Solving a Binary Puzzle (original) (raw)

Abstract

A Binary puzzle is a Sudoku-like puzzle with values in each cell taken from the set \(\{ 0, 1\}\). Let \(n \ge 4\) be an even integer, a solved binary puzzle is an \(n \times n\) binary array that satisfies the following conditions: (1) no three consecutive ones and no three consecutive zeros in each row and each column; (2) the number of ones and zeros must be equal in each row and in each column; (3) there can be no repeated row and no repeated column. This paper proposes three approaches to solve the puzzle. The first method is based on a complete backtrack-based search algorithm. The idea is to propagate and fill an unsolved binary puzzle according to the three constraints, followed by a random guess if the puzzle remains unsolved. The second method of solving a binary puzzle is by representing it as an instance of a Boolean satisfiability problem which allows the solution for a binary puzzle to be obtained using SAT solvers. The third approach is based on expressing a binary puzzle as a system of polynomial equations over the binary field \(\mathbb {F}_2\). The set of solutions for the equation system implies the solutions for the binary puzzle and it is obtained by computing a Gröbner basis of the ideal generated by the polynomials. We experimentally compare the three approaches with binary puzzles of various sizes and different numbers of empty cells using a computer algebra system.

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  1. Eindhoven University of Technology, Eindhoven, The Netherlands
    Putranto H. Utomo
  2. Mathematical Institute, University Leiden and Centrum Wiskunde en Informatica, Amsterdam, The Netherlands
    Rusydi H. Makarim

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  1. Putranto H. Utomo
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  2. Rusydi H. Makarim
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Utomo, P.H., Makarim, R.H. Solving a Binary Puzzle.Math.Comput.Sci. 11, 515–526 (2017). https://doi.org/10.1007/s11786-017-0322-4

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