Design and Analysis of a Modified 3D Sudoku Solver

Sudoku is a complicated multidimensional mathematical structure with several applications in various computer science domains. 3D Sudoku, compared to 2D, has one more dimension that can potentially provide an extra edge in the field of different application areas. Several researchers have developed various types of 2D Sudoku solvers using different methodologies. However, there is very limited research in the area of developing 3D Sudoku solvers. Thus, two different solvers for solving 3D Sudoku puzzles of size $9\times 9 \times 9$ are proposed in this work. Both solvers provide all possible solutions for solving a 3D Sudoku puzzle. 2D Sudoku puzzles are applied in different research domains with different purposes. Recently, 3D structure of Sudoku has been applied in several areas to achieve more effectiveness compared to 2D Sudoku. Additionally, it can also be used to solve problems in 3D space. Again, solving an NP-complete puzzle by considering its 3D structure is a challenging job. Thus, we endeavoured to achieve all probable solutions for a 3D Sudoku instance in this work. In the first version of our proposed algorithm, all possible values for each blank cell are computed and stored. Subsequently, a few elimination-based methods are used to reduce the number of probable values (if possible) for each blank cell. Finally, the solutions are computed using the backtracking method. In the second version of our proposed algorithm, the nine 2D Sudoku puzzles, lying in the $xz$ -plane one above the other, which form the 3D puzzle are fed as the input. All possible solutions are obtained for each of the nine puzzles. Then, the obtained solutions are mapped to achieve one or more solutions for the 3D Sudoku instance. Thus, our proposed techniques provide a new approach for solving 3D Sudoku. In addition, applying the obtained solutions provides us with an advantage over 2D Sudoku, in solving problems in the 3D space and where more data is required.