Abstract
Sudoku, which is one of the most popular puzzles in the world, can be considered as a kind of combinatorial problem. Considering a Sudoku puzzle as a singleton set constraint, we define a purely mathematical hierarchy of Sudoku puzzles in terms of a Boolean polynomial ring. We also introduce a sufficiently practical symbolic computation method using Boolean Gröbner bases to determine the hierarchy level of a given Sudoku puzzle. According to our experiments through our implementation, there exists a strong positive correlation between our hierarchy and the levels of difficulty of Sudoku puzzles usually assigned by a heuristic analysis. Our mathematical hierarchy would be a universal tool which ensures the mathematical correctness of the level of a Sudoku puzzle given by a heuristic analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chang, C., Fan, Z., Sun, Y.: Hsolve: A Difficulty Metric and Puzzle Generator for Sudoku (2008), http://web.mit.edu/yisun/www/papers/sudoku.pdf
Gago-Vargas, J., Hartillo-Hermoso, I., Martín-Morales, J., Ucha-Enríquez, J.M.: Sudokus and Gröbner Bases: Not Only a Divertimento. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 155–165. Springer, Heidelberg (2006)
Inoue, S.: On the Computation of Comprehensive Boolean Gröbner Bases. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 130–141. Springer, Heidelberg (2009)
Nagai, A., Inoue, S.: An Implementation Method of Boolean Gröbner Bases and Comprehensive Boolean Gröbner Bases on General Computer Algebra Systems. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 531–536. Springer, Heidelberg (2014)
Challenge your brain: is this the world’s hardest Sudoku?, http://www.efamol.com/efamol-news/news-item.php?id=10
Introducing the World’s Hardest Sudoku, http://www.efamol.com/efamol-news/news-item.php?id=43
Kandri-Rody, A., Kapur, D., Narendran, P.: An Ideal-Theoretic Approach for Word Problems and Unification Problems over Commutative Algebras. In: Jouannaud, J.-P. (ed.) RTA 1985. LNCS, vol. 202, pp. 345–364. Springer, Heidelberg (1985)
Noro, M., et al.: A Computer Algebra System Risa/Asir (2014), http://www.math.kobe-u.ac.jp/Asir/asir.html
Rudeanu, S.: Boolean functions and equations. North-Holland Publishing Co., American Elsevier Publishing Co., Inc., Amsterdam-London, New York (1974)
Sato, Y., et al.: Boolean Gröbner bases. Journal of Symbolic Computation 46(5), 622–632 (2011)
Sato, Y., Nagai, A., Inoue, S.: On the Computation of Elimination Ideals of Boolean Polynomial Rings. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, pp. 334–348. Springer, Heidelberg (2008)
Gohnai, K.: Cross Word editorial desk (2008). Number Placement Puzzles (Basic, Middle, High, SuperHigh, Hard, SuperHard, UltraHard). Kosaido Publishing Co. (2008) (in Japanese)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Inoue, S., Sato, Y. (2014). A Mathematical Hierarchy of Sudoku Puzzles and Its Computation by Boolean Gröbner Bases. In: Aranda-Corral, G.A., Calmet, J., Martín-Mateos, F.J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2014. Lecture Notes in Computer Science(), vol 8884. Springer, Cham. https://doi.org/10.1007/978-3-319-13770-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-13770-4_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-13769-8
Online ISBN: 978-3-319-13770-4
eBook Packages: Computer ScienceComputer Science (R0)