Far East Journal of Applied Mathematics

The Far East Journal of Applied Mathematics publishes original research papers and survey articles in applied mathematics, covering topics such as nonlinear dynamics, approximation theory, and mathematical modeling. It encourages papers focusing on algorithm development.

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$S_\infty$-RANK AND LAC INVARIANT FOR THE EVALUATION OF DIFFICULTY LEVEL OF SUDOKU PUZZLES BY BOOLEAN GROEBNER BASES

Authors

  • Tetsuo Nakano
  • Miku Shindou
  • Tsukasa Yoshihara

Keywords:

boolean Groebner bases, Sudoku, Inoue algorithm, $S_\infty$-rank, LAC

DOI:

https://doi.org/10.17654/0972096025012

Abstract

The Sudoku is a very popular puzzle all over the world and it is also an interesting subject in discrete mathematics. Recently, an excellent algorithm called the Inoue algorithm has been developed for solving Sudoku puzzles based on boolean Groebner bases, and successfully applied to the mathematical evaluation of the difficulty level of Sudoku puzzles.

Received: August 2, 2025
Accepted: September 3, 2025

References

[1] W. Bosma, J. J. Cannon, C. Fieker and A. Steel, eds., Handbook of Magma Functions, accessed 10 April 2025. http://magma.maths.usyd.edu.au/magma/.

[2] A. Inkala, AI Sudoku, accessed 10 April 2025. www.aisudoku.com/en/.

[3] S. Inoue, Efficient singleton set constraint solving by boolean Gröbner bases, Communications of JSSAC 1 (2012), 27-37.

[4] S. Inoue and Y. Sato, A mathematical hierarchy of Sudoku puzzles and its computation by boolean Groebner bases, LNCS 8884, Springer, 2014, pp. 88-98.

[5] T. Nakano, Y. Maruyama and S. Ohki, On the mathematical evaluation of difficulty level of Sudoku puzzles by boolean Groebner bases, Far East J. Appl. Math. 106(1-2) (2020), 43-70.

[6] T. Nakano, S. Minami, S. Harikae, K. Arai, H. Watanabe and Y. Tonegawa, On the Inoue invariants of the puzzles of Sudoku type II, Bulletin of JSSAC 24 (2018), 77-90.

[7] T. Nakano, M. Shindou, N. Mikoshiba and T. Yoshihara, The SMY invariant and the MDSL conjecture in the CII algorithm for solving Sudoku puzzles, Far East J. Appl. Math. 114 (2022), 25-48.

[8] T. Nakano, M. Shindou and T. Yoshihara, On the correlation of some mathematical indicators of difficulty level of Sudoku puzzles in terms of boolean Groebner bases, Computer Algebra: Foundations and Applications, Kyoto University, 2022, pp. 1-13 (in Japanese).

[9] T. Nakano and Y. Tonegawa, Introduction to boolean Groebner bases and their applications to puzzles of Sudoku type, J. Algebra Appl. Math. 12 (2014), 1-31.

[10] Y. Sato, S. Inoue, A. Suzuki, K. Nabeshima and K. Sakai, Boolean Gröbner bases, J. Symbolic Comput. 46 (2011), 622-632.

[11] D. Tarek, Accessed June 6, 2025.

http://forum.enjoysudoku.com/the-hardest-sudokus-new-thread-t6539.html.

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Published

2025-09-23

Issue

Vol. 118 No. 2 (2025)

Section

Articles

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How to Cite

$S_\infty$-RANK AND LAC INVARIANT FOR THE EVALUATION OF DIFFICULTY LEVEL OF SUDOKU PUZZLES BY BOOLEAN GROEBNER BASES. (2025). Far East Journal of Applied Mathematics, 118(2), 229-245. https://doi.org/10.17654/0972096025012

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