In this article
Killer Sudoku looks scary at first: an empty grid with dotted shapes and small numbers. But it uses the Sudoku rules you already know, plus one extra idea: the digits in each dotted cage must add up to the number shown. In this guide you’ll learn the Killer Sudoku rules, the cage combinations worth memorising, the famous “rule of 45”, and how to make your first moves on a real puzzle.
What is Killer Sudoku?
Killer Sudoku rules
- The classic rule: each row, column and 3×3 box contains the digits 1 to 9 exactly once.
- The cage rule: the digits in a cage add up to the small number in its top-left corner.
- No repeats in a cage: a digit can appear only once in each cage, even if the cage crosses boxes.
Here is a real Killer Sudoku. It has 29 cages, no given digits and exactly one solution.
Cage combinations cheat sheet
The heart of Killer Sudoku is knowing which digits can make a sum. Small and large sums are the most useful because they can only be made a few ways. Here is every two-cell cage:
| Sum | Digits | Sum | Digits |
|---|---|---|---|
| 3 | 1+2 | 11 | 2+9, 3+8, 4+7, 5+6 |
| 4 | 1+3 | 12 | 3+9, 4+8, 5+7 |
| 5 | 1+4, 2+3 | 13 | 4+9, 5+8, 6+7 |
| 6 | 1+5, 2+4 | 14 | 5+9, 6+8 |
| 7 | 1+6, 2+5, 3+4 | 15 | 6+9, 7+8 |
| 8 | 1+7, 2+6, 3+5 | 16 | 7+9 |
| 9 | 1+8, 2+7, 3+6, 4+5 | 17 | 8+9 |
| 10 | 1+9, 2+8, 3+7, 4+6 |
Notice that a two-cell 10 can never be 5+5, because digits can’t repeat in a cage. The real gems are the sums that can only be made one way:
| Cells | Sum → digits |
|---|---|
| 2 | 3 = 1+2 · 4 = 1+3 · 16 = 7+9 · 17 = 8+9 |
| 3 | 6 = 1+2+3 · 7 = 1+2+4 · 23 = 6+8+9 · 24 = 7+8+9 |
| 4 | 10 = 1+2+3+4 · 11 = 1+2+3+5 · 29 = 5+7+8+9 · 30 = 6+7+8+9 |
| 5 | 15 = 1+2+3+4+5 · 16 = 1+2+3+4+6 · 34 = 4+6+7+8+9 · 35 = 5+6+7+8+9 |
You don’t need to learn every combination. Remember the extremes: the smallest and largest sums for each cage size are always a single, fixed set of digits.
The rule of 45: innies and outies
The digits 1 to 9 add up to 45. So every row, every column and every box in a Killer Sudoku sums to 45. That simple fact is the most powerful tool in the game.
Look at the bottom-right box of our puzzle. Three cages (12, 21, 10) sit entirely inside it and cover eight of its nine cells. Together they add up to 12 + 21 + 10 = 43. The one remaining cell must make up the difference: 45 − 43 = 2.
A cell like this, inside the region but in a cage that pokes out of it, is called an innie. The opposite, a single cell sticking out of a region whose cages otherwise fill it, is an outie: its value is the cages’ total minus 45. The same trick works on two neighbouring rows or columns, which together sum to 90.
How to start a Killer Sudoku, step by step
With no digits on the board, where do you begin? Use this order:
- Mark the one-way cages. In our puzzle, the 3-cell 23 can only be 6, 8, 9, the 2-cell 4 can only be 1, 3 and the 3-cell 23 can only be 6, 8, 9. Pencil those digits into their cages.
- Apply the rule of 45 to every row, column and box, looking for innies and outies. That’s how we found the 2 above.
- Follow the chain. The 2 sits in a two-cell cage of 9, so its partner at row 6, column 8 must be 9 − 2 = 7.
- Combine with classic Sudoku. Once a few digits are in, use hidden singles, pairs and the other Sudoku strategies as usual. Cage combinations act like extra pencil marks.
Tips and common mistakes
| Do | Avoid |
|---|---|
| Start with the smallest and largest cage sums | Starting with a middle sum like 15 in three cells, which can be made eight ways |
| Use the rule of 45 on every box, row and column | Forgetting that two neighbouring rows sum to 90 |
| Check that no digit repeats inside a cage | Writing 5+5 in a two-cell cage of 10 |
| Pencil in possible digits for each cage | Doing all the sums in your head |
| Double-check additions before placing a digit | Trusting a quick sum: one slip breaks the whole grid |
| Mix cage logic with classic techniques | Only thinking about sums once digits are on the board |
Where Killer Sudoku comes from
Killer Sudoku comes from Japan, where it is known as samunanpure, “sum number place”. Its invention is usually credited to the Tokyo puzzle setter Tetsuya Nishio, who developed it from an idea by one of his students, Miyuki Nisawa. It first appeared outside Japan in The Times in London in 2005, during the great Sudoku craze. You may also see it called sumdoku or addoku.
The puzzle also borrows from Kakuro, another Japanese number puzzle built on sums. For the wider story of how Sudoku itself spread, read our history of Sudoku, and for other styles to try, see our guide to Sudoku variants.
Answer to the example puzzle
Frequently asked questions
Can numbers repeat in a Killer Sudoku cage?
No. A digit can appear only once in each cage, as well as once in each row, column and box. That is why a two-cell cage of 10 can be 1+9, 2+8, 3+7 or 4+6, but never 5+5.
Does Killer Sudoku have any given numbers?
Usually none. A classic Killer Sudoku starts with an empty grid; the cage sums are the only clues. A well-made puzzle still has exactly one solution.
What is the rule of 45 in Killer Sudoku?
The digits 1 to 9 add up to 45, so every row, column and box sums to 45. If the cages inside a box cover all but one cell, that cell must be 45 minus their total.
Is Killer Sudoku harder than normal Sudoku?
It usually feels harder at first because you start with no digits and need a little addition. Once you know the common cage combinations and the rule of 45, easy Killer puzzles are very approachable.
Do I need to be good at maths for Killer Sudoku?
Only simple addition up to 45. Most of the work is logic: knowing which digit sets can make a sum and combining that with the normal Sudoku rule.
Sources
- Killer sudoku Wikipedia
- Sudoku: Variants Wikipedia