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A JIGSAW SUDOKU GENERATOR AND SOLVER PROJECT

A JIGSAW SUDOKU GENERATOR AND SOLVER PROJECT

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A JIGSAW SUDOKU GENERATOR AND SOLVER PROJECT

This report discusses the game of Jigsaw Sudoku. The main aim of the project is to create a program which can generate and solve Jigsaw Sudoku. Algorithms are discussed dealing with the creation of Jigsaw Sudoku puzzles. Strategies for solution of Sudoku and Jigsaw Sudoku are defined and a method of linking these strategies is dealt with. A discourse of the mathematics associated with Jigsaw Sudoku is provided, including sections on the Four-Colour Theorem and setting of hints. An analysis of the effects of the number of hints on solvability is carried out. Evaluations of the success of the project are made and extensions are indicated. A conclusion and summary of the project as a whole is provided.

BACKGROUND

This section introduces the reader to the games of Sudoku and Jigsaw Sudoku. The rules of the games are explained and uniqueness is discussed. A range of mathematical facts about Sudoku are imparted and the idea of using strategies to solve a grid is presented.

 

AIMS

Here, a more specific discussion of the project shall be carried out. The main aim of the project is announced, as well as a consideration of the merits of project. Challenges identified within the project are also described.

THE CREATION

This chapter addresses the main challenges associated with the creation of a playable Jigsaw Sudoku puzzle. The problem of creating a grid of randomly shaped pieces is considered, followed by the question of setting the hints.

THE SOLUTION

Here I shall explain a selection of the strategies used in the report.To begin, there shall be an explanation of the difficulty of a strategy and the term possibility. A selection of strategies will then be outlined.

USABILITY

Usability covers the extra bits of implementation which makes the game usable and playable. Firstly, the functionality of the game itself is discussed. Following this is a discourse of the Four-Colour Threorem, which I have used in the game.

 

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Figures............................................................................................................................................ 6

Excursuses......................................................................................................................................7

Précis............................................................................................................................................... 8

§1: Background............................................................................................................................10
1.1: Sudoku.............................................................................................................................. 10
1.2: Jigsaw Sudoku................................................................................................................. 11
1.3: Mathematics of Sudoku..................................................................................................12
1.4: The Strategies of Sudoku............................................................................................... 12

§2: Aims and Challenges............................................................................................................13
2.1: The Project........................................................................................................................ 13
2.2: Why?................................................................................................................................. 13
2.3: The Challenges................................................................................................................ 14
2.4: Extras.................................................................................................................................14
2.5: Thoughts...........................................................................................................................14

§3: The Creation.......................................................................................................................... 16
3.1: Shuffling........................................................................................................................... 16
3.2: Setting the Hints.............................................................................................................. 19
3.3: Counting Solutions......................................................................................................... 20

§4: The Solution........................................................................................................................... 22
4.1: Levels of Difficulty..........................................................................................................22
4.2: Narrowing the Possibilities........................................................................................... 22
4.3: In the Game...................................................................................................................... 23
4.4: Novice Sudoku Strategies.............................................................................................. 23
4.4.1: Only-Possible........................................................................................................... 24
4.4.2: Singles....................................................................................................................... 24
4.4.3: Conjugate Pairs........................................................................................................ 24
4.5: Selected Harder Strategies............................................................................................. 25
4.5.1: Pointing Couplets.................................................................................................... 25
4.5.2: X-Wings.....................................................................................................................26
4.6: Jigsaw Sudoku Specific Strategies................................................................................ 27
4.6.1: Little Law of Leftovers............................................................................................27
4.6.2: Big Law of Leftovers............................................................................................... 28
5: Usability................................................................................................................................. 30
5.1: The Game......................................................................................................................... 30
5.1.1: Saving and Loading............................................................................................... 30
5.1.2: Clues.......................................................................................................................... 30
5.1.3: User-Inputted Grids................................................................................................ 31
5.2: Four-Colouring................................................................................................................ 31

§6: Analysis.................................................................................................................................. 33
6.1: Hints vs. Solvability........................................................................................................ 33
6.2: The 3x3 Experiment........................................................................................................ 34
6.3: The 4x4 Experiment........................................................................................................ 35
6.4: N - 1 Hints........................................................................................................................36
6.5: N - 2 Hints........................................................................................................................38
6.6: Permutations of Non-Unique Grids............................................................................. 38

§7: Evaluation and Extensions.................................................................................................. 40
7.1: Generating Large Grids..................................................................................................40
7.2: Further Strategies............................................................................................................ 40
7.3: Choice of Difficulty......................................................................................................... 41
7.4: Superfluity........................................................................................................................41

§8: The Conclusion......................................................................................................................43
8.1: Achievements.................................................................................................................. 43
8.2: Issues and Revisions....................................................................................................... 44
8.3: Summary.......................................................................................................................... 44

Reading List................................................................................................................................. 45

Appendix A: Results of Experiments.......................................................................................47

FIGURES

Figure 1.1: A Sudoku Puzzle..................................................................................................... 11

Figure 1.2: A Jigsaw Sudoku Puzzle........................................................................................ 11

Figure 3.1: Pieces Sharing a Single Edge................................................................................. 17

Figure 3.2: Shuffling a 4x4 Grid................................................................................................ 18

Figure 4.1: The possibilities....................................................................................................... 23

Figure 4.2: A Sketched X-Wing................................................................................................. 26

Figure 5.1: A Comparison of a Graduated vs. a Four-Coloured Grid.................................31

Figure 6.1: Number of Hints Placed vs. Unique Solutions Generated (3x3)...................... 35

Figure 6.2: Number of Hints Placed vs. Unique Solutions Generated (4x4)...................... 36

Figure 6.3: Permutations of a and b in a Grid of Size 3......................................................... 39

Figure 7.1: Superfluous Hints................................................................................................... 42

EXCURSUSES

Excursus 3.1: The Shuffling Algorithm....................................................................................17

Excursus 3.2: Checking Completeness of a Given Piece....................................................... 18

Excursus 3.3: Arbitrary Numbers............................................................................................. 19

Excursus 3.4: To Place the Hints............................................................................................... 20

Excursus 3.4: Counting Solutions – An Example................................................................... 21

Excursus 4.1: Conjugate Pairs – An Example......................................................................... 24

Excursus 4.2: Pointing Couplets............................................................................................... 25

Excursus 4.3: X-Wings – An Example...................................................................................... 27

Excursus 4.4: The Little Law of Leftovers............................................................................... 28

Excursus 4.5: The Big Law of Leftovers................................................................................... 29

Excursus 5.1: A Proof of the Four-Colour Theorem.............................................................. 32

Excursus 6.3: Proof of N – 1 Hints Leading to a Unique Solution....................................... 37

Excursus 6.4: Proof that N – 2 Hints will not Lead to a Unique Solution...........................38

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