There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem

TL;DR

This paper proves that the minimum number of clues in a Sudoku puzzle is 17 by exhaustively searching for 16-clue puzzles and developing a novel enumeration method for hitting sets.

Contribution

The authors developed a new algorithm for efficiently enumerating hitting sets, enabling an exhaustive search that confirms 17 clues as the minimum in Sudoku puzzles.

Findings

No 16-clue Sudoku puzzles exist.

Minimum clues for a valid Sudoku puzzle is 17.

Developed a novel enumeration method for hitting sets.

Abstract

The sudoku minimum number of clues problem is the following question: what is the smallest number of clues that a sudoku puzzle can have? For several years it had been conjectured that the answer is 17. We have performed an exhaustive computer search for 16-clue sudoku puzzles, and did not find any, thus proving that the answer is indeed 17. In this article we describe our method and the actual search. As a part of this project we developed a novel way for enumerating hitting sets. The hitting set problem is computationally hard; it is one of Karp's 21 classic NP-complete problems. A standard backtracking algorithm for finding hitting sets would not be fast enough to search for a 16-clue sudoku puzzle exhaustively, even at today's supercomputer speeds. To make an exhaustive search possible, we designed an algorithm that allowed us to efficiently enumerate hitting sets of a suitable size.