Ease and Toil - Analyzing Sudoku

# Ease and Toil - Analyzing Sudoku - Page 1 of 62 MCM 2008...

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Page 1 of 62 MCM 2008 Team #3780 Ease and Toil: Analyzing Sudoku February 18, 2008 Look at any current magazine, newspaper, computer game package or handheld gaming device and you likely find sudoku, the latest puzzle game sweeping the nation. Sudoku is a number-based logic puzzle in which the numbers 1 through 9 are arranged in a 9 × 9 matrix, subject to the constraint that there are no repeated numbers in any row, column, or designated 3 × 3 square. In addition to being entertaining, sudoku promises valuable insight into computer sci- ence and mathematical modeling. In particular, since sudoku solving is an NP-Complete problem, algorithms to generate and solve sudoku puzzles may offer new approaches to a whole class of computational problems . Moreover, we can further explore mathematical modeling techniques through generating puzzles since sudoku construction is essentially an optimization problem. The purpose of this paper is to propose an algorithm that may be used to construct unique sudoku puzzles with four different levels of difficulty. We attempted to minimize the complexity of the algorithm while still maintaining separate difficulty levels and guar- anteeing unique solutions. In order to accomplish our objectives, we developed metrics with which to analyze the difficulty of a given puzzle. By applying our metrics to published control puzzles with spe- cific difficulty levels we were able to develop classification functions for specific difficulty ratings. We then used the functions we developed to ensure that our algorithm gener- ated puzzles with difficulty levels analogous to those currently published. We also sought out to measure and reduce the computational complexity of the generation and metric measurement algorithms. Finally, we worked to analyze and reduce the complexity involved in generating puzzles while maintaining the ability to choose the difficulty of the puzzles generated. To do so, we implemented a profiler and performed statistical hypothesis testing to streamline the algorithm .

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Page 2 of 62 MCM 2008 Team #3780 Contents 1 Introduction 3 1.1 Statement of Problem . . . . . . . . . 3 1.2 Relevance of Sudoku . . . . . . . . . 3 1.3 Goals . . . . . . . . . . . . . . . . . . 3 1.4 Rules of Sudoku . . . . . . . . . . . . 3 1.5 Terminology and Notation . . . . . . 3 1.6 Indexing . . . . . . . . . . . . . . . . 4 1.7 Formal Rules of Sudoku . . . . . . . 5 1.8 Example Puzzles . . . . . . . . . . . 5 2 Background 5 2.1 Common Solving Techniques . . . . 5 2.1.1 Naked Pair . . . . . . . . . . 5 2.1.2 Naked Triplet . . . . . . . . . 5 2.1.3 Hidden Pair . . . . . . . . . . 6 2.1.4 Hidden Triplet . . . . . . . . . 6 2.1.5 Multi-Line . . . . . . . . . . . 6 2.2 Previous Works . . . . . . . . . . . . 7 2.2.1 SudokuExplainer . . . . . . . 7 2.2.2 QQWing . . . . . . . . . . . . 7 2.2.3 GNOME Sudoku . . . . . . . 7 3 Metric Design 10 3.1 Overview . . . . . . . . . . . . . . . . 10 3.2 Assumptions . . . . . . . . . . . . . . 10 3.3 Mathematical Basis for WNEF . . . 10 3.3.1 Complexity . . . . . . . . . . . 10 4 Metric Calibration and Testing 11 4.1 Control Puzzle Sources . . . . . . . . 11 4.2 Testing Method . . . . . . . . . . . . 12 4.2.1 Defining Difficulty Ranges . . 12 4.2.2 Information Collection . . . . 12 4.2.3 Statistical Analysis of Con- trol Puzzles . . . . . . . . . . . 12 4.3 Choice of Weight Function . . . . . . . 12 5 Generator Algorithm 12 5.1 Overview . . . . . . . . . . . . . . . . 12 5.2 Detailed Description . . . . . . . . . 14 5.2.1 Completed Puzzle Generation 14 5.2.2 Cell Removal . . . . . . . . . . 14 5.2.3 Uniqueness Testing . . . . . . 15 5.3 Pseudocode . . . . . . . . . . . . . . . 15 5.3.1 Completed Board Generation 15 5.3.2 Random Masking . . . . . . . 16 5.3.3 Tuned Masking . . . . . . . . 17 5.3.4 Uniqueness Testing . . . . . .
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