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:
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
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. Proof sketch: Condition 2 forces exactly one of