Killer Sudoku

The puzzles here start with very few given digits, between none and ten, so the cage sums do most of the work. There are three levels, Easy, Medium and Hard, and cage arithmetic is used in every stored puzzle.

How do cages change the puzzle?

Each cage is a small connected group of cells, up to five on this site, with a small number in its corner showing the total. The digits in the cage must add up to that total, and they must all be different. A cage of one cell simply tells you its digit. The ordinary Sudoku rules still hold across the board, so a cage can also be constrained by the row, column and box that its cells sit in. See the rules page for the exact statement and the controls page for input.

Here, cages come from the solved grid: the generator divides a finished grid into groups of adjacent cells holding different digits and then computes each cage’s sum. That guarantees the puzzle has at least one solution, and a solution counter then confirms it has exactly one.

Why do the puzzles have so few givens?

In the stored puzzles the number of givens runs from 0 to 10 and averages 3.1 on Easy, 3.6 on Medium and 3.4 on Hard. Thirteen of the 270 stored puzzles have none at all. The information is in the cage sums, so you start by working out what each cage can hold. A puzzle that begins with an empty grid is not unfair, because the sums are enough to force the solution.

Which combinations are forced?

The first skill in killer Sudoku is knowing which sums have only one possible set of digits. A few are worth memorizing:

• Two cells that sum to 3 must be 1 and 2, and two cells that sum to 17 must be 8 and 9.
• Three cells that sum to 6 must be 1, 2 and 3, and three cells that sum to 24 must be 7, 8 and 9.
• A cage of two cells that sums to 4 must be 1 and 3, and one that sums to 16 must be 7 and 9.

Beyond these, a sum has several possible sets, and the cage’s position narrows them down. If a three-cell cage with a sum of 7 can only be 1, 2 and 4, then each of those digits is confined to the cage’s cells, which removes them as candidates from other cells in a shared row or box.

What techniques does the board use?

The solver has two cage techniques alongside the usual ones. In the stored sample:

• Cage sums is used in every puzzle at every level. It removes candidates that no combination of digits for the whole cage could use.
• Innie and outie sums appears at Hard, in 86% of the 70 Hard puzzles. It rests on the fact that every row, column and box adds up to 45. If the cages inside a house leave a cell or two sticking in or out, that small group has a known sum.

The ordinary techniques also appear. Pointing pairs turn up in nearly every Medium and Hard puzzle, and at Hard, naked pairs are used in 83% of puzzles and box/line reduction in 80%. Singles are needed everywhere. The lessons on naked pairs and pointing pairs apply on this board unchanged.

How many puzzles and levels are there?

The site stores 100 Easy, 100 Medium and 70 Hard killer puzzles. There is no Expert or Evil level, because the generator’s killer puzzles did not reach those techniques. Each has exactly one solution, checked before every release. Because digits matter to the sums, the game varies killer puzzles by moving rows, columns and cages around but never by relabeling digits.

A way to start a killer puzzle

1. Scan for one-cell cages and fill them in.
2. Find the two- and three-cell cages with forced combinations and write those digits into the cells as notes.
3. Use the 45 total: if the cages lying entirely inside a box cover all but one of its cells, that cell’s digit is 45 minus the sums of those cages. This is the simplest innie.
4. Apply the cage-sum rule to every other cage, removing candidates that no combination allows.
5. Only then look for ordinary patterns such as pairs.

The pencil marks guide will help you keep the candidates tidy, since killer puzzles lean heavily on notes.

Where does killer fit?

Killer is a good second board once you can finish classic Easy and Medium. If you want a different extra rule, try diagonal, anti-knight or the jigsaw regions of irregular Sudoku. If you want to understand the grading used here, read how Sudoku difficulty is graded, and return to the games hub for the other boards.