129 — Sudoku

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.

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. sudoku 129

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. In Sudoku 129, the pattern of 1,2,9 in

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: Let base pattern for row ( r )

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