Killer Sudoku: Rules, Cage Combinations and the Rule of 45

The rules and the notation

The grid is a normal 9x9 sudoku. Every row, every column and every 3x3 box holds the digits 1 to 9 exactly once.

On top of that, the grid is divided into cages. A cage is outlined with a dashed or dotted line, and the small number in its top-left corner is the cage total. The digits in the cage must add up to that total, and no digit may appear twice in the same cage, even when the cage spans two boxes.

Most killer puzzles give no starting digits at all. Everything comes from the cage totals. That sounds harder than it is: the totals carry a lot of information once you know how to read them.

Cells are usually named by row and column, so r3c7 is row 3, column 7. Boxes are numbered 1 to 9 from the top left, reading across.

Cage-sum math in one minute

Because a cage cannot repeat digits, its total has hard limits. The smallest possible sum of n cells is 1 + 2 + ... + n, and the largest is the top n digits added together.

Two cells range from 3 to 17. Three cells range from 6 to 24. Four cells range from 10 to 30. Five cells range from 15 to 35.

A sum at or one step away from either limit has only one possible set of digits. Those are the unique cages, and they are where almost every killer begins.

Two more facts save time. A two-cell cage with an even total can never hold half of it: a 10 cannot be 5 + 5, so 5 is excluded, and a 12 excludes 6. And a two-cell cage with any total is a strict pair, so once you know one digit the other follows instantly.

Unique cage combinations

Learn this table and you will spot an opening in most puzzles within seconds. Every sum listed has exactly one possible set of digits; the order inside the cage is still for you to find.

The pattern extends to five cells: 15 = {1,2,3,4,5}, 16 = {1,2,3,4,6}, 34 = {4,6,7,8,9} and 35 = {5,6,7,8,9}.

Almost-unique cages and the complement trick

The next ring of sums has exactly two combinations, and they often share digits. That shared digit is a certainty even before you know which combination is right.

When a cage covers most of a house, think about what is missing instead of what is there. The digits not in the cage add up to 45 minus the cage total. A seven-cell cage of 38 inside one box leaves two cells that sum to 7, which is a far smaller search.

• 2 cells, 5: {1,4} or {2,3}. 2 cells, 6: {1,5} or {2,4}.
• 2 cells, 14: {5,9} or {6,8}. 2 cells, 15: {6,9} or {7,8}.
• 3 cells, 8: {1,2,5} or {1,3,4}, so the cage must contain 1.
• 3 cells, 22: {5,8,9} or {6,7,9}, so the cage must contain 9.
• 4 cells, 12: {1,2,3,6} or {1,2,4,5}, so it must contain 1 and 2.
• 4 cells, 28: {4,7,8,9} or {5,6,8,9}, so it must contain 8 and 9.

The rule of 45: innies and outies

Every row, column and box contains 1 to 9 once, so its cells always add up to 45. This simple fact is the most powerful technique in killer sudoku.

Pick a house, say a box. Add up the cages that lie entirely inside it. If those cages cover all but one cell, that last cell is 45 minus their total. A cell found this way is called an innie.

It works the other way too. Suppose the cages that touch a box cover all nine of its cells plus one cell outside it. Add those cages up and subtract 45: the result is the value of the cell that sticks out, called an outie.

The rule scales. Two adjacent rows add up to 90, three rows or a band of three boxes to 135. Large regions often have just one or two cells sticking in or out, even when a single box does not.

When exactly one cell sticks in and one sticks out, the rule gives you their difference rather than their values. That is still useful: if the innie must be 4 more than the outie, the innie is at least 5 and the outie at most 5, which strips half the candidates from both cells.

How classic sudoku techniques work in killer

Everything from classic sudoku still applies, and cages add a new kind of locked set.

A cage that lies inside a single box and has a known digit set behaves like a naked subset for that box. A three-cell cage of 24 in box 5 holds 7, 8 and 9, so no other cell in box 5 can hold any of them. The same holds for a row or column when the cage sits entirely in that line.

Pointing works through cages as well. If a cage must contain a 9 and all its cells lie in one row, that row's 9 is inside the cage, so the 9 can be removed from the rest of the row.

Hidden singles become easier to see once cages have narrowed the candidates. And because most killers start empty, pencil marks are not optional: write the candidate set for each unique cage straight away.

A worked line of reasoning

Here is a typical chain from the opening of a killer, step by step.

Look at box 1. Suppose the cages lying wholly inside it total 41, and the only cell not covered is r3c3, which belongs to a two-cell cage of 7 shared with r4c3 just below. By the rule of 45, r3c3 = 45 - 41 = 4. The cage total then gives r4c3 = 7 - 4 = 3.

Now follow the 3. It sits in column 3, so no other cell of column 3 can be 3. Further down, a two-cell cage of 4 covers r7c2 and r7c3. A sum of 4 in two cells can only be {1,3}. The 3 cannot go in r7c3, because column 3 already has its 3, so r7c3 = 1 and r7c2 = 3.

Keep going. The 4 in r3c3 also removes 4 from the rest of column 3. If box 7 still needs its 4 and the only free cells there are r7c1, r8c2, r8c3 and r9c3, the 4 cannot be in r8c3 or r9c3, so it is confined to r7c1 or r8c2. One more crossing line will settle it.

Notice how little arithmetic was involved: one subtraction from 45, one subtraction inside a cage, and one unique combination. The rest was ordinary sudoku.

A solving routine that works

Killer rewards a fixed order of attack. Run through it, and when a step produces a digit, go back to the top.

• Mark every unique cage with its digit set.
• Mark the almost-unique cages with the digits they must contain.
• Apply the rule of 45 to each box, then each row and column, then to two and three rows or columns together.
• Use cages with known sets as locked sets inside their box or line.
• Run classic techniques: singles, pairs, pointing, box-line reduction.
• Revisit the larger cages. Every new digit shrinks their possible combinations.

Common mistakes

Most errors in killer come from forgetting one of its two constraints.

• Repeating a digit inside a cage that spans two boxes. The no-repeat rule applies to the whole cage, not just the part in one box.
• Writing 5 into a two-cell cage of 10, or 6 into a cage of 12.
• Miscounting cells when applying the rule of 45. Count the cells of the house and the cells of the cages before you subtract.
• Ignoring outies. When a box will not give an innie, the cell that sticks out often gives the answer instead.
• Skipping pencil marks because the grid starts empty. In killer, candidates are the puzzle.