Logic Masters Deutschland e.V.

Chaos Construction messing with double-chaos-X-Sums and a pair of negative constraints.

• Normal Chaos Construction: Fill each row, column, & region with the digits 1-9 once each. Regions consist of 9 orthogonally-connected cells and must be determined by the solver.
  
• Chaos X-Sums: Black and red clues outside the grid give the sum of the first N cells in that row/column, where N is the first digit seen from the clue. These N cells all belong to the same region, and if the N+1th cell is inside the grid, it must be a different region.
• Double Chaos X-Sums: Red clues outside the grid apply again starting from the N+1th cell. (The same clue indicates the sum of the M cells starting from cell N+1, where M is the value in the N+1th cell. Those M cells all belong to the same region, and the subsequent N+M+1th cell belongs to a different region, if present.)
• A '?' clue can indicate a series of cells with any sum. The red '?' clue may have different sums for the first and second seen regions.
  
• Squares: A digit in a square indicates the number of cells in the same region seen orthogonally, including itself. Region borders block vision.
  
• Negative Constraint: All outside edge positions that could have a valid Chaos X-Sum clue have one.
• Negative Constraint: All valid squares are given.
  
• Two region borders are given between r3c1:r4c1 and between r4c2:r4c3.

Example for Double Chaos X clues and the negative constraint:

• In row 4 with a red clue, the 3 and 2 give the extents of those first two regions in the row. As the clue is a question mark, 3+4+5=12 and 2+7=9 may have different sums.
• In row 5 with a black clue, the 4 gives the extent of the first region of its row.
• Similarly for row 2 and the 2. This could have been a red question mark, because the 1 in r2c3 correctly describes the extent of that second region, but a black clue is sufficient to satisfy the negative constraint. A black clue gives no indication about digits and regions beyond the first.
• In column 4, the 3 in r2c4 correctly describes the extent of that second region in the column, but because r1c4 contains a 6, no Chaos X clue is possible.
• The corner-marked 1 in r1c1 cannot actually be a 1, because it would correctly describe the extent of the first region in its row and no Chaos X clue is present for the row.

SudokuPad: https://sudokupad.app/zd3sn059zf

Chaos Constructions:

1. Impostor Chaos
2. HeX-Sums
3. Double+Negative X-Sums