How to Open an Empty Killer Sudoku Grid (First Five Moves)

Step 1: Scan for 'Forced Pair' Two-Cell Cages

Your very first action is a global scan. Look for any two-cell cage with a target sum of 3, 4, 16, or 17. These sums have only one possible digit combination in standard Sudoku, making them 'forced pairs.' A sum of 3 must be {1,2}, a sum of 4 must be {1,3}, a sum of 16 must be {7,9}, and a sum of 17 must be {8,9}. Lightly pencil these digit pairs into the corners of the corresponding cells. These are your anchor points for the entire puzzle.

• Do not place the digits yet. Just mark the candidate pair in the cell corners to keep the grid clean for the next steps.
• If a cage is marked with a dashed line, it may still be a two-cell cage. Pay close attention to the cage outlines.

Step 2: Mark Min-Max Candidate Ranges for All Cages

With your forced pairs noted, systematically work through every cage. For each cage, calculate the minimum and maximum possible digit for any cell within it. This is done using the cage sum and the number of cells. For a two-cell cage summing to 10, the minimum digit in a cell is 1, but the maximum is 9 (since the other cell could be a 1). For a three-cell cage summing to 15, the minimum possible digit is 1 (e.g., 1+5+9) and the maximum is 9 (e.g., 9+1+5). Write these small min-max ranges (e.g., 1-9, 2-8) in the center of each cell. This step transforms the blank grid into a map of possibilities.

Step 3: Find Cages That Interact with Forced Pairs

Now, look for cages that share a row, column, or 3x3 box with your marked forced pairs. The digits in a forced pair act as powerful eliminators. For example, if a forced pair {1,2} is in a row, no other cell in that row can be a 1 or 2. Examine the min-max ranges of cells in cages in that same row. You can often immediately reduce a cell's range or even lock in a digit, creating what is effectively a Hidden Single for that cage's context. This interaction is where your first concrete placements usually come from.

Step 4: Place Your First Concrete Digits

Based on the interactions in Step 3, you will find cells where all candidates but one are eliminated. Place these digits firmly in the grid. Once a digit is placed, it becomes a new eliminator. Immediately re-scan the cages and rows/columns/boxes affected by this new digit. A placed digit can turn a multi-candidate cell in a neighboring cage into a Naked Single. This chain reaction of logic is the goal of your opening. Don't just place one digit and stop. Use each new placement to trigger the next scan.

• After placing a digit, cross out its candidate from the min-max marks in all cells that see it in the same row, column, and box.
• If two cells in a cage end up with the same two candidates, you've found a Naked Pair for that cage, allowing for further eliminations.

Step 5: Shift Focus to the Smallest or Largest Cages

Once the immediate interactions with forced pairs are exhausted, change your strategy. Look for the cages with the fewest number of cells (often two or three cells) or the most extreme sums (very low or very high). Small cages have fewer combinations, making them easier to analyze. A three-cell cage with a very low sum like 6 has very limited options ({1,2,3}). Similarly, a high-sum cage like 22 for three cells is restrictive. Analyzing these cages systematically often reveals the next set of placements, continuing the momentum from your strong opening.