Solving arbitrary N-th Rank Sudoku in Polynomial Time using the "Cell-To-Symbol Tally" Method

Sudoku is a widely studied constraint satisfaction problem whose generalized form on an (n^2 x times n^2) grid (referred to as the n-th rank Sudoku) has been proven to be NP-complete. As the grid size increases, conventional solving techniques such as brute-force search and backtracking experience rapid growth in computational complexity, making large instances increasingly difficult to solve efficiently. This paper presents a structured solving approach termed Cell-to-Symbol Tally for generalized Sudoku puzzles. The method begins by constructing an options table for each empty cell that lists all candidate symbols consistent with Sudoku constraints. By applying a tally-based filtering process, invalid candidates are systematically eliminated, allowing deterministic placement of symbols for the majority of cells. In cases where two candidate symbols remain, a limited prediction step is applied to a small subset of cells, referred to as key cells. For well structured puzzles with a unique solution, experimental observations indicate that the number of such key cells does not exceed (N-1), where (N) denotes grid size. These key cells exhibit a structural dependency that can be interpreted as forming a Hamiltonian-like path. The proposed method provides a logical framework for solving the remaining cells while incorporating an error-tracking mechanism to handle prediction steps. Experimental results suggest that the approach significantly reduces the search space and improves the efficiency of solving generalized Sudoku puzzles.