Sudoku Grids That Require Many Clues

2026-07-06

AI Summary

This study investigates the minimum number of clues required to guarantee a unique solution for $n^2 \times n^2$ Sudoku puzzles in the worst case. Through combinatorial analysis and constructive proofs, it rigorously establishes that typical large-scale Sudoku solution grids can tolerate only a logarithmic number of empty cells while preserving uniqueness. The work further presents explicit constructions of extremal instances requiring 18 and 80 clues for the standard $9 \times 9$ and $16 \times 16$ Sudoku grids, respectively. These results demonstrate that, for most completed grids, nearly all entries must be retained as clues to ensure a unique solution, thereby significantly advancing the understanding of the intrinsic complexity of Sudoku as a constraint satisfaction problem.

Abstract

Motivated by worst-case algorithmic time bounds for solving sudoku, we prove that a majority of filled-in $n^2\times n^2$ sudoku grids require all but a logarithmic fraction of cells to be filled by clues. For $9\times 9$ and $16\times 16$ sudoku, we construct grids that require $18$ clues and $80$ clues.

Problem

• Sudoku
• clues
• worst-case
• uniqueness
• grid

Innovation

• Sudoku
• clue minimality
• worst-case complexity
• combinatorial construction
• uniquely solvable grids