Killer Sudoku: How It Works and How to Solve One

Killer sudoku is a sudoku variant where the grid is divided into dotted-outline "cages," each labeled with a target sum, and the digits inside a cage must add up to that sum without repeating. You still fill the 9×9 grid so every row, column, and 3×3 box holds 1–9 once each — killer sudoku just replaces most of the starting clues with arithmetic.

Most killer grids start with zero given digits. That sounds impossible the first time you see it, and then you learn two or three cage tricks and it stops being impossible almost immediately. The arithmetic never gets harder than adding single digits. What you're really learning is which sums are so restrictive that they hand you the answer.

This guide covers the rules, the combination tables worth memorizing, the rule of 45, innies and outies, and a worked opening so you can see the logic in motion.

The rules of killer sudoku

Three layers, stacked:

1. Standard sudoku rules. Every row, column, and 3×3 box contains 1–9 exactly once.
2. Cage sums. The digits in each dotted cage add up to the small number printed in its top-left corner.
3. No repeats within a cage. A cage cannot contain the same digit twice, even when its cells are in different rows, columns, and boxes.

That third rule is the one people miss, and it's doing enormous work. Without it, a two-cell cage summing to 10 could be 5+5. With it, 10 in two cells means exactly one of {1,9}, {2,8}, {3,7}, {4,6} — four options instead of five, and more importantly, four pairs of distinct digits you can reason about.

A note on variants: the no-repeat rule is standard and near-universal, but a small number of publishers allow repeats in cages that span different units. If a puzzle doesn't say so explicitly, assume no repeats.

Cages come in all shapes — L-shapes, zigzags, single cells. A one-cell cage is just a given digit written in an unusual way: a cage labeled 7 containing one square is a 7.

The two facts that make killer sudoku solvable

Before any technique, two numbers to burn into memory.

45. The digits 1 through 9 sum to 45. So every row sums to 45, every column sums to 45, and every 3×3 box sums to 45. This is the backbone of the entire game.

Cage sums at the extremes are nearly deterministic. A two-cell cage summing to 17 can only be 8+9. A three-cell cage summing to 7 can only be 1+2+4. When a puzzle hands you one of these, it's handing you a solved set of digits — you just don't know their order yet, which is often enough.

Cage combination tables

These are worth having in front of you until they're automatic. Every table below is exhaustive.

Two-cell cages (complete)

Note what's missing: there is no two-cell cage summing to 2 or 18, and none summing to 1. Also note that 3, 4, 16, and 17 each have exactly one solution — those are free digits.

Three-cell cages worth knowing

Four- and five-cell cages worth knowing

If you memorize nothing else, memorize the single-combination cages: 2-cell 3, 4, 16, 17; 3-cell 6, 7, 23, 24; 4-cell 10, 11, 29, 30; 5-cell 15, 16, 34, 35. Puzzle setters plant these deliberately as entry points, and spotting one is usually your first move.

The rule of 45, and how to actually use it

The rule of 45 says that because every row, column, and box sums to 45, you can find a missing digit by subtracting the cages you know from 45.

The mechanics are easier than the description. Two cases.

Innies

An innie is a cell inside a unit that isn't covered by any cage lying entirely within that unit.

Suppose you're looking at box 1 (rows 1–3, columns 1–3). Four cages sit completely inside it, covering eight of the nine cells, and their sums are 12, 9, 11, and 6.

`` 12 + 9 + 11 + 6 = 38 45 − 38 = 7 ``

The ninth cell — the innie — is a 7. Not "could be." Is. You've placed a digit in a grid with no given numbers, using nothing but addition.

Outies

An outie is the mirror image: a cell outside the unit, belonging to a cage that mostly sits inside it.

Suppose the cages that cover box 4 total 48 across ten cells — nine inside the box, one poking out into box 5.

`` 48 − 45 = 3 ``

The single cell sticking out is a 3.

Scaling it up

The rule of 45 doesn't stop at one unit. Two rows sum to 90. Three rows — a full band of the grid — sum to 135. An entire 3×3 band of boxes sums to 135 too, which means you can run the innie/outie trick across a third of the board at once.

This is the technique that opens most killer grids. When a puzzle looks like a wall of dotted lines and nothing else, start by outlining a box, adding up the cages, and looking for the leftover.

Cage-to-unit elimination

Here's the second big idea: a cage confined to a single row, column, or box tells you about every other cell in that unit.

Because a cage never repeats digits, and because the row it sits in also never repeats digits, the cage's possible digits are simply removed from everything else in that row.

Concrete example. A two-cell cage summing to 17 sits at R3C1 and R3C2. From the table, 17 in two cells is only 8+9. So:

• R3C1 and R3C2 are 8 and 9, in some order.
• Every other cell in row 3 loses both 8 and 9.
• Both cells are also in box 1, so every other cell in box 1 loses 8 and 9 too.

That's fourteen candidate eliminations from a single label. It works exactly like a naked pair in standard sudoku, except the cage handed it to you for free.

The same logic applies to a three-cell cage summing to 7 ({1,2,4}) sitting inside one box: 1, 2, and 4 vanish from the other six cells of that box.

The complement trick

A related move that catches people out pleasantly. A four-cell cage summing to 30 must be {6,7,8,9}. If that cage sits entirely inside box 6, then the box's other five cells hold the remaining digits — {1,2,3,4,5} — and they must sum to 45 − 30 = 15.

You've now constrained five cells you haven't even looked at.

A worked opening

Let's put it together on a plausible corner of a grid. Box 1 (rows 1–3, columns 1–3) contains these cages, all fully inside the box:

• Cage A: R1C1 + R1C2, sum 4
• Cage B: R1C3 + R2C3, sum 16
• Cage C: R2C1 + R2C2, sum 13
• Cage D: R3C1 + R3C2 + R3C3, sum 12

Step 1 — check the arithmetic. 4 + 16 + 13 + 12 = 45. The cages cover all nine cells and the total matches, which confirms the box is fully accounted for. (If it hadn't summed to 45, I'd have miscounted a cell.)

Step 2 — cash in the forced cages. Cage A sums to 4 in two cells, so it's 1+3, no alternative. Cage B sums to 16 in two cells, so it's 7+9.

Step 3 — squeeze cage C. Cage C sums to 13 in two cells: 4+9, 5+8, or 6+7. But cage B already claims the 9 and the 7 for box 1, and no digit repeats in a box. That kills 4+9 and 6+7. Cage C is 5+8.

Step 4 — cage D falls out. Box 1 has used 1, 3, 7, 9, 5, 8. The three digits left are 2, 4, 6 — and 2+4+6 = 12, which matches cage D's label exactly. Consistent.

We've gone from an empty box to knowing all nine digits, split into four known pairs and triples, using only sums. Ordering them requires the row and column constraints from neighboring boxes, but the hard part is done.

Step 5 — export the information. Cage D occupies all of row 3 inside box 1 and holds {2,4,6}. So R3C4 through R3C9 all lose 2, 4, and 6. Cage A sits in row 1 holding {1,3}, so the rest of row 1 loses 1 and 3.

That's the rhythm of killer sudoku: solve a cage, then immediately push what you learned outward into the neighboring units.

Killer sudoku vs. standard sudoku

The techniques overlap more than the tables suggest. Once cages have given you enough candidates, killer sudoku becomes ordinary sudoku — naked pairs, pointing pairs, X-Wings, all of it applies. The cages are how you get in the door.

Mistakes that cost people the most time

• Not writing candidates down. In standard sudoku you can hold an easy puzzle in your head. In this variant you cannot, because the information arrives as sets of digits rather than placements. Write the candidate list into every cell of a solved cage.
• Forgetting the no-repeat rule when a cage bends across boxes. An L-shaped cage crossing from box 2 into box 5 still can't contain two 3s, even though those two cells share no row, column, or box.
• Adding cages that don't fully cover the unit. The rule of 45 only works if you know exactly which cells your cages cover. Trace the dotted outline with a finger. A cage that pokes out by one cell is an outie, not an error — but you have to notice it.
• Chasing large cages first. A five-cell cage summing to 25 has dozens of combinations and tells you almost nothing. Work the extremes: the very small sums and the very large ones. A 3-cell cage of 7 is worth more than a 5-cell cage of 25.
• Ignoring the parity check. If your cage sums for a box total 46 instead of 45, stop. You've either misread a label or miscounted a cell, and everything after that point will be wrong.

Where to practice

Killer sudoku rewards volume more than most puzzle types, because the combination tables only become automatic through repetition. After perhaps thirty puzzles you stop looking up that 16 in two cells is 7+9 and simply see it.

If you're building up to killer sudoku and want to shore up the standard-sudoku half first, our free puzzle maker generates classic 9×9 sudoku at easy, medium, and hard, one to six per page, with an answer key on every sheet. Getting fast at crosshatching and naked pairs on plain grids makes the killer variant substantially less intimidating, because you'll only be learning one new thing instead of two.

Frequently asked questions

Is killer sudoku harder than regular sudoku?

Generally yes, but not because the arithmetic is difficult — it's addition of single digits. It's harder because it usually starts with no given numbers, so your first move has to be constructed rather than spotted. Once a few cages are resolved, the difficulty drops toward that of a hard standard sudoku. Experienced solvers often find a medium killer easier than a hard classic, because cage sums give more precise information than scattered given digits do.

Can digits repeat inside a killer sudoku cage?

No, not in the standard version. Digits cannot repeat within a cage even when the cage's cells lie in different rows, columns, and boxes. This is what makes the combination tables useful — it's why a two-cell cage summing to 10 has four possibilities rather than five. A handful of publishers use a repeats-allowed variant, but they always say so in the instructions.

What is the rule of 45 in killer sudoku?

The rule of 45 is the observation that the digits 1–9 sum to 45, so every row, column, and 3×3 box in the grid also sums to 45. You use it by adding the cages inside a unit and subtracting from 45: if cages covering eight of a box's nine cells total 38, the ninth cell is a 7. The same subtraction run in reverse finds "outies" — cells outside the unit whose value you can deduce from a cage that overhangs.

Do you need to be good at math for killer sudoku?

You need to add numbers under 45 comfortably, and that's the entire mathematical requirement. There's no multiplication, no fractions, no algebra. The skill that actually matters is combinatorial pattern recognition — knowing that 24 in three cells means {7,8,9} — and that's memorization plus practice, not mathematical talent.

What's the best first move in a killer sudoku?

Scan the whole grid for single-combination cages first: two-cell 3, 4, 16, or 17; three-cell 6, 7, 23, or 24; four-cell 10, 11, 29, or 30. Mark their digit sets straight into the cells. Then run the rule of 45 on the box or row that has the most cages fully contained inside it, looking for an innie. Between those two passes you'll almost always have a foothold.

Why is it called "killer" sudoku?

The name comes from the puzzle's British publication history in the mid-2000s, when it appeared in newspapers alongside standard sudoku and was marketed as the brutal older sibling. It's also known as sumdoku, sum sudoku, and addoku. The Japanese original, samunamupure, translates roughly as "sum number place," which is a more accurate description and a considerably less catchy one.