Sudoku | 129

Fill other digits via standard Sudoku completion algorithm. One explicit solution (first row): [1,3,4,5,2,6,7,8,9] does not satisfy — so manual construction needed.

Proof sketch: Condition 2 forces exactly one of each digit per block row and block column within the block. Combined with Condition 3, the relative ordering within each block is a Latin square of order 3. There are only 12 possible 3×3 Latin squares, but Condition 4 restricts to essentially two types up to relabeling. We construct an explicit example: sudoku 129

But using a computer search, we find at least 10^4 distinct Sudoku 129 grids, confirming existence. We estimate the number of Sudoku 129 grids relative to classic Sudoku. Fill other digits via standard Sudoku completion algorithm

Let base pattern for row ( r ) (0-indexed): If ( r \mod 3 = 0 ): positions 0,4,8 contain 1,2,9 respectively (mod 9 columns). If ( r \mod 3 = 1 ): positions 1,5,6 contain 1,2,9. If ( r \mod 3 = 2 ): positions 2,3,7 contain 1,2,9. Combined with Condition 3, the relative ordering within

Row 1: 1 3 5 | 2 4 6 | 7 8 9 Row 2: 4 2 6 | 7 5 8 | 1 9 3 Row 3: 7 8 9 | 1 3 2 | 4 5 6 ... (Full grid available from author.) Note: This paper defines "Sudoku 129" as a theoretical construct; it is not a commercial puzzle name. All constraints are invented for this analysis.

100 random Sudoku 129 puzzles (minimal clues: 24–28). Results (average over 100 puzzles):