1 of 3
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Level 2 · Sums and pairs
How to solve killer sudoku
A dashed cage adds up to its small clue, and digits never repeat inside it.
In killer sudoku, dashed outlines called cages group cells together. The small number in a cage's corner is the total of its digits, and no digit can repeat inside a cage. Normal sudoku rules still apply to every row, column and box.
Many killer puzzles start with few or no given digits, so the sums do the work. You solve by knowing which digit sets make each total and by adding up whole rows and boxes.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits in a cage sum to the small clue in its corner and can’t repeat.
- Killer cage rules apply.
- Numbers inside a dashed outline add up to the total shown.
Worth memorizing
- Some cages have only one combination. Two cells: 3 is 1+2, 4 is 1+3, 16 is 7+9, 17 is 8+9. Three cells: 6 is 1+2+3, 24 is 7+8+9.
- No repeats means no 5+5. A two-cell cage of 10 can’t hold a 5.
- Every full row, column and box adds up to 45. Solvers call this the 45 rule, and it cracks most killers.
Solving tips
Learn the one-way cages
Some totals can be made only one way. A two-cell 3 is 1 + 2, a two-cell 17 is 8 + 9, a three-cell 6 is 1 + 2 + 3 and a three-cell 24 is 7 + 8 + 9. Mark those digits in the cage right away.
Use the 45 rule
Every row, column and box adds up to 45. If the whole cages inside a box cover eight of its cells and total 38, the last cell is 45 - 38 = 7.
Find innies and outies
Sometimes the cages that cover a box also cover one cell outside it. If those cages total 52, the outside cell (the outie) is 52 - 45 = 7. An innie works the same way for a cell left uncovered inside the box.
No repeats cuts combinations
A two-cell cage of 10 can't be 5 + 5, so 5 is out. A two-cell 4 can only be 1 + 3, since 2 + 2 would repeat.
A worked example: the 45 rule in one box
We name cells by row and column, so row 1, column 1 is r1c1. In the top-left box, a two-cell cage of 17 covers r1c1 and r1c2, and r1c3 belongs to a cage that runs out of the box. The middle row of the box is a three-cell cage of 6, and the bottom row is a three-cell cage of 15.
- The cage of 17 in two cells can only be 8 + 9. Write 89 in the center of both cells.
- The cage of 6 in three cells can only be 1 + 2 + 3. The middle row of the box holds 1, 2 and 3 in some order.
- Use the 45 rule. The box adds to 45, and the three cages fully inside it add to 17 + 6 + 15 = 38. The one cell left over, r1c3, is 45 - 38 = 7.
- Now the bottom cage. A three-cell 15 has eight possible sets, but 1, 2, 3, 7, 8 and 9 are already used in this box. Only 4, 5 and 6 remain, and 4 + 5 + 6 = 15. The bottom row holds 4, 5 and 6.
- The 7 at r1c3 also helps the cage that runs out of the box. Subtract 7 from that cage's total, and the rest of it becomes a smaller cage with a new, shorter list.
Practice
2 of 3
What goes in the last cell?
3 of 3
Tap every digit that could go here.
Select all that apply
Reference table
| Cells | Total | Only combination |
|---|---|---|
| 2 | 3 | 1 + 2 |
| 2 | 4 | 1 + 3 |
| 2 | 16 | 7 + 9 |
| 2 | 17 | 8 + 9 |
| 3 | 6 | 1 + 2 + 3 |
| 3 | 7 | 1 + 2 + 4 |
| 3 | 23 | 6 + 8 + 9 |
| 3 | 24 | 7 + 8 + 9 |
| 4 | 10 | 1 + 2 + 3 + 4 |
| 4 | 11 | 1 + 2 + 3 + 5 |
| 4 | 29 | 5 + 7 + 8 + 9 |
| 4 | 30 | 6 + 7 + 8 + 9 |
Going deeper
How to count the other combinations
Two tricks tell you how many ways a cage can be filled. First, totals mirror. Swap every digit d for 10 - d, and a cage of n cells totaling T becomes one totaling 10 times n minus T. A two-cell 5 (1 + 4 or 2 + 3) mirrors a two-cell 15 (9 + 6 or 8 + 7), so both have two combinations.
Second, a cage mirrors the digits it leaves out. A cage of n cells totaling T leaves 9 - n digits that total 45 - T. A five-cell 15 has one combination, 1 + 2 + 3 + 4 + 5, because the four digits left over must total 30, which is only 6 + 7 + 8 + 9. The same logic gives one combination for five-cell totals of 16, 34 and 35.
Counts peak in the middle. Two-cell totals of 9, 10 and 11 have 4 combinations each. Three-cell totals of 14, 15 and 16 have 8 each. The four-cell total of 20 has 12, the most of any four-cell cage.
Cages with two combinations
The next easiest cages have exactly two options. Two cells: 5 is 1 + 4 or 2 + 3, 6 is 1 + 5 or 2 + 4, 14 is 5 + 9 or 6 + 8, and 15 is 6 + 9 or 7 + 8. Three cells: 8 is 1 + 2 + 5 or 1 + 3 + 4, and 22 is 5 + 8 + 9 or 6 + 7 + 9. Four cells: 12 is 1 + 2 + 3 + 6 or 1 + 2 + 4 + 5, and 28 is 4 + 7 + 8 + 9 or 5 + 6 + 8 + 9.
Look for the digits both options share. A three-cell 8 always holds a 1. A three-cell 22 always holds a 9. A four-cell 12 always holds 1 and 2, and a four-cell 28 always holds 8 and 9. You can mark those digits in the cage before you know the rest.
The 45 rule across several houses
The 45 rule scales up. Two whole rows add to 90, and three boxes in a band add to 135. Add every cage that sits fully inside the area, subtract from the total, and what's left is the sum of the cells that stick out or stay uncovered.
When two cells are left over, their sum works like a hidden cage. If they share a row, column or box, they can't repeat, and the combination lists apply. If they share no house, they can be the same digit.
Common mistakes
Using half of an even total
A two-cell cage can never hold half its total, since that would repeat a digit. A 10 can't hold a 5, a 12 can't hold a 6, and a 14 can't hold a 7.
Counting cages that stick out
The 45 rule only uses cages that sit fully inside the row, column or box. If a cage pokes out, its total includes cells outside the area, and you have to treat those cells as outies.
Deciding the order too early
Knowing a cage is 8 + 9 tells you the set. It doesn't tell you which cell is 8. Use rows, columns and other cages to settle the order.
In mixed-rule puzzles
Killer cages mix well with other sum rules. A cage that overlaps an arrow or a little killer diagonal gives two sums for some of the same cells, and subtracting one from the other can isolate a single cell. Kropki dots and XV marks inside a cage cut its list of combinations. An X inside a three-cell cage of 17 means the third cell is 7. In a killer with sandwich clues, small and large cages help you place the 1s and 9s, which fix where each sandwich starts and ends. Some setters drop the no-repeat rule for cages, so check the puzzle's rules.
What the video shows
- The small number is the cage total. These three digits add up to 6.
- 2 + 2 + 2 is 6, but digits can’t repeat inside a cage.
- 1 + 3 + 2 = 6 with no repeats. Here the cage must hold 1, 2 and 3.
Questions solvers ask
What is the 45 rule in killer sudoku?
Every row, column and 3x3 box holds 1 to 9 once, so its digits always add up to 45. Subtract the cage totals you know from 45 to find the cells left over.
Can numbers repeat in a killer sudoku cage?
No. Digits inside one cage never repeat, even when the cage covers cells in more than one box.
What does the small number in a killer sudoku cage mean?
It is the total of all the digits inside that dashed cage.
How is killer sudoku different from regular sudoku?
Killer sudoku adds cages with sum clues on top of the normal rules. Many killer puzzles give few or no starting digits.
Is there a killer sudoku combinations chart?
Yes. The table on this page lists every cage of 2, 3 or 4 cells with only one combination. For other totals, use the mirror trick. A cage of n cells totaling T has as many combinations as one totaling 10 times n minus T.
How many combinations does a 3-cell cage of 15 have?
Eight: 1 + 5 + 9, 1 + 6 + 8, 2 + 4 + 9, 2 + 5 + 8, 2 + 6 + 7, 3 + 4 + 8, 3 + 5 + 7 and 4 + 5 + 6. Three-cell totals of 14 and 16 also have eight each.
What does a killer cage with no number mean?
Some killer puzzles include cages without a total. The digits still can't repeat, but the sum is unknown. Check the puzzle's rules to confirm.
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