Sudokus and Gröbner Bases: Not Only a Divertimento

Abstract

Abstract. 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.

Modeling Sudoku Puzzles with Python

“…A graph model for Sudoku is presented in [Var05]. In this model, every cell of the Sudoku grid is represented by a node of the graph.…”

“…Polynomial system models The graph model above is introduced in [Var05] as a prelude to modeling Sudoku puzzles as systems of polynomial equations. The polynomial system model in [Var05] involves variables x i for i ∈ {1, . .…”

“…Specifying a Sudoku puzzle requires extending this model by adding polynomials to represent clues. According to the model from [Var05], if D is the set of fixed cells (i.e. cell label, value pairs) then to the polynomial system we need to add polynomials The sympy implementation of a Groebner basis algorithm can be used to find solutions of this polynomial system.…”

Computational analysis of LED circuits based on partial quasi‐group rings

“…Further, Kumar et al 14 dealt with the completion problem of partial Latin squares motivated by its possible application to light path assignments and switch configurations. This approach has already been used in the literature to analyze the solvability of other games based on combinatorial designs [29][30][31][32] This paper is organized as follows. 16 In the literature, (partial) quasi-groups and (partial) Latin squares have motivated the introduction and later analysis of different types of combinatorial games.…”

“…More specifically, we make use of computational algebraic geometry to deal with such a problem. This approach has already been used in the literature to analyze the solvability of other games based on combinatorial designs [29][30][31][32] This paper is organized as follows. In Section 2, we expose some preliminaries on electric circuits and partial quasi-group rings.…”