Sudokus and Gröbner Bases: Not Only a Divertimento

  • Jesús Gago-Vargas
  • Isabel Hartillo-Hermoso
  • Jorge Martín-Morales
  • José María Ucha-Enríquez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Gröbner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide information on systems that are —in spite of their origin— hard to solve.


Polynomial Equation Algebraic System Polynomial System Hilbert Scheme Colouring Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Adams, W.W., Loustaunau, P.: An introduction to Gröbner bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)zbMATHGoogle Scholar
  2. 2.
    Bayer, D.: The division algorithm and the Hilbert scheme. Ph. D. Thesis. Harvard University (June 1982)Google Scholar
  3. 3.
    Becker, T., Weispfenning, V.: Gröbner bases. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993); A computational approach to commutative algebra, In cooperation with Heinz KredelGoogle Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Springer, Berlin (1997)Google Scholar
  5. 5.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Berlin (1998)zbMATHGoogle Scholar
  6. 6.
    Felgenhauer, B., Jarvis, F.: Enumerating possible sudoku grids (2005) (online, accessed December 30, 2005),
  7. 7.
    Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005),
  8. 8.
    Kaye, R.: Minesweeper is NP-complete. Math. Intelligencer 22(2), 9–15 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kelley, C.T.: Iterative methods for linear and nonlinear equations. Frontiers in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995); With separately available softwareGoogle Scholar
  10. 10.
    Knuth, D.E.: Dancing links (preprint)Google Scholar
  11. 11.
    Kreuzer, M., Robbiano, L.: Computational commutative algebra, vol. 1. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  12. 12.
    Pegg, E.: Sudoku variations (2005) (online, accessed December 12, 2005),
  13. 13.
    Sturmfels, B.: Solving Systems of Polynomial Equations. Amer. Math. Soc., CBMS Regional Conferences Series, No. 97, Providence, Rhode Island (2002)Google Scholar
  14. 14.
    Verschelde, J.: PHCcpack: A general-purpose solver for polynomial systems by homotopy continuation, 2005. ACM Transactions on Mathematical Software 25(2), 251–276 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jesús Gago-Vargas
    • 1
  • Isabel Hartillo-Hermoso
    • 2
  • Jorge Martín-Morales
    • 3
  • José María Ucha-Enríquez
    • 1
  1. 1.Dpto. de ÁlgebraUniv. de SevillaSevillaSpain
  2. 2.Dpto. de MatemáticasUniv. de CádizPuerto Real, CádizSpain
  3. 3.Depto. de MatemáticasUniv. de ZaragozaZaragozaSpain

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