Learn how to play Killer Sudoku: cage combinations, the 45 rule with worked examples, innies and outies, and a step-by-step solving workflow for beginners.
Killer Sudoku looks intimidating the first time you open a puzzle: an empty 9×9 grid, no starting numbers anywhere, just dotted outlines with little sums printed in their corners. Most people who bounce off this variant don't lack math skills — they lack a starting routine. This guide gives you one: the three rules, the handful of cage combinations worth memorizing, the 45 rule that does the heavy lifting, and a step-by-step workflow that tells you exactly what to try next when you get stuck.
If you've never solved classic Sudoku before, spend a week or two there first. Killer Sudoku is built on top of those rules, and everything below assumes you know them.
What Is Killer Sudoku? (Rules in 30 Seconds)
Killer Sudoku — also called Sumdoku or Addoku — was created by Japanese puzzle designer Tetsuya Nishio in the late 1990s and spread worldwide through publications like The Times of London. It uses the same 9×9 grid as classic Sudoku, but instead of pre-filled digits, the grid is carved into irregular groups of cells called cages, each marked with a small target sum.
Three rules govern every puzzle:
- Standard Sudoku rules apply. Each row, column, and 3×3 box must contain the digits 1–9 exactly once.
- Cage sums must match. The digits inside each cage must add up to the number printed in its corner.
- No repeats inside a cage. Even if a cage spans two rows where a repeated digit would otherwise be legal, the same digit cannot appear twice within one cage.
The third rule is the one beginners forget, and it's quietly the most important. It's the difference between Killer Sudoku and a simple arithmetic exercise — it dramatically cuts down which digit combinations are even possible.
Notice what's missing: givens. A well-constructed Killer Sudoku usually starts completely blank. The cage sums alone pin the grid down to exactly one solution, reachable through pure logic.
Cage Combinations You Should Memorize
Because digits cannot repeat inside a cage, many cage sums have only one possible set of digits. These "locked" cages are where every solve begins. The extremes lock up automatically: a cage can't contain anything smaller than 1+2 or larger than 8+9, so sums at either end of the range have no alternatives.
Here are the combinations that pay off most often:
| Cage size | Sum | Only possible digits |
|---|---|---|
| 2 cells | 3 | {1, 2} |
| 2 cells | 4 | {1, 3} |
| 2 cells | 16 | {7, 9} |
| 2 cells | 17 | {8, 9} |
| 3 cells | 6 | {1, 2, 3} |
| 3 cells | 7 | {1, 2, 4} |
| 3 cells | 23 | {6, 8, 9} |
| 3 cells | 24 | {7, 8, 9} |
| 4 cells | 10 | {1, 2, 3, 4} |
| 4 cells | 30 | {6, 7, 8, 9} |
If you memorize nothing else, learn the four 2-cell combinations. They show up constantly, and each one instantly gives you a candidate pair to work with.
One example of how a locked cage becomes a real placement: suppose a 2-cell cage sums to 17, so you know it holds {8, 9}. Now look at where those two cells sit. If one of them shares a row with a placed 9 (or a column with one), that cell can't be 9 — so it must be 8, and its partner takes the 9. You didn't guess; you intersected the cage constraint with the standard row/column/box rules. That intersection is the whole game.
The 45 Rule: Your Most Powerful Weapon
Here's the arithmetic engine behind Killer Sudoku: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. Every complete row, column, and box contains exactly those digits, so every row, column, and box sums to exactly 45. Solvers call each of these units a house.
The 45 rule exploits a simple observation: if the cages covering most of a house have known sums, the missing piece is just 45 minus what you've accounted for.
Worked example. Imagine one 3×3 box where the cages sitting entirely inside it sum to 12, 15, and 10 — a total of 37. The whole box must equal 45, and there's exactly one cell not covered by those cages. That cell must contain 45 − 37 = 8. One subtraction, one digit placed, on a grid where you otherwise haven't confirmed anything yet.
The same logic handles cages that straddle a house boundary:
- An innie is a cell inside a house that isn't covered by the cages you're tallying. Innie value = 45 − (sum of fully contained cages).
- An outie is a cell that sticks out of a house from a cage that mostly sits inside it. Outie value = (total cage sum) − 45.
Boxes are usually the best place to hunt for these, not rows or columns. Cage outlines tend to stay locally compact, so a single box is more often covered by whole cages plus one or two protruding cells — exactly the geometry the 45 rule feeds on. Rows and columns work too; the rule even scales (two full rows total 90, three total 135) once you're comfortable with the basics.
A Step-by-Step Killer Sudoku Solving Workflow
Strong Killer Sudoku solvers don't scan randomly. They run the same loop every puzzle, in the same order, until something breaks open. Steal this routine:
- Scan for locked cages. Mark candidates for every cage with a unique combination (the table above), plus any cage whose options have already been narrowed by the grid.
- Check one-cell remainders. Any cage where all but one cell is already solved is free money — subtract to get the last digit.
- Run the 45 rule on every box. Add up the cages in each 3×3 box and look for innies and outies. Then do rows and columns.
- Update candidates. Every placement and every narrowed cage changes the options around it. Cross out digits in affected rows, columns, boxes, and cages.
- Apply classic techniques. Naked singles, hidden singles, naked pairs, pointing pairs — everything from standard Sudoku works unchanged here. Cage-derived candidates often create pairs that pure Sudoku logic wouldn't have spotted.
- Repeat. Each pass is faster than the last because the grid is denser with information.
Two habits keep the loop honest. First, after placing a digit, update four things: its row, its column, its box, and every cage it touches. Many wasted restarts come from updating the Sudoku side but forgetting that a cage's remaining sum has changed. Second, when you subtract from a partially solved cage, say the result out loud with its cell count — "this is now 11 in two cells" — instead of scribbling a bare number. That small verbal check prevents arithmetic drift, the slow error that poisons everything downstream.
And when you're genuinely stuck? Change the question. Stop asking "what is this cage?" and ask "what can this cage no longer be?" A placed 5 in the same row kills the {5, 9} option in any 2-cell cage sharing that row. Narrowing a cage from four combinations to two is real progress, even when nothing gets filled in yet. If a whole grid stays frozen, look for the area where the most constraints overlap — a cage crossing two nearly-complete houses is worth far more attention than an open corner. What you should never do is guess. Well-formed Killer Sudoku puzzles are designed to solve by logic alone; the feeling of "I'll just try an 8 here" almost always means an earlier subtraction went wrong.
5 Common Killer Sudoku Mistakes (and How to Avoid Them)
1. Forgetting the no-repeat rule. A 2-cell cage summing to 10 is not 5+5 — it's {1,9}, {2,8}, {3,7}, or {4,6}. Always run combination checks through the no-repeat filter first.
2. Over-noting at the start. Writing every possible combination into every cage before using any grid logic turns the puzzle into visual soup. Note only forced digits and narrow cages; wait for the grid to earn fuller notes.
3. Ignoring rows, columns, and boxes. Beginners stare at cage math so hard they stop scanning houses. Killer Sudoku is still Sudoku first — the cage rule adds pressure, it doesn't replace classic constraints.
4. Letting cage arithmetic go stale. After a digit is placed inside a 3-cell cage summing to 15, the cage is now a 2-cell cage summing to (15 − that digit). Recompute immediately, not five moves later.
5. Treating combinations as ordered. A cage sum of 3 tells you the cells hold {1, 2} — not which cell is which. Order comes later, from row, column, and box elimination.
Killer Sudoku vs. Classic Sudoku: Which Should You Play First?
| Feature | Classic Sudoku | Killer Sudoku |
|---|---|---|
| Starting digits | Pre-filled clues (17–45 givens) | Usually none |
| Extra constraint | None | Cage sums + no in-cage repeats |
| Entry point | Scanning for singles | Locked cages + the 45 rule |
| Math required | None | Basic addition and small combinations |
| Difficulty curve | Easy to Expert | Medium to Very Hard |
Classic Sudoku remains the better training ground: it teaches the scanning and elimination instincts that Killer Sudoku assumes you have. But don't fear the jump. Killer Sudoku's cage sums add constraints as well as difficulty, and plenty of solvers find an easy Killer puzzle friendlier than an expert classic one — the sums tell you things that givens never would. The honest answer: switch once classic Medium puzzles stop feeling like work.
FAQ
Can numbers repeat in a Killer Sudoku cage? No. Digits must be unique within each cage, even across row or column boundaries where a repeat would normally be legal. This rule is what makes combination tables possible.
Do I need to be good at math? You need addition and the ability to list small combinations of the digits 1–9. After a few puzzles you'll recognize {8, 9} in a 17-cage as a pattern, not a calculation.
What's the best first move in Killer Sudoku? Mark the locked cages — unique combinations from the table — then run the 45 rule across all nine boxes looking for innies and outies. Those two scans generate almost every strong opening.
Is Killer Sudoku harder than regular Sudoku? Generally yes for beginners, because it layers arithmetic over standard logic. But the extra constraints also provide extra information, so experienced solvers often find it more interesting rather than strictly harder.
Ready to Try One?
The fastest way to make all of this stick is to solve an easy Killer Sudoku right now and run the workflow deliberately: locked cages, one-cell remainders, 45 rule, then classic scanning. You can play Killer Sudoku online free on iQiQGame — no download, no signup — and it's a good idea to warm up with classic Sudoku if your scanning feels rusty. Start on easy difficulty, keep your notes clean, and let the sums do the talking.
