TWO CONSTRAINTS, ONE GRID
Every cell in Killer Sudoku obeys two rules at the same time: the cage sum, and Sudoku's row/column/box uniqueness. Neither rule alone is enough to solve most cells — you need both constraints to intersect.
The practical consequence: when you eliminate a digit from a cell via Sudoku logic, check whether the remaining cage candidates have changed. When you enumerate cage candidates, check whether any are ruled out by Sudoku. Every step feeds the other.
Treat cages and rows/columns/boxes as two overlapping filter systems — not as two separate puzzles.
START WITH FORCED CAGES
Some cages have only one possible combination of digits regardless of arrangement:
- A 2-cell cage summing to 3 must be {1, 2} — no other pair sums to 3 with distinct digits.
- A 2-cell cage summing to 17 must be {8, 9}.
- A 3-cell cage summing to 6 must be {1, 2, 3}.
- A 3-cell cage summing to 23 or 24 has very few options.
Find these locked cages first. You can't place specific digits in specific cells yet, but you can eliminate those digits from other cells in the same row, column, or box immediately — which is often enough to unlock a neighboring cage.
THE 45-RULE
Every row, column, and 3×3 box contains the digits 1–9 exactly once, so each sums to 45. This gives you a powerful tool: if you know the sum of every cage fully inside a row except one, you know what that last cage must contribute to the row total.
The most useful case: when all but one cage in a row are fully contained within that row, the remaining cage's cells in that row must sum to (45 minus the sum of the fully-contained cages). If only one cell of that cage sits in the row, you know that cell's value exactly.
Apply the same logic to columns and boxes. The 45-rule often cracks a puzzle section that looks locked at first glance.
ENUMERATE CANDIDATES, THEN CROSS-REFERENCE
For cages that aren't forced, list all valid combinations. A 2-cell cage summing to 10 could be {1, 9}, {2, 8}, {3, 7}, or {4, 6}. Write them down.
Then apply Sudoku: if 9 already appears in the same row as both cells of that cage, eliminate the {1, 9} combination. If 8 is also present, eliminate {2, 8}. Suddenly the cage might be forced to {3, 7} or {4, 6} — and you know those four digits are excluded from the rest of that row.
The intersection of cage combinations and Sudoku elimination is where most Killer Sudoku progress actually comes from.
SUBTRACTION CAGES ARE ALWAYS PAIRS
In a 2-cell cage, if you know the sum, you can determine which digit goes where once Sudoku constraints narrow the candidates to one arrangement. The pair is symmetric — the cage doesn't tell you the order.
To place individual digits within a cage, you need an external constraint: one of the cage's cells is also constrained by a row/column/box that already contains one of the two candidates. That's the signal to look for.
THE ORDER OF ATTACK
- Mark all forced cages (single-combination sums).
- Apply the 45-rule to any row, column, or box where most cages are contained.
- Enumerate candidates for remaining cages — cross-reference with Sudoku eliminations.
- Look for naked Sudoku singles or hidden singles — cage work often creates them.
- Repeat: each placement changes cage candidates and Sudoku eliminations together.
Alternating between cage arithmetic and Sudoku scanning every few steps — rather than finishing all cage work before any Sudoku scanning — is what keeps the board moving on hard difficulties.