Cage Eliminations
Use the no-repeat-in-cage rule to eliminate candidates. When combined with standard Sudoku constraints, this technique reveals hidden singles and forces placements.
The No-Repeat Rule
Every Killer Sudoku cage has a fundamental constraint: all digits within the cage must be different. This is in addition to the standard Sudoku uniqueness rules for rows, columns, and boxes.
Once you place a digit in one cell of a cage, that digit is eliminated as a candidate for all other cells in the same cage. This creates powerful deductions, especially when combined with:
Row/column/box constraints (Sudoku rules)
Cage sum constraints (arithmetic)
Unique cage combinations (from Technique 1)
Key insight: Cage eliminations work in both directions. Placing a digit eliminates it from the cage, and knowing a cage's digit set eliminates those digits from intersecting regions.
Example 1: Simple Elimination from Placement
Let's see how placing one digit cascades through a cage:
3-cell cage, sum 15:
Step 1: Place a digit via Sudoku logic
Suppose row/column analysis forces Cell A to be 7. We now know Cell A = 7.
Step 2: Eliminate from cage
Since Cell A = 7, no other cell in this cage can be 7 (no repeat in cage).
Step 3: Use arithmetic constraint
Cage sums to 15. Cell A contributes 7. Cells B and C must sum to 15 − 7 = 8.
Cells B and C must be two different digits summing to 8. Possible pairs: {1,7}, {2,6}, {3,5} (we can check which are compatible with current candidates).
Cell B candidates: {4, 6, 8}. Cell C candidates: {1, 2, 4, 6}. The pair must be from the intersection where B+C=8. Checking: if B=6, then C=2. If B=4, then C=4 (invalid, same digit). If B=8, then C=0 (invalid, not 1-9).
Only valid: B=6, C=2. Both cells are now determined!
Placing a single digit (Cell A = 7) led to determining the entire cage through elimination + arithmetic. This cascading logic is the heart of Killer Sudoku solving.
Example 2: Elimination from Known Cage Sets
When a cage has a unique combination (from Technique 1), you can eliminate those digits from intersecting rows/columns/boxes.
2-cell cage, sum 17:
What we know:
The cage must contain exactly 8 and 9 (unique combination for sum 17). We don't know which cell gets which digit yet, but we know both digits are used.
Sudoku elimination:
Row 5 contains these two cells. Therefore, no other cell in Row 5 can be 8 or 9 (already used by this cage).
Box 4 contains these two cells. Therefore, no other cell in Box 4 can be 8 or 9.
Eliminate 8 and 9 from all other cells in Row 5 and Box 4. This can immediately solve cells that had only {8, 9, X}as candidates—they become X!
Example 3: Naked Pairs Within Cages
When two cells in a cage can only be two specific digits (a "naked pair"), you can eliminate those digits from other cells in the cage.
4-cell cage, sum 18:
Observation:
Cells A and B together must be {1, 5} in some order. No other combination is possible for them.
Elimination:
Since A and B use up 1 and 5, cells C and D cannot use those digits (no repeat in cage).
Now check the arithmetic: A+B (which is 1+5=6) + C + D = 18, so C+D = 12. With C in {3,7} and D in {3,4,7}, if C=7 then D=5 (invalid, 5 is used by A or B). If C=3 then D must be 9 (but 9 isn't a candidate for D). Re-examine: we may have a contradiction, or D's candidates need revision based on other constraints.
This example shows how naked pairs within a cage eliminate candidates, but you must always verify against the cage sum to ensure consistency.
Hidden Singles Revealed by Elimination
One of the most powerful uses of cage elimination is finding hidden singles: digits that can only go in one cell of a region, even though that cell has multiple candidates.
Scenario:
Row 7 needs a 9 somewhere. Most cells in Row 7 are in cages that cannot contain 9 (e.g., a cage {1,2,3} or a cage that already has its digit set determined without 9).
After eliminating all cells where 9 is impossible (due to cage constraints or Sudoku rules), only one cell remains as a candidate for 9.
Result: That cell must be 9, even if it had other candidates. This is a hidden single revealed by elimination.
Scan for hidden singles frequently. After each placement or elimination, check rows, columns, and boxes for digits that now have only one possible position.
When to Use Cage Eliminations
This technique is most effective:
1.
After placing any digit
Every placement eliminates that digit from its cage. Check the cage immediately for cascading deductions.
2.
When you know a cage's digit set
Unique combinations (e.g., {1,2} for sum 3) let you eliminate those digits from intersecting regions.
3.
When scanning for hidden singles
Regularly check if a digit has only one possible cell in a row, column, or box after eliminations.
Common Mistakes to Avoid
❌ Forgetting to eliminate from the cage
When you place a digit in a cell, immediately eliminate it from all other cells in the same cage. Skipping this step loses information and can lead to contradictions later.
❌ Eliminating from the wrong regions
Cage eliminations only affect cells within the same cage. Standard Sudoku eliminations (row/column/box) are separate. Don't confuse the two.
❌ Ignoring arithmetic after elimination
After eliminating candidates, always re-check the cage's sum constraint. The remaining candidates must still be able to sum to the target.
The Elimination Workflow
Integrate cage eliminations into your solving routine:
Identify unique cages and mark their digit sets.
Place a digit using Sudoku or cage logic.
Eliminate that digit from:
- Other cells in the same cage
- Other cells in the same row
- Other cells in the same column
- Other cells in the same 3×3 box
Check for forced placements: naked singles (cell has one candidate left) or hidden singles (digit has one possible cell in a region).
Repeat until the puzzle is solved.
Why Eliminations Are Essential
Cage eliminations bridge the gap between arithmetic (cage sums) and logic (Sudoku rules). Without tracking eliminations, you'll miss most of the puzzle's constraints and struggle to make progress beyond the initial unique cages.
Advanced solvers use pencil marks (candidate notation) to track which digits are possible in each cell. As eliminations narrow the candidates, forced placements emerge naturally. This technique is not optional—it's foundational for Medium difficulty and above.