Killer Sudoku: rules, cage combinations and free puzzles

Killer sudoku swaps given digits for arithmetic. The grid is divided into dashed cages, each with a small total in its corner, and you work out the digits from the sums alone. Play killer sudoku online free here, with hints that teach the next step.

Hoot’s killer puzzles start with very few digits, often none at all, so the cages do all the talking. Two tools carry you a long way: knowing which totals can only be made one way, and the rule of 45.

The rules

• Normal sudoku rules apply: 1 to 9 once in every row, column and 3×3 box.
• The digits in each dashed cage add up to the total in its top-left corner.
• A digit cannot repeat within a cage, even when the cage spans more than one box.
• Every cell belongs to exactly one cage.

How to start

Killer sudoku cage combinations

Most totals can be made in several ways, but the smallest and largest totals for each cage size are forced. Learn these few and you will spot them instantly:

So a two-cell cage of 17 is always 8 and 9, and a three-cell 6 is always 1, 2 and 3. Our complete combination table lists every forced total from two cells to nine.

Use the rule of 45

Every row, column and box contains 1 to 9, which add up to 45. Add up the cages that sit entirely inside a house; whatever is missing must be in the cells left over.

The same idea scales up: two rows total 90, three boxes 135. A cell that pokes out of a region is an outie; a cell left inside is an innie. Finding them is the heart of advanced killer solving.

Then think about what each cage cannot hold

A two-cell cage of 10 can never contain a 5 (that would need 5 + 5). A two-cell 9 cannot contain a 9. Ruling digits out like this is often enough to start a hidden single in a busy box.

Common mistakes

• Allowing a repeat in a cage, such as making 4 from 2 + 2. Digits never repeat inside a cage.
• Forgetting that cages crossing box borders still can’t repeat a digit.
• Miscounting which cells are inside a region when using the rule of 45. Shade them if it helps.
• Assuming two different cages can’t share a digit. They can, as long as sudoku rules allow.

How the ranks scale difficulty

Every puzzle type on Hoot’s Puzzles comes in four ranks, named after the stages of a mathematical idea. Lemma is the warm-up: about a dozen more digits are given than logic strictly needs, and singles plus the variant’s own rule will carry you through. Proposition needs a real idea or two, typically pointing and claiming. Theorem is near-minimal, so almost every clue is load-bearing and you may need pairs. Conjecture gives the fewest digits and sparser rule clues: expect to think hard.

In killer sudoku rank also controls cage size. Lemma cages are mostly two or three cells, where the combinations are few; at Conjecture they can stretch to six cells, where the arithmetic gets far more interesting.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Keep exploring

• Little Killer SudokuOutside clues add up the diagonals their arrows point along.
• Sandwich SudokuOuter clues sum the digits trapped between the 1 and the 9.
• CalcudokuA Latin square with arithmetic cages: + − × ÷. Sizes 5×5 to 8×8.
• Solving techniquesSingles, pairs, cage arithmetic, line logic and Latin squares, all in one guide.
• All puzzle guidesEvery puzzle type on Hoot’s Puzzles, with rules and tips.