Killer sudoku

Killer sudoku rules

Killer sudoku is sudoku with added sums. The grid and the basic rules are the same: fill every row, column and 3×3 box with the digits 1 to 9, once each. The difference is that the grid is covered with cages: groups of cells drawn with a dashed outline, each with a small number in the corner. That number is the sum of the digits in the cage.

• Every row, column and box contains 1–9 exactly once, as usual.
• The digits in each cage add up to the cage’s sum.
• A digit cannot repeat inside a cage.

Most killer puzzles start with an empty grid: every cell belongs to exactly one cage, and the sums are all the information you get. Some easier versions include a few givens. A good killer puzzle, like a regular sudoku, has exactly one solution.

Why the sums are enough

A cage of two cells with the sum 3 can only be 1 and 2, because no other pair of different digits adds to 3. A cage of two cells with the sum 10 could be 1 and 9, 2 and 8, 3 and 7, 4 and 6. Most of killer sudoku is about narrowing down those combinations, using the cage sums together with the normal sudoku rules. The combinations cheat sheet lists every possibility for every cage size, so you do not have to work them out by hand.

How to start: cages with only one combination

Some cage sums can be made in only one way. These are the best places to start, because they tell you exactly which digits go in the cage (though not yet in which order). The smallest and largest sums for each cage size are always unique:

• 2 cells: 3 = 1+2, 4 = 1+3, 16 = 7+9, 17 = 8+9
• 3 cells: 6 = 1+2+3, 7 = 1+2+4, 23 = 6+8+9, 24 = 7+8+9
• 4 cells: 10 = 1+2+3+4, 11 = 1+2+3+5, 29 = 5+7+8+9, 30 = 6+7+8+9
• 5 cells: 15 = 1+2+3+4+5, 16 = 1+2+3+4+6, 34 = 4+6+7+8+9, 35 = 5+6+7+8+9

For example, a 3-cell cage of 7 must contain 1, 2 and 4. If two of its cells are in the same row, then that row’s 1, 2 and 4 are partly accounted for, and the remaining digit sits in the third cell. Write the combination as notes in each cell of the cage, then use rows, columns and boxes to sort out the order.

The 45 rule

The digits 1 to 9 add up to 45. Every row, column and box contains each digit once, so each of them sums to 45. That simple fact is the most powerful tool in killer sudoku.

Suppose the cages that lie completely inside a box add up to 38, and one more cell of the box belongs to a cage that sticks out. That cell must be 45 − 38 = 7. Cells that belong to the unit but sit in cages that cross its border are called innies; cells of those cages that lie outside the unit are outies. The 45 rule works for any group of whole units too: two adjacent rows sum to 90, three to 135, and a whole band of three boxes to 135.

A worked example

Imagine a box with four cages. Three of them lie completely inside it: a 2-cell cage of 3, a 3-cell cage of 24 and a 3-cell cage of 12. That accounts for eight cells, and their sums add up to 3 + 24 + 12 = 39. The ninth cell belongs to a cage that crosses into the next box, so its cage sum does not help directly. But the whole box must add up to 45, so the ninth cell is 45 − 39 = 6. You can check that this fits: the 3-cage holds 1 and 2, the 24-cage holds 7, 8 and 9, and the only three remaining digits other than 6 are 3, 4 and 5, which do add up to 12.

Combinations and overlaps

When a cage has several possible combinations, look at which digits are common to all of them. A 3-cell cage of 15 can be 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6. Every combination that leaves out 5 contains a 4 or a 6, so if the cage’s row already contains both a 4 and a 6 elsewhere, the cage must contain the 5. Removing combinations that clash with digits already placed is called combination elimination, and the combinations lookup lets you filter by digits that must or must not appear.

Another useful fact: a cage that sits entirely inside one row, column or box removes its digits from the rest of that unit, once you know them. A 2-cell cage of 3 inside a box means no other cell of that box can be 1 or 2. That is the killer equivalent of a naked pair.

Tips for solving killer sudoku

• Fill in every unique cage first, as notes. Then look at boxes where a few cages fit completely inside.
• Apply the 45 rule to every row, column and box that has a single innie or outie. Do the same for two or three adjacent units.
• Use complements: the digits of an 8-cell cage are 1–9 minus one digit, and that missing digit is 45 minus the cage sum. A 7-cell cage of 45 − x has the same missing digits as a 2-cell cage of x.
• Keep the regular sudoku techniques in play. Once digits appear, hidden singles and pointing pairs work exactly as in a normal puzzle.
• Remember that a cage can cross box borders, but its digits still may not repeat.
• Check your additions. Most killer mistakes come from a miscounted 45-rule sum, not from bad logic.

Is killer sudoku maths?

A little. You need small additions and subtractions, and it helps to know a few combinations by heart. But the core of the puzzle is still logic: every digit is placed because there is no other option. If you can solve a medium sudoku, you have the skills to start on killer puzzles; the combinations table fills in the arithmetic.

Playing killer sudoku on NineByNine

Playable killer puzzles are not available on NineByNine yet; they are planned as a later update. Until then, the killer sudoku combinations cheat sheet works fully: it lists every cage combination by size and sum and lets you look up a cage in a second. For regular sudoku, play unlimited puzzles or try today’s daily puzzle.