Killer Sudoku rules: make every sum earn its place

Two extra checks for every cage

A Killer grid can look almost empty, yet a tiny number beside a dotted outline may carry several digits’ worth of information. Start by identifying the cells inside that outline. The label belongs to their total, not to the top-left cell alone.

Ordinary Sudoku constraints remain active. Each cage must also reach its printed sum without repeating a digit, including when the cage crosses a row or box boundary. [1]

A cage is therefore not interchangeable with a full Sudoku house. A three-cell cage contains three distinct digits; it does not need every digit from 1 to 9. DKM’s own rules explicitly preserve cage uniqueness even when its cells do not share a conventional house. [2]

A sum identifies a set before it identifies a cell

Consider two empty cells, A and B, in a cage marked 17. Using distinct digits no larger than 9, the only available set is {8,9}. That statement leaves two possible orders. If A already sees a fixed 9 in its row, A becomes 8 and B becomes 9. The arithmetic supplied the pair; the row supplied its orientation.

The distinction prevents a subtle error. If a larger cage crosses a row containing 9, only its cells in that row lose 9. A cell in another row might still take it. Reject an entire combination only when its digits cannot be assigned to the cage’s actual cells in any legal order.

The 45 rule is an accounting rule

A complete house contains 1+2+3+4+5+6+7+8+9, so its total is 45. The publisher’s strategy guide uses this to compare a house with cages entirely contained inside it. [3]

In our row illustration, three disjoint cages cover eight cells. Their totals are 6, 15 and 15, giving 36. The only uncovered cell is therefore 45−36=9. This proof does not require knowing the order of the covered digits; they are shown here so that you can inspect the arithmetic.

Before subtracting, trace every cage boundary. If a counted cage extends outside the house, its label includes an outside cell. If a cage was counted twice, the sum is also wrong. The number 45 is reliable; the bookkeeping is where most false deductions enter.

When a cage crosses the boundary

A crossing cage can still be useful if you describe the counted region exactly. Suppose a collection of disjoint cages covers every cell of one box, plus exactly one cell outside it. If their labels add to 52, the outside cell is 52−45=7. This is an outie calculation. It works because there is one extra cell and no missing box cell.

If the same collection leaves a hole inside the box as well, subtraction instead gives the outside value minus the missing inside value. A difference of 7 is then a relationship between two cells, not permission to write 7 anywhere. Sketching the boundary is often quicker than repairing a confident arithmetic mistake.

Keep useful notes, then update them

Start with a restrictive cage and list its possible sets separately from the notes in individual cells. A sum-4 pair must be {1,3}; {2,2} is forbidden. A sum-10 pair has more possibilities, so row and column restrictions become more important. Our companion combinations guide works through that filtering process.

After a placement, subtract its digit from the cage’s remaining total and remove that digit from the remaining cage cells. Then revisit the cell’s row, column and box. A sum can expose a single, and that single can resolve a different cage. Alternating these checks keeps the arithmetic connected to the grid.

If a cage has no surviving assignment, stop before filling another cell. Recheck the label, the cage membership and the candidate you most recently removed. A contradiction is useful evidence that one assumption needs review.

A one-cell cage is especially direct: its total is the cell’s value. For a longer cage, distinguish a remaining sum from a repeated clue. A sum of 12 across two unresolved cells does not mean either cell is 6, and a no-repeat cage specifically rules out 6 plus 6. Use the total to create possibilities, then let the constraints remove them.

Check the exact claim

The following questions distinguish sums, distinctness and cell coverage. Explain the answer before selecting it; the explanation is the skill you can reuse on a new board.

The Killer page is the relevant ZUDOKU destination. Use its printed cages as the authority for each puzzle. This guide’s small diagrams are rule demonstrations, not extracted game positions.