Cross-Hatching: The Scan Behind Every Easy Solve

Step 1: Choose a Digit and Identify Starting Constraints

Begin by selecting a single digit from 1 to 9. Look at the puzzle's givens and any numbers you have already placed to find every instance of that digit on the grid. Each existing digit acts as a constraint, blocking its entire row and column. For a generic example, imagine you choose the digit 5. You would first locate all the 5s already in the puzzle. Each of these 5s prevents another 5 from being placed in its row and column. These constraints are your starting map for the cross-hatching process, which is a core part of Sudoku scanning.

Step 2: Scan All Nine Boxes for the Chosen Digit

The next step is to examine each of the nine 3x3 boxes, one by one, for your chosen digit. For each box, check if the digit is already placed inside it. If it is, you can skip that box. If the digit is not in the box, you must determine where it could go. Look at the rows and columns that intersect the box. If any of those rows or columns already contain your digit (from your constraints map), you can cross off those entire rows or columns within the box. The remaining empty cells, after applying these row and column blockades, are the candidate cells for your digit in that particular box.

Continuing our generic digit 5 example, take a box in the top-left. If a row crossing that box already has a 5, you eliminate that entire row within the box. If a column crossing that box also has a 5, you eliminate that column. The intersection of these eliminations may leave only one possible cell for the 5 in that box, which is a hidden single specific to the digit you are tracking. This systematic elimination across all boxes is more reliable than hoping to spot such a placement by glancing at the grid.

• Always scan all nine boxes for your chosen digit. Do not skip boxes where the digit isn't initially present.
• Mentally or with light pencil marks, track the rows and columns blocked by existing digits.

Step 3: Record Placements and Update Your Constraints

When your scan of a box reveals only one possible cell for the digit, you have found a placement. Write the digit into that cell. This new placement is now a powerful additional constraint. It blocks a new row and a new column. You must immediately incorporate this new information into your ongoing scan for the current digit. This often creates a chain reaction, revealing another placement for the same digit in a different box that was previously ambiguous. Our step-by-step solving philosophy emphasizes this iterative updating.

In our example, if placing a 5 in a box now blocks a column for an adjacent box you haven't scanned yet, that box may now yield a placement. Work through all nine boxes for digit 5, updating your mental map after each placement, until no more placements for that digit can be found. Only then is your scan for that digit complete.

Step 4: Move to the Next Digit and Repeat the Full Process

Once you have exhausted all possible placements for your first digit, move on to the next digit. Start again from Step 1. The critical rule is to incorporate all constraints from previously placed digits, not just the original givens. The 5 you placed in the previous step now acts as a given for all subsequent digit scans. This layered approach is why cross-hatching finds hidden singles that are invisible to cell-by-cell scanning. A cell might look open to many numbers, but when you scan for, say, the digit 3, you may find it is forced into that very cell because all other options in its box are eliminated by 3s in intersecting rows and columns.

A complete pass involves scanning all nine digits in sequence, 1 through 9. After a full pass, the puzzle will have fewer empty cells. You then start a new pass from digit 1 again, using the updated grid. This repeats until the puzzle is solved or you need more advanced techniques like Box-Line Reduction.

Why This Method Finds Hidden Singles

Cell-by-cell scanning looks at one empty cell and considers which digits could fit there. This often overlooks a key fact: a digit may have only one possible home in a large unit like a row, column, or box. Cross-hatching inverts this logic. It asks, 'For this specific digit, where must it go in this box?' By applying all row and column constraints for that single digit, the answer often becomes obvious. A hidden single for a digit exists when, within a row, column, or box, all but one cell is eliminated for that digit by seeing it in other units. Cross-hatching is designed to spot this condition box-by-box for every digit systematically.

How to Tell a Finished Pass from a Rushed One

A finished cross-hatching pass is methodical and complete. You have a clear record, mental or written, of having scanned all nine boxes for digit 1, found all possible placements, updated constraints, and then moved to digit 2, and so on through digit 9. A rushed pass is haphazard. It might jump between digits, skip boxes, or fail to use newly placed digits as immediate constraints for the rest of the scan. The hallmark of a proper pass is that after scanning a digit, you are confident there are no more obvious placements for it on the current grid. If you find a placement for digit 5 later in the same pass after you thought you were done with it, your initial scan was incomplete.