How to play Killer Sudoku

Sudoku with no given digits. The grid is carved into dashed cages, each labelled with the total of its squares, and every puzzle has exactly one answer.

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Killer Sudoku is Sudoku crossed with Kakuro. It keeps the nine-by-nine grid and its three constraints, throws away every printed digit, and replaces them with an arithmetic partition of the board. This page sets out the rules exactly, including the parts that are easy to get wrong, and the vocabulary you will meet in anything else written about the puzzle.

Setup

The board is nine squares by nine, divided as usual into nine three-by-three boxes. Overlaid on it is a second, unrelated division: a partition of all 81 squares into cages. A cage is a set of orthogonally connected squares — joined edge to edge, never only at a corner — drawn with a dashed outline. Cages never overlap and every square belongs to exactly one.

Each cage carries a single small number in the corner of its top-left square. That is the cage total. There is no other printed information anywhere on the board, and in particular there are no given digits. A one-square cage would be a given digit wearing a dashed border, so the puzzles here never contain one; the smallest cage is two squares.

The three constraints

A completed grid must satisfy all three of the following at once.

First, the Sudoku constraint: every row, every column and every three-by-three box contains each of the digits one to nine exactly once. Any of these nine-square groups is called a nonet, or a unit.

Second, the sum constraint: the digits in each cage add up to that cage's total, exactly. Not at most, not approximately.

Third, the cage constraint: no digit appears twice within one cage. This is a real, separate rule and it is the one people forget. A cage can bend across a box boundary so that two of its squares share no row, no column and no box — the Sudoku rules would happily allow the same digit in both, and the cage rule forbids it.

The rule of forty-five

Because every nonet holds one to nine, every nonet sums to 45. Any set of k whole nonets therefore sums to 45k. This is the lever that opens a board with no givens.

Take a nonet — say the top row. Add up the totals of every cage that lies entirely inside it. Subtract that from 45. What is left is the total of the squares in that row that belong to cages sticking out of it. Those squares are the innies. If there is exactly one innie, you have just read a digit straight off the board.

The mirror image works too. Add up the totals of every cage that touches the row at all. Subtract 45. What is left is the total of the squares those cages occupy outside the row — the outies. One outie, one free digit.

Neither trick is limited to a single nonet. A band of three rows sums to 135, a stack of two columns to 90, and the same subtraction applies. Wider regions tend to produce cleaner innies, because more cages fall wholly inside them.

Cage combinations

A cage total plus a cage size is often a very small piece of information. Two squares adding to seventeen must be eight and nine. Three squares adding to seven must be one, two and four. Four squares adding to eleven must be one, two, three and five. These forced sets are called combinations, and knowing the extreme ones by sight is most of the speed in this puzzle.

The useful half of a combination is frequently not which digits are in it but which are out. A three-square cage totalling twenty-three can only be six, eight and nine, which means the whole cage is a wall that no one, two, three, four, five or seven can cross.

Difficulty, and what it means here

The three grades are defined by the reasoning a board needs, not by how it looks. Easy needs cage combinations, singles and single innies or outies. Medium additionally needs locked candidates, naked pairs, or a hidden single inside a cage. Hard additionally needs innie or outie sets of two or more squares, or naked triples.

Crucially, a board is only given a grade if the grade below genuinely fails on it. A Hard board is one where the Medium toolkit, run to exhaustion, leaves squares empty. That is checked on every board before it is dealt, so the label is a measurement rather than an intention.

Ending the game

There is one way to finish and it is the obvious one: fill all 81 squares so that no nonet repeats a digit, every cage totals exactly its corner number, and no cage repeats a digit. Because every board has exactly one solution, there is no partial credit and no alternative answer — a full grid is either the answer or it is wrong.

A cage that is complete and correct is drawn in teal. A cage that has become impossible — a repeat, an overshoot, or a total the remaining squares can no longer reach — is drawn in red as soon as that is true, rather than waiting until the grid is full. Hints fill one square correctly and add a thirty-second penalty to the recorded time.

Controls

Pick a square, then type a digit or tap the pad.

Keyboard

1–9 writes a digit into the selected square.
Arrow keys move the selection, and wrap round at the edges.
Backspace or Delete clears the square.
N toggles pencil-notes mode, U undoes, H takes a hint.

Touch

Tap a square to select it, then tap a digit on the pad below or beside the grid.
Tap the same digit again to clear the square, or use the Erase key.
The Notes button turns the pad into pencil marks.

Strategy tips

Start with the extreme cages, not the interesting ones. A two-square seventeen and a three-square six are certainties, and certainties are what the rule of forty-five needs to bite on.
Add up a whole box before you look at any square in it. Boxes are the friendliest nonet for innies, because cages tend to sit inside a box rather than straddle two.
Widen the region when a single nonet gives you nothing. Two adjacent rows total ninety and three total one hundred and thirty-five, and a wider net catches more cages whole.
Remember that a cage forbids repeats even where Sudoku would allow them. An L-shaped cage crossing a box boundary is the classic place to lose an hour.
Write the combination, not the candidates, when a cage is forced. "6-8-9" pencilled once tells you more than three squares of candidate lists.
A cage that spans two boxes is doing double duty. Its digits are constrained by both, so it is usually the most informative thing on that part of the board.
Watch the parity of a long cage. Five squares totalling fifteen must be one to five; five totalling thirty-five must be five to nine. The extremes are always forced.
If you are stuck, count the cages you have completed rather than the squares. A region where every cage is still open is a region you have not actually attacked yet.
Never guess. Every board here is proved solvable by pure deduction before it is dealt, so a dead end is always a mistake several moves back, not bad luck.