Anti-Chess Sudoku: A Practical Solving Order

Step 1: Immediately Apply All King-Move Exclusions

Your first action in any Anti-Chess Sudoku puzzle should be to apply the king-move constraint to every given digit. For each number already placed in the grid, eliminate that candidate from all cells a chess king's move away. This includes all orthogonally and diagonally adjacent cells. Doing this first is critical because it creates immediate 'naked singles' or severely restricts candidate lists, setting up the rest of the solve. This rule is the defining extra constraint of the variant, so applying it upfront is the most powerful move you can make.

• Scan the grid systematically, perhaps box by box, to ensure you don't miss any king-attack relationships.
• Use pencil marks to note these eliminations clearly; they form the foundation for all later logic.

Step 2: Target Cells Surrounded by Givens

After the initial king-move exclusions, shift your focus to empty cells that are orthogonally surrounded by given digits or numbers you just placed. These cells are 'constrained by proximity' because their row, column, and box neighbors are largely determined. Check the standard Sudoku rules for these cells first; the combination of a nearly full local group and the pre-applied king exclusions frequently leaves only one possible digit. This step exploits the local density of information. It is more efficient than looking at cells in open rows, as explained in our article Why a Full Anti-Chess Grid Is Impossible, which discusses the puzzle's inherent constraints.

• Look for empty cells that have givens to the left, right, above, and below.
• If a cell is in a row or column with 7 or 8 digits already placed, it's a prime candidate for this step.

Step 3: Settle Box Constraints Before Long Row Scans

Before examining full rows or columns, concentrate on completing individual 3x3 boxes. The box is a compact, manageable unit. Use standard Sudoku techniques within the box, always cross-referencing with the active king-move bans from digits in neighboring boxes. Because a box only has nine cells, it's easier to spot hidden singles or pairs here than in a nine-cell row that may be interrupted by the king constraint. Solving boxes first builds stable 'blocks' of solved cells that then heavily restrict the rows and columns passing through them. This mirrors effective strategy in related puzzles like Anti-Knight Sudoku Rules, where local constraints are also king.

• Work on the box with the most givens or the fewest remaining empty cells.
• Remember that a digit solved in one box bans itself via king moves from cells in adjacent boxes.

Step 4: Perform Row and Column Scans with the King Constraint

Only after progressing through the previous steps should you perform comprehensive row and column scans. At this point, many rows and columns will already be partially solved from your box work. When scanning, you must consider both the standard Sudoku rule (no repeats in the row/column) and the anti-king rule. A candidate might be valid for a cell based on the row alone but eliminated because it's a king's move away from the same digit in an adjacent row. This integrated check is the final layer of deduction. For a broader look at how chess moves define Sudoku variants, explore Chess Sudoku Rules.

• Use candidate pencil marks systematically to track possibilities across rows and columns.
• Pay special attention to where a row passes through two adjacent boxes, as king attacks can come from above or below.

Step 5: Iterate and Apply Advanced Techniques

The solve proceeds by iterating through this order: apply new king exclusions from any digit you place, then look for newly constrained cells, then tackle the next most promising box, and so on. As the grid fills, you may need to employ more advanced Sudoku techniques like X-Wings or Unique Rectangles, but always with the king constraint in mind. The anti-king rule can sometimes break the symmetry these techniques rely on in classic Sudoku, so double-check all logic. The priority remains: let the chess rule and local constraints guide you before expanding to global scans.

• If you feel stuck, cycle back to Step 1—a newly placed digit may have created a king exclusion you missed.
• Advanced techniques are a last resort; most puzzles crack under the repeated application of Steps 1-4.

Worked Example: Applying the Order

Imagine a puzzle where the center cell (r5c5) is given as '5'. Following Step 1, you immediately eliminate '5' from all eight cells surrounding it (the king's move). Now, suppose cell r4c5 (directly above the center) is empty but has givens in cells r4c4, r4c6, r5c5, and r3c5. This makes it a perfect target for Step 2. Checking its row, column, and box, you might find only one digit fits that also isn't banned by a king move from another '5' elsewhere. Placing that digit then triggers more king exclusions (back to Step 1). You then (Step 3) look at the central box; the new placement and exclusions may reveal a hidden single. By repeatedly cycling through this focused order, you avoid wasted effort on rows that are still too loosely constrained.