ON SPECIFIC 17-CLUE SUDOKU PUZZLES

Abstract

Since Sudoku was invented, it has been interesting to find Sudoku puzzles with the minimum number of clues. Royle collected 49,151 17-clue Sudoku puzzles between 2005 and 2009, each of which is not isomorphic to any other, and McGuire claimed in 2012 that no 16-clue Sudoku puzzle exists. Since 2009, no new 17-clue Sudoku puzzles have been found. This paper proposes an algorithm to find 17-clue Sudoku puzzles based on the number of clues in the topmost 9x3 cells, which is called a top-block. Using this algorithm, we prove that (1) no 17-clue puzzles exist of which the top-block has fewer than three clues and (2) precisely 95 17-clue puzzles exist of which the top-block has three and only three clues. Moreover, we compared these 95 puzzles to the 49,151 collected 17-clue Sudoku puzzles and found that these 95 puzzles are already collected by Royle. Thus, a conjecture from this paper is that most of the 17-clue Sudoku puzzles have been found.