Killer Sudoku is one of the most popular and rewarding Sudoku variants, combining the familiar logic of classic Sudoku with arithmetic-based cage constraints. If you enjoy standard Sudoku and want a deeper challenge that exercises both your logical reasoning and mental math, Killer Sudoku is the natural next step. This comprehensive guide covers the rules, essential cage sum combinations, and strategies from beginner to advanced level.
What Is Killer Sudoku?
Killer Sudoku (sometimes called Sum Sudoku) is a puzzle played on the standard 9×9 Sudoku grid divided into nine 3×3 boxes. Like classic Sudoku, every row, column, and box must contain the digits 1 through 9 exactly once. The critical difference is that the grid is also divided into cages—groups of cells outlined by dotted lines—each with a target sum displayed in the corner.
The key rules are:
- Standard Sudoku rules apply. Each digit 1–9 appears exactly once in every row, column, and 3×3 box.
- Cage sum rule. The digits within each cage must add up to the cage’s target number.
- No repeats in cages. A digit cannot appear more than once within a single cage, even if the cells are in different rows, columns, or boxes.
- No given digits. Unlike standard Sudoku, Killer Sudoku puzzles typically start with an empty grid—all information comes from the cage constraints.
How Killer Sudoku Differs from Classic Sudoku
| Feature | Classic Sudoku | Killer Sudoku |
|---|---|---|
| Grid Size | 9×9 | 9×9 |
| Given Digits | Yes (17–40+) | Usually none |
| Cage Sums | No | Yes |
| Arithmetic Required | None | Addition |
| No-Repeat Rule | Rows, columns, boxes | Rows, columns, boxes, AND cages |
| Typical Difficulty | Easy to Evil | Medium to Expert |
| Average Solving Time | 5–60+ min | 15–90+ min |
The absence of given digits means you start every Killer Sudoku with a completely blank grid. All your information comes from the cage sums and the standard Sudoku constraints working together.
Cage Sum Combinations
Memorizing the possible digit combinations for common cage sums is essential for solving Killer Sudoku efficiently. Below are the combinations for cages of 2 to 5 cells.
2-Cell Cage Combinations
| Sum | Possible Combinations |
|---|---|
| 3 | {1,2} |
| 4 | {1,3} |
| 5 | {1,4} {2,3} |
| 6 | {1,5} {2,4} |
| 7 | {1,6} {2,5} {3,4} |
| 8 | {1,7} {2,6} {3,5} |
| 9 | {1,8} {2,7} {3,6} {4,5} |
| 10 | {1,9} {2,8} {3,7} {4,6} |
| 11 | {2,9} {3,8} {4,7} {5,6} |
| 12 | {3,9} {4,8} {5,7} |
| 13 | {4,9} {5,8} {6,7} |
| 14 | {5,9} {6,8} |
| 15 | {6,9} {7,8} |
| 16 | {7,9} |
| 17 | {8,9} |
3-Cell Cage Combinations
| Sum | Possible Combinations |
|---|---|
| 6 | {1,2,3} |
| 7 | {1,2,4} |
| 8 | {1,2,5} {1,3,4} |
| 9 | {1,2,6} {1,3,5} {2,3,4} |
| 10 | {1,2,7} {1,3,6} {1,4,5} {2,3,5} |
| 11 | {1,2,8} {1,3,7} {1,4,6} {2,3,6} {2,4,5} |
| 12 | {1,2,9} {1,3,8} {1,4,7} {1,5,6} {2,3,7} {2,4,6} {3,4,5} |
| 23 | {6,8,9} |
| 24 | {7,8,9} |
4-Cell Cage Combinations (Selected)
| Sum | Possible Combinations |
|---|---|
| 10 | {1,2,3,4} |
| 11 | {1,2,3,5} |
| 12 | {1,2,3,6} {1,2,4,5} |
| 13 | {1,2,3,7} {1,2,4,6} {1,3,4,5} |
| 29 | {2,8,9,10} — wait, max is 9! → {3,8,9,…} → {5,7,8,9} |
| 30 | {6,7,8,9} |
Note: For a complete table, the minimum sum for a 4-cell cage is 10 (1+2+3+4) and the maximum is 30 (6+7+8+9).
5-Cell Cage Combinations (Extremes)
| Sum | Possible Combinations |
|---|---|
| 15 | {1,2,3,4,5} |
| 35 | {5,6,7,8,9} |
For 5-cell cages, sums range from 15 to 35, with many possible combinations in between. Most solvers reference a lookup table or app for mid-range 5-cell cage sums.
The 45 Rule
The 45 rule is the single most powerful technique specific to Killer Sudoku, and it derives from a simple fact: the digits 1 through 9 sum to 45. Therefore, every complete row, column, and 3×3 box must have digits that total exactly 45.
This means if you know the sum of all cages within a row except one cell (or a small group of cells), you can calculate the value of those unknown cells by subtracting the known cage sums from 45.
Example: If a row contains three complete cages summing to 12, 15, and 11 (totaling 38), and one cell remains outside those cages, that cell must contain 45 − 38 = 7.
The 45 rule extends to combinations of rows, columns, and boxes. For two rows, the total is 90. For a row plus a box, you can subtract overlapping cages. Mastering these calculations is essential for higher-level Killer Sudoku solving. For a deeper exploration, see our dedicated article on the 45 rule in Sudoku.
Basic Strategies for Beginners
Start with Extreme Cage Sums
Cages with the minimum or maximum possible sum for their size have only one combination. A 2-cell cage summing to 3 must be {1,2}. A 2-cell cage summing to 17 must be {8,9}. A 3-cell cage summing to 6 must be {1,2,3}. These locked combinations immediately restrict candidates and give you a starting foothold.
Use the No-Repeat Rule Aggressively
Remember that digits cannot repeat within a cage. If a 3-cell cage sums to 15, the possible combinations are {1,5,9}, {2,4,9}, {2,6,7}, {3,5,7}, {3,4,8}, {4,5,6}. But if the cage spans two cells in the same row, any digit that already appears in that row can be eliminated from the cage’s candidates.
Apply Standard Sudoku Techniques
Once you’ve established candidates from cage constraints, all standard Sudoku techniques apply. Naked singles, hidden singles, naked pairs, and hidden pairs work exactly as they do in classic Sudoku. The cage constraints simply provide an additional source of eliminations on top of the standard row/column/box constraints.
Cross-Reference Cages with Regions
When a cage is entirely contained within a single box, row, or column, the no-repeat rule within the cage is already enforced by the Sudoku constraint. But when a cage spans multiple regions, the cage’s no-repeat rule provides extra information beyond what standard Sudoku gives you. Pay special attention to these cross-region cages.
Advanced Strategies
Innies and Outies
An “innie” is a cell that sticks into a region from a cage that mostly lies outside it. An “outie” is a cell that sticks out of a region from a cage mostly inside it. Using the 45 rule, you can often determine the exact value of innies and outies.
For example, if all cages within a box sum to 48 (3 more than 45), the outie cell (the one cell belonging to those cages but lying outside the box) must equal 3.
Cage Splitting
When a cage spans two regions and you know the sum contributed to each region (using the 45 rule), you can treat the cage as two smaller virtual cages. This effectively reduces the problem and often reveals new eliminations.
Combination Overlap
When two cages in the same row share several possible combinations, compare them to find digits that must or cannot appear in each cage. If one cage’s only valid combinations all include a 5, then a 5 is locked into that cage and can be eliminated from the rest of the row.
Tips for Beginners
- Memorize 2-cell combinations. These are the most common and most useful to know by heart.
- Always apply the 45 rule early. Check every row, column, and box for opportunities to calculate unknown sums.
- Write candidates in every cell. Full notation is even more important in Killer Sudoku than in classic Sudoku because of the additional constraints.
- Start with small cages. 2-cell and 3-cell cages with extreme sums offer the most immediate progress.
- Be patient. Killer Sudoku is harder and slower than standard Sudoku. Don’t expect to match your classic Sudoku solving times.
- Check your arithmetic. The most common errors in Killer Sudoku are simple addition mistakes. Take an extra moment to verify sums.
If you’re looking for a gentler introduction to cage-based Sudoku, consider practicing standard Sudoku rules until they’re second nature, then transition to Killer Sudoku with the techniques fresh in your mind.
Frequently Asked Questions
What is Killer Sudoku?
Killer Sudoku is a Sudoku variant that overlays cage-sum constraints onto the standard 9×9 grid. Cells are grouped into cages with target sums, and digits cannot repeat within a cage. Unlike classic Sudoku, the grid typically starts empty with no given digits—all information comes from the cage sums.
Is Killer Sudoku harder than regular Sudoku?
Generally, yes. Killer Sudoku requires all the logical deduction of standard Sudoku plus arithmetic reasoning with cage sums. Easy Killer Sudoku puzzles are roughly comparable to medium-difficulty standard Sudoku. The added layer of cage constraints makes the solving process more complex and typically takes longer.
What is the 45 rule in Killer Sudoku?
Because digits 1 through 9 sum to 45, every complete row, column, and 3×3 box has a digit sum of exactly 45. This allows you to calculate unknown cell values by subtracting known cage sums from 45. It’s the most important technique specific to Killer Sudoku. Learn more in our 45 rule guide.
Can digits repeat in a Killer Sudoku cage?
No. Within a single cage, every digit must be unique. This rule applies in addition to the standard Sudoku constraints that prevent repeats in rows, columns, and 3×3 boxes. The cage no-repeat rule is a critical source of eliminations that helps narrow down candidates.
How do I get started with Killer Sudoku?
Start by mastering standard Sudoku if you haven’t already. Then learn the 2-cell cage sum combinations, understand the 45 rule, and try an easy Killer Sudoku puzzle. Use full notation and be prepared for puzzles to take longer than classic Sudoku. Work through the basic strategies in this article before attempting advanced techniques.