Sudoku 129 Here

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.

In Sudoku 129, the pattern of 1,2,9 in block ( B_ij ) (block row i, block col j) is uniquely determined by the row pattern offset and column pattern offset modulo 3. sudoku 129

| Metric | Classic Sudoku | Sudoku 129 | |----------------------------|----------------|------------| | Avg. backtracks (millions) | 0.2 | 1.4 | | Avg. time (ms) | 15 | 98 | | Min clues needed (observed)| 17 | 24 | 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. | Metric | Classic Sudoku | Sudoku 129

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

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.

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: