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### Heuristic Reasoning on Graph and Game Complexity of Sudoku

The Sudoku puzzle has achieved worldwide popularity recently, and attracted great attention of the computational intelligence community. Sudoku is always considered as Satisfiability Problem or Constraint Satisfaction Problem. In this paper, we propose to focus on the essential graph structure underlying the Sudoku puzzle. First, we formalize Sudoku as a graph. Then a solving algorithm based on heuristic reasoning on the graph is proposed. The related r-Reduction theorem, inference theorem and their properties are proved, providing the formal basis for developments of Sudoku solving systems. In order to evaluate the difficulty levels of puzzles, a quantitative measurement of the complexity level of Sudoku puzzles based on the graph structure and information theory is proposed. Experimental results show that all the puzzles can be solved fast using the proposed heuristic reasoning, and that the proposed game complexity metrics can discriminate difficulty levels of puzzles perfectly.

### Difficulty Rating of Sudoku Puzzles: An Overview and Evaluation

How can we predict the difficulty of a Sudoku puzzle? We give an overview of difficulty rating metrics and evaluate them on extensive dataset on human problem solving (more then 1700 Sudoku puzzles, hundreds of solvers). The best results are obtained using a computational model of human solving activity. Using the model we show that there are two sources of the problem difficulty: complexity of individual steps (logic operations) and structure of dependency among steps. We also describe metrics based on analysis of solutions under relaxed constraints -- a novel approach inspired by phase transition phenomenon in the graph coloring problem. In our discussion we focus not just on the performance of individual metrics on the Sudoku puzzle, but also on their generalizability and applicability to other problems.

### Generating and Solving Logic Puzzles through Constraint Satisfaction

Solving logic puzzles has become a very popular past-time, particularly since the Sudoku puzzle started appearing in newspapers all over the world. We have developed a puzzle generator for a modification of Sudoku, called Jidoku, in which clues are binary disequalities between cells on a 9 9 grid. Our generator guarantees that puzzles have unique solutions, have graded difficulty, and can be solved using inference alone. This demonstration provides a fun application of many standard constraint satisfaction techniques, such as problem formulation, global constraints, search and inference. It is ideal as both an education and outreach tool. Our demonstration will allow people to generate and interactively solve puzzles of user-selected difficulty, with the aid of hints if required, through a specifically built Java applet.

### Difficulty Rating of Sudoku Puzzles by a Computational Model

We discuss and evaluate metrics for difficulty rating of Sudoku puzzles. The correlation coefficient with human performance for our best metric is 0.95. The data on human performance were obtained from three web portals and they comprise thousands of hours of human solving over 2000 problems. We provide a simple computational model of human solving activity and evaluate it over collected data. Using the model we show that there are two sources of problem difficulty: complexity of individual steps (logic operations) and structure of dependency among steps. Beside providing a very good Sudoku-tuned metric, we also discuss a metric with few Sudoku-specific details, which still provides good results (correlation coefficient is 0.88). Hence we believe that the approach should be applicable to difficulty rating of other constraint satisfaction problems.

### Utterly addictive! Pit your wits against the puzzle masters of Japan

The pencil-and-paper logic puzzle is arguably Japan's most successful cultural export of recent years. Look inside almost any daily newspaper and you will find at least one number puzzle with a Japanese name; sudoku most commonly, but there are many others, such as kakuro and futoshiki, to mention only the ones that appear regularly in the Guardian. Shelves stuffed full of these exotic-sounding, square-gridded, numerical brain-teasers fill every newsagent and bookstore. I visited Tokyo to try to understand why Japan dominates the puzzle world. I discovered a country with a unique puzzle culture.