Killer Sudoku Cage Combinations: The Complete Chart
Every Killer Sudoku cage has a fixed set of possible digit combinations for its size and sum, and memorising the unique ones instantly places or splits cages. This reference chart, organized by cage size, lists the critical sums and their digit combinations. Use it to quickly deduce possibilities and eliminate candidates, turning a complex puzzle into a logical deduction game. Understanding these combinations is a core part of any [easy killer sudoku strategy](/blog/easy-killer-sudoku-strategy).
Two-Cell Cages (Doubles)
A two-cell cage, or double, is the most common and often the most telling. Each sum from 3 to 17 is possible, but only four sums produce a single, unique pair of digits. Knowing these allows you to immediately write two numbers into the grid. The unique sums are 3 (1+2), 4 (1+3), 16 (7+9), and 17 (8+9). Every other sum from 5 to 15 has at least two possible pairs. Sums 9 and 10 are the most flexible, each with four possible combinations: 9 can be (1+8, 2+7, 3+6, 4+5) and 10 can be (1+9, 2+8, 3+7, 4+6).
- Unique sums: 3 (1+2), 4 (1+3), 16 (7+9), 17 (8+9).
- Most flexible sums: 9 and 10, each with 4 digit pairs.
- Cage sums of 5, 6, 7, 8, 11, 12, 13, 14, and 15 all have 2 or 3 possible pairs.
Three-Cell Cages (Triples)
Three-cell cages provide more information but also more possible combinations. The unique sums that lock in a single set of digits are 6 (1+2+3), 7 (1+2+4), 23 (6+8+9), and 24 (7+8+9). On the other end of the spectrum, sums 14, 15, and 16 are the most flexible, with eight possible combinations each. For example, sum 15 can be made from (1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6). Other notable sums include 12 (7 combos) and 21 (3 combos).
- Unique sums: 6 (1+2+3), 7 (1+2+4), 23 (6+8+9), 24 (7+8+9).
- Most flexible sums: 14, 15, 16 (8 combos each).
- Sum 12 has 7 combinations.
- Sum 21 has 3 combinations: (4+8+9, 5+7+9, 6+7+8).
Four-Cell Cages (Quads)
With four cells, the range of possible sums widens. The unique, locked-in combinations are for sums 10 (1+2+3+4), 11 (1+2+3+5), 29 (5+7+8+9), and 30 (6+7+8+9). These are powerful to find. The middle sums, like 18 through 22, have many combinations. For instance, sum 22 has 11 different ways to combine four distinct digits from 1 to 9. As cages grow, using the 45 rule alongside combination knowledge becomes critical. If a four-cell cage is entirely within one row, column, or 3x3 box, its sum can help deduce other values in that region.
- Unique sums: 10 (1+2+3+4), 11 (1+2+3+5), 29 (5+7+8+9), 30 (6+7+8+9).
- Sum 22 is highly flexible with 11 combinations.
- Sums in the low 20s are common and offer many possibilities.
Five-Cell and Larger Cages
Five-cell cages have four unique sums: 15 (1+2+3+4+5), 16 (1+2+3+4+6), 34 (4+6+7+8+9), and 35 (5+6+7+8+9). For six-cell and larger cages, unique combinations are extremely rare. Instead of memorizing lists, focus on digit restrictions. A six-cell cage must contain the digits 1 through 9, but will be missing three numbers. Its sum will be 45 minus the sum of those three missing numbers. This logic, an extension of the 45 rule, is more powerful than trying to recall a specific combination. For very large cages spanning seven or eight cells, you can often deduce the missing one or two digits directly.
- Unique 5-cell sums: 15 (1-5), 16 (1-4 & 6), 34 (4,6-9), 35 (5-9).
- For 6+ cell cages, use the 45 rule: Cage Sum = 45 - Sum(Missing Digits).
- A 7-cell cage's sum reveals the sum of the two digits *not* inside it.
How to Use This Combination Chart
First, scan the puzzle for cages with unique sums (like 3, 4, 7, 10, 15, 16, 24, 29, 30, 35) and pencil in those digits immediately. Next, for flexible cages, list the possible combinations and cross-reference them with other solved cells in the same row, column, or box. For example, if a three-cell cage with sum 12 is in a row that already contains a 5 and a 9, you can eliminate any combination containing a 5 or 9. Always combine cage logic with standard Sudoku rules. A killer sudoku calculator can generate combinations, but internalizing the unique sums speeds up solving. Remember, the goal is to use the combinations to create Naked Pairs or Pointing Pairs within rows, columns, and boxes.
- Start by solving cages with unique sums.
- For flexible sums, write combo lists and eliminate based on solved neighbors.
- Use cage combos to create standard Sudoku patterns like Naked Pairs.
- Combine the 45 rule with cage sums for larger deductions.
Key Facts
- ▪In a two-cell Killer Sudoku cage, only four sums have a unique digit pair: 3 (1+2), 4 (1+3), 16 (7+9), and 17 (8+9).
- ▪A three-cell cage with a sum of 6 must contain the digits 1, 2, and 3, while a sum of 24 must be 7, 8, and 9.
- ▪The sums 14, 15, and 16 for a three-cell cage are the most flexible, each having eight possible combinations of digits.
- ▪Four-cell cages have unique combinations only for sums 10 (1+2+3+4), 11 (1+2+3+5), 29 (5+7+8+9), and 30 (6+7+8+9).
- ▪For a five-cell cage, a sum of 15 is always the digits 1 through 5, and a sum of 35 is always 5 through 9.
- ▪Cages with six or more cells rarely have a single unique digit combination; solvers should use the 45 rule to find missing digits instead.
- ▪The 45 rule states that the total of all digits in any row, column, or 3x3 box is 45, allowing you to deduce missing values from partial sums.
- ▪Combination knowledge is most powerful when cross-referenced with solved cells in the same row, column, or box to eliminate possibilities.
Frequently Asked Questions
What is the most important Killer Sudoku combination to know?
The unique sums for small cages. Memorize the pairs for 3, 4, 16, 17; the triples for 6, 7, 23, 24; and the quads for 10, 11, 29, 30. These instantly reveal digits.
How do I solve a cage with a very common sum, like 10 in two cells?
List all combinations (1+9, 2+8, 3+7, 4+6). Then, check the cage's row, column, and box. If a 7 is already present in that row, you can eliminate the 3+7 combo, narrowing the possibilities.
What is the 45 rule in Killer Sudoku?
Each row, column, and 3x3 box must sum to 45 (1+2+...+9). If you know the sum of some cells in a region, you can find the sum or value of the remaining cells. Learn more in our cage rules guide.
Are there unique combinations for six-cell cages?
Almost never. For cages of six or more cells, shift your strategy. Use the 45 rule: the cage sum equals 45 minus the sum of the digits from 1-9 that are *not* in the cage. Focus on which digits are missing.
Where can I learn the basic rules of Killer Sudoku?
If you're new to the puzzle, start with our guide on what is killer sudoku. It explains the dotted cages, sums, and how the rules combine classic Sudoku with math.