Foundations
Killer Sudoku & the Rule of 45
Killer Sudoku is Sudoku with cages: every row, column and 3×3 box still holds 1 to 9 once each, and the digits inside each dashed cage must add up to the small number in its corner, without repeating a digit. Your first tool is the rule of 45: a row, column or box always adds up to 1 + 2 + … + 9 = 45, so if the cages lying fully inside a box total 40, the one cell left over must be 5. Use it together with the tiny cages that have only one possible combination, such as 3 = 1 + 2 or 17 = 8 + 9.
What you need to see on the grid
These are the first things to look for on a fresh Killer grid, before you write a single pencil mark. Four terms carry the whole page:
- Cage
- A group of cells inside a dashed outline, with its sum printed in the corner. Its digits add up to that sum and are all different, even when the cage runs across two boxes. The cages cover the whole grid, so their sums are the puzzle's clues.
- Combination
- A set of different digits that fills a cage to its sum. Small and large sums have very few: two cells making 3 can only be 1 + 2, and two cells making 17 only 8 + 9. The cage sum chart lists every combination by size and sum.
- The rule of 45
- Each row, column and box holds 1 to 9 once, so its nine cells always add up to 45. Subtract the cages that lie fully inside it, and what is left belongs to the cells those cages do not cover.
- Innie
- A cell of the row, column or box that no inside cage covers, because its own cage crosses the edge. When a box has just one innie, the rule of 45 gives its digit straight away.
How to apply it, step by step
One Killer grid runs through all four steps, at the very start of the puzzle: no digit is placed yet. Every diagram shows the whole grid; the cages a step uses are outlined with their sum in the corner. The cells a step needs are drawn; the others are left blank and hatched, so you know you can ignore them.
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Step 1Know the three rules
Rule 1 is plain Sudoku: every row, column and 3×3 box holds the digits 1 to 9, each exactly once. Row 1 and box 3 below are two of the 27 rows, columns and boxes that must each end up with 1 to 9.
Rule 2 is the cage sum: the digits inside a dashed cage add up to the number in its corner. The cage drawn here covers R1C6, R1C7 and R2C7, so those three digits total 22.
Rule 3: no digit repeats inside a cage, even when the cage runs across two boxes. R1C6 and R2C7 share no row, column or box, so plain Sudoku would let both be 9 — and 9 + 4 + 9 does make 22. But they are in the same cage, so they must differ, which leaves the cage only 5 + 8 + 9 or 6 + 7 + 9.
Row 1 and box 3 in full, with the 22-cage that runs from box 2 into box 3. No pencil marks: this step only reads the rules. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 22 R2 R3 R4 R5 R6 R7 R8 R9 - Row 1 and box 3: 1 to 9, once each
- No shared row, column or box — but the same cage
- Cage, sum in its corner
- Not needed for this step
Filled in: row 1; box 3; the cage. The other 66 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Start with the cages that have only one combination
Before any pencil marks, look for the smallest and largest sums. Two different digits that add up to 3 can only be 1 and 2, so the 3-cage in R1C1 and R1C2 holds exactly 1 and 2, in some order: cross 3 to 9 out of both cells. The 17-cage just below it, in R2C1 and R3C1, works the same way: 17 is only 8 + 9.
This is the simplest case of reading a cage's combinations. Most sums have several, and the cage sum chart lists every one of them by size and sum; the Cage Combinations guide shows how to cross them with the grid. For now, pencil in the cages that give themselves away.
Technique: Cage combination elimination
Only the 3-cage matters here. Two different digits make 3 only as 1 + 2, so 3 to 9 leave both cells. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 3 R2 R3 R4 R5 R6 R7 R8 R9 - Cell of the 3-cage
- Candidates 1 and 2, the digits studied
- Candidate removed
- Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 79 cells are left blank: they are not needed for this step. Removed: 3 from R1C1 and R1C2; 4 from R1C1 and R1C2; 5 from R1C1 and R1C2; 6 from R1C1 and R1C2; 7 from R1C1 and R1C2; 8 from R1C1 and R1C2; 9 from R1C1 and R1C2.
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Step 3Add up the cages inside a box: the rule of 45
Every box holds 1 to 9 once, and 1 + 2 + … + 9 = 45. Four cages touch box 1. Three of them lie fully inside it — the 3-cage, the 17-cage and the 20-cage — and together they make 3 + 17 + 20 = 40. The fourth, the 11-cage, crosses the edge: R3C3 is in box 1, but its partner R3C4 is in box 2.
So R3C3 is the one cell of the box that no inside cage covers — its innie — and it must make up the difference: 45 − 40 = 5. Write R3C3 = 5. You did not need a single pencil mark, only the sums, which is why the rule of 45 is so often the first move of a Killer puzzle. Boxes with two innies, and the cells a crossing cage pushes outside, are the subject of the Innies & Outies guide.
Technique: Rule of 45 (innies/outies)
Box 1 and every cage that touches it, without pencil marks. The 3, 17 and 20 cages stay inside the box and make 40; the 11-cage crosses into box 2, leaving R3C3 as the innie: 45 − 40 = 5. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 3 20 R2 17 R3 115 R4 R5 R6 R7 R8 R9 - Box 1: its nine cells make 45
- The innie: 45 − 40
- Digit placed
- Cage, sum in its corner
- Not needed for this step
Filled in: box 1; the 4 cages. The other 71 cells are left blank: they are not needed for this step. Placed: R3C3 = 5.
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Step 4Follow the new digit into its cage
A digit you place in Killer works twice: in its row, column and box as usual, and in its cage. The 11-cage now holds the 5 in R3C3, so its other cell must be 11 − 5 = 6. Until now R3C4 could be 2, 3, 4, 6, 7, 8 or 9 — a 1 would need a 10 beside it, and the 5 now sits in the same row and cage. Every candidate but 6 goes, and you write R3C4 = 6.
A cage with one empty cell has a single combination left: exactly what its sum still misses. Box 1 has moved on too: with 1 and 2, 8 and 9, and the 5 accounted for, the 20-cage must hold exactly 3, 4, 6 and 7. Keep alternating the two tools — cage sums and the rule of 45 — and each new digit hands you the next.
Technique: Cage combination elimination
The 11-cage after step 3: R3C3 = 5, so R3C4 = 11 − 5 = 6. Every other candidate leaves R3C4. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 115 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- The cage's last empty cell
- Candidate 6, the digit studied
- Candidate removed
- Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 79 cells are left blank: they are not needed for this step. Removed: 2 from R3C4; 3 from R3C4; 4 from R3C4; 7 from R3C4; 8 from R3C4; 9 from R3C4.
Common mistakes
Repeating a digit in a cage that spans two boxes
A cage that runs across a box border looks like two separate pieces, and it is tempting to fill each piece by the rules of its own box. The cage still holds each digit at most once: in the 22-cage of step 1, R1C6 and R2C7 may not both be 9, although nothing else keeps them apart. Treat every cage as one unit, whatever boxes it touches.
Adding in a cage that sticks out of the box
The rule of 45 counts only cages whose every cell is inside the box. Count a crossing cage by mistake and the arithmetic breaks — sometimes visibly, like 3 + 17 + 20 + 11 = 51 for box 1, more than 45; sometimes as a believable wrong digit. Trace each dashed outline all the way round before you add its sum.
Using the rule of 45 on boxes only
Rows and columns add up to 45 too, and long, thin cages often fit inside one of them better than inside a box. When no box has a single innie, try every row and column; and when one line is not enough, the same count works across two or three neighbouring rows or columns — the subject of the Rule of 45 across Multiple Regions guide.
Keep going
- Cage CombinationsList a cage's combinations, cross them with the grid, and clear every candidate no surviving combination can use.
- Forced Cells in CagesCross a cage's combinations with its candidates until one cell has a single digit left — then place it.
- Innies & OutiesA row, column or box totals 45: take away the cages inside it, and the cells left over must make the rest.
- Rule of 45 across Multiple RegionsTwo whole rows hold 90: add up every cage touching them, and the cells sticking out make up the difference.
Every cage combination, for every size and sum: Killer Sudoku Cage Sum Chart — all combinations
Learn the move, not just the digit
In Killer mode, the hint in the app shows the next cell and explains the move behind it rather than just filling it in: on the grid above, once R3C3 = 5, it points to R3C4 and works out what the 11-cage still needs, 11 − 5 = 6.