one empty square on a row or column, fill it with the missing number

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Too specific, so we used more general rules

Computer SimulationInput: A Sudoku position POutput: A set of squares {SP1,…, SPn} that were selected based on the our rules

Note: No actual numbers were suggested

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7 3 1 6

6 1 2 8

8 2 4 9

6 9 8

3 1

2 5 4

5 8 7 4

3 6 8 1

1 4 6 9

7 3 1 6

6 1 2 8

8 2 4 9

6 9 8

3 1

2 5 4

5 8 7 4

3 6 8 1

1 4 6 9

Simulation vs. Human BehaviorCompare the output of the program to the output of the subjectsOutput ratio

A measure of how many unnecessary candidates are produced by the rulesShould be as small as possible (1 is ideal)

Cover ratioNumber of positions where a square selected by the human solver was generated by the programShould be as close to 100% as possible

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Simulation vs. Human Behavior

Output ratioSingle position

CP is the total number of candidate squares, BP is the total number of blank squares

Whole puzzle

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100P

PP B

CO

100

1

1

n

PP

n

PP

B

CO

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

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100S

RC c

Rule Set and Comparison

Likely strategies of human subjectsGive 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

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Rule Set and Comparison

Rule 1Generate all squares with corresponding subgrid, row or column having less than T1 empty squares

Rule 2Generate all squares in a subgrid with less than T2 empty squares

Rule 3Generate all squares in a row or column with less than T3 empty squares

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Results for Sudoku Puzzle 1

Individual rule results

Ordered rule results1. Rule 1 with T1 = 2

2. Rule 2 with T2 = 1

3. Rule 3 with T3 = 3

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1 8 2 4 6 3

4 9 3 5

5 3 6 2 4

4 6 7 8

8 9 3 5 4 7

2 9 6

7 4 5 6 1 9

3 4 8

6 1 7 9 4 2

Rule Threshold Output ratio Cover ratio

1 9 54.6% 96.8%

2 4 54.0% 94.7%

3 3 52.8% 95.8%

Output ratio: 40.0%

Cover ratio: 92.1%

Results for Sudoku Puzzle 2

Individual rule results

Ordered rule results1. Rule 2 with T2 = 1

2. Rule 1 with T1 = 2

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Rule Threshold Output ratio Cover ratio

1 12 66.8% 97.1%

2 8 46.7% 91.8%

3 5 99.9% 98.0%

Output ratio: 20.8%

Cover ratio: 98.4%

7 3 1 6

6 1 2 8

8 2 4 9

6 9 8

3 1

2 5 4

5 8 7 4

3 6 8 1

1 4 6 9

Conclusions and Future WorkConclusions

The rules suggest that human Sudoku solvers prefer rows and columns over subgridsGeneral rules that can be applied to all puzzles are hard to find

Future workLook at more complex rulesInclude search to decide between multiple candidatesAdd suggestions for numbers to pencil in

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