Killer Sudoku

• Each row contains 1–9, each digit once.
• Each column contains 1–9, each digit once.
• Each 3 × 3 box contains 1–9, each digit once.
• Each dashed cage adds up to the small number in its top-left corner.
• No digit repeats inside a cage — this is the rule most implementations leave out, and it is what makes the totals usable.
• A correct puzzle has exactly one solution and never requires guessing.

That fifth rule does more work than it looks. Without it, 16 across two cells could be 8+8; with it, 16 has exactly one pair — 7 and 9 — and so does 17. The no-repeat rule is why a cage total is information rather than an arithmetic curiosity.

Every row, every column and every 3 × 3 box holds the digits 1 to 9 exactly once, so every one of those 27 units adds up to 45. That number is the lever. Pick a box, add up the totals of the cages lying entirely inside it, and the difference from 45 is what the cells those cages miss must come to. On a grid with no printed digits, this is the only opening move there is — and it is enough.

Worked example: a box has cages of 11, 9 and 18 lying wholly inside it, and between them they cover eight of its nine cells. 11 + 9 + 18 = 38, and 45 − 38 = 7, so the ninth cell is a 7 — with nothing else on the board and nothing else needed. If the arithmetic ever lands outside 1 to 9, the cells you thought were covered are not, and a cage is reaching out of the box.

They are the two directions of the 45 rule, and they are the technique that makes a clueless grid tractable. An innie is a cell of a unit that no wholly-contained cage covers: add the contained cages, subtract from 45, and when exactly one cell is left over you have its digit. An outie runs the other way: take every cage that touches the unit, including the ones that spill past its edge, add their totals, and subtract 45. What remains is the total of the cells sticking out. When exactly one cell sticks out, that is its digit.

Outies are the harder habit to acquire and the more valuable, because a cage straddling a box boundary looks like an obstacle and is in fact a free digit on the far side of it. The hint button on this board applies the rule in both directions, and says which direction it went and which row, column or box it worked on.

Four for every cage size: the two smallest totals that size can make and the two largest. Everything between them is ambiguous. That is a small enough table to simply learn, and it is where almost every killer sudoku starts.

The ambiguity peaks in the middle, and knowing how badly is worth as much as knowing the certainties. Nine across two cells is four different pairs. Fourteen across three cells is eight different sets — the worst any three-cell total gets. Twenty across four cells is twelve. A cage in that range tells you almost nothing on its own and everything once one of its cells is settled.

Yes, and that is the form the puzzle was invented in. An ordinary sudoku needs clues — a 2012 exhaustive search by McGuire, Tugemann and Civario proved that no sudoku with a unique solution has fewer than 17. A killer needs none, because the cages carry the information instead: thirty-odd totals, each constraining two to four cells, is more than enough to pin down all 81.

The No clues level here prints nothing. It carves the grid into around thirty cages of two to four cells, where the middle two levels use about twenty-six cages running up to five — smaller cages mean more totals, and the totals are now the whole puzzle. Then a solver runs over the layout, and the board deals it only once it has proved there is a single completion. Not once it looks likely: once it is proved.