Sudoku Written by Brian Hamilton and Karan Hamilton.
Brian and Karan are the puzzle lovers and game makers behind Sudoku Online Puzzles. You can learn more on the About page.
Guides are reviewed for clarity, accuracy, and usefulness for solvers. See our editorial standards, puzzle testing process, or send a correction.
Killer Sudoku hands you a grid cut into dotted shapes, each with a small number in its corner, and few digits or none. The rules fit in four lines. The hard part for a beginner is reading them on a real board: which cells a shape covers, why a digit can’t appear twice in it even across a box line, and where the first digit comes from. Every board below was dealt by the generator behind our Killer Sudoku game, and every move shown is one our solver makes on it.
What are the rules of Killer Sudoku?
Fill the 9×9 grid with the digits 1 to 9 so that:
- every row holds each digit once;
- every column holds each digit once;
- every 3×3 box holds each digit once;
- the digits in every cage add up to its total, and no digit appears twice in the same cage.
The first three are the rules of classic Sudoku, unchanged. The fourth replaces the printed clues: instead of being shown some of the digits, you are told what groups of them add up to. A total tells you which digits a cage can hold, never which of its cells each one goes in. The rows, columns and boxes settle that.
A cage is the group of cells inside one dotted outline. Its total, or sum, is the small number in its corner. A candidate is a digit a cell could still hold, usually pencilled in small until only one is left; our guide to pencil marks covers the habit.
How do you read the dotted lines and the small numbers?
Each dotted outline is one cage, and its total sits in the cage’s top-left cell: the highest one, or the leftmost of them if two share the top row. In our game each cage also gets a pale tint, picked only so that it differs from the cages touching it. The colours mean nothing else.
Three things hold on every board we deal, and on most you will meet elsewhere:
- Every cell belongs to exactly one cage. Cages never overlap and never leave a gap.
- A cage can be any joined-up shape, and it can run straight across the thick box lines. A quarter of the two-cell cages on our Easy boards do; on Expert it is nearly half of all cages larger than one cell.
- The totals add up to 405. Each of the nine rows holds 1 to 9, which comes to 45, and nine 45s are 405.
Can a number repeat in a Killer Sudoku cage?
No. A digit appears at most once in a cage, even when the cage crosses a box line and even when two of its cells sit in different rows, columns and boxes. A total that only works with a repeat, such as 5 + 5 in a two-cell cage of 10, is not a way to fill it.
Most of the time ordinary Sudoku forbids the repeat anyway: the two cells of a two-cell cage always share a row or a column. The cage rule earns its keep in bent cages of three or more cells, where two cells can share nothing at all.
How often does a cage have such a pair? We counted on 300 boards from each of our levels.
| Level | Largest cage | Cages crossing a box line | Cages with two cells sharing no row, column or box |
|---|---|---|---|
| Easy | 2 cells | 24% | none |
| Medium | 3 cells | 34% | 11% |
| Hard | 4 cells | 42% | 20% |
| Expert | 5 cells | 46% | 27% |
Percentages of the cages with two or more cells, on 300 boards per level from our generator, dealt from fixed seeds. One-cell cages are left out.
So on Easy the cage rule never tells you anything Sudoku doesn’t, while on Expert more than a quarter of the cages contain a pair of cells that only the cage rule keeps apart.
The earliest killer puzzles didn’t always spell the no-repeat rule out, and a few variants today drop it on purpose; a puzzle like that says so in its instructions. Ours never allow a repeat, and our Killer Sudoku solver rejects a cage that holds one.
Does Killer Sudoku have any given numbers?
Traditionally none: the totals are the only clues. Our boards bend that a little. The generator draws its cages over a finished grid, and wherever a cage is a single cell, its total is the digit, a given in all but name. In our game such a cage looks like any other, an empty cell with a small number in the corner, and you type the digit in yourself.
How many you get depends on the level. Easy is built from cages of one or two cells on purpose and averages about 36 one-cell cages out of 58. From Medium up the generator aims for cages of two cells or more, but one that runs out of room to grow stays at a single cell, and about six remain on a typical board. A puzzle printed elsewhere may have none, so be ready to start from a truly empty grid.
How do you start a Killer Sudoku? The first moves on a real Easy board
Go back to the Easy board in the first picture. A good opening has three steps:
- Type in the one-cell cages. That is 35 digits on this board, each one free.
- Find the totals that allow only one set of digits. For two cells those are 3 (1 + 2), 4 (1 + 3), 16 (7 + 9) and 17 (8 + 9); our Killer Sudoku combinations chart lists them for every cage size. This board has two, a 16 and a 3.
- Let the digits already placed decide the order. A pair of digits fills a two-cell cage two ways round, and the rows, columns and boxes rule one of them out.
That 7 in R5C4 is the first digit our solver places outside a one-cell cage, after only 11 of the free digits, because the two 9s that decide it are among them. Each digit placed then clears itself from its row, column, box and cage, and the board keeps opening: with all 35 free digits in, 10 cells are already down to one candidate. Every one of this board’s 81 digits goes in that way, as a naked single.
We ran our solver over 300 Easy boards. Singles and cage arithmetic, the two moves on this page, finished all 300. On 201 of them it read the cage totals once at the start and placed every digit after that as a single. With the one-cell cages in, the median board had 10 cells down to one candidate, and the leanest had one. Hidden singles, a digit with one place left in a row, column or box, hardly feature: 197 across the 300 boards, against 24,103 naked singles.
What do you need to learn after the rules?
Easy needs nothing beyond what you have just seen. The bigger cages from Medium up hide more, and three ideas carry most of the rest:
- Combinations. Which sets of different digits make each total, which digits every set needs and which digits no set can use. The combinations guide lists them all, worked out by computer rather than typed by hand.
- Digits a cage locks into a row or box. If a cage must hold a 9 and all its cells that can take one share a row, the rest of that row can’t have a 9. It is the killer form of locked candidates, and naked pairs work just as they do in classic Sudoku.
- The rule of 45. Every row, column and box adds up to 45, so the cages lying inside one tell you what its leftover cells must make. Our rule of 45 guide shows it on real boards, with how often each level needs it.
Our Killer Sudoku tips and strategies guide puts these in the order a solver reaches for them, level by level.
Stuck on a Killer Sudoku? Check these first
A beginner’s stall is usually one of four things:
- A one-cell cage left blank. In our game it looks like any other empty cell with a small number in its corner.
- A cage with one cell to go. Its last digit is the total minus the rest.
- A total read once and never again. Two cells making 13 can be 4 + 9, 5 + 8 or 6 + 7. If both cells lie in a row that already has its 9 and its 8, only 6 + 7 is left, and the cage has its digits.
- A repeat that slipped through. Two equal digits in one cage are wrong even when they make the total.
Add up its cage totals before you start. They must come to 405; if not, one was copied wrong. If they do and the puzzle still won’t move, enter it in the Killer Sudoku solver, which tells you whether it has exactly one answer. It fills the grid, though, without showing the steps.
For a new board every day, the daily Killer Sudoku comes from the same generator: Easy on Mondays and Tuesdays, Medium midweek, Hard on Fridays and Expert at the weekend. Paper solvers can print a set from our printable Killer Sudoku page.
Killer Sudoku
Four levels, every board proved to have exactly one answer. Easy starts with about 36 free digits, Expert with five or six.
▶ Play Killer Sudoku