Killer Sudoku Combinations: The Cheat Sheet (and How to Use It)
In Killer Sudoku, each dotted cage shows a target sum its cells must reach using different digits, never repeating inside the cage. Killer Sudoku combinations are the sets of digits that hit each target — and knowing which cages have only one possible set (a 2-cell cage of 17 must be {8, 9}) tells you exactly where to start.
What Is a "Combination" in Killer Sudoku?
Killer Sudoku adds dotted cages to the standard grid — each cage shows a target sum its digits must reach, and no digit may repeat within the same cage. That no-repeat rule is what keeps the option list short: a 2-cell cage summing to 17 can only be 8+9, while a cage summing to 10 has four valid pairs. A combination is the unordered set of distinct digits that hits a given cage sum — knowing it tells you exactly which candidates to pencil in and which to rule out.
If you're brand new to Killer Sudoku and want the full rules first, start with how to play Killer Sudoku and come back here for the reference tables.
The Combinations That Have Only One Answer (Start Here)
These "unique" cages give you free digits — no guessing, no pencil marks needed. Scan the whole grid for them before doing anything else.
When a cage has exactly one possible combination, the digits are confirmed the moment you see the cage sum. These are the highest-value squares on any Killer Sudoku grid.
The table below lists every unique combination by cage size. Source for cage rules and combination structure: SudokuWiki.org — Killer Combinations.
| Cells | Sum | 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} |
| 5 | 15 | {1, 2, 3, 4, 5} |
| 5 | 16 | {1, 2, 3, 4, 6} |
| 5 | 34 | {4, 6, 7, 8, 9} |
| 5 | 35 | {5, 6, 7, 8, 9} |
These are the cages to scan for first. Every unique cage you find locks in those digits before you've done any real deduction — which in turn narrows down every row, column, and box those cells touch.
Worked mini-example: You spot a 2-cell cage with sum 17. From the table: only {8, 9}. Write 8 and 9 as candidates in those two cells immediately. Now every row, column, and 3×3 box touching those cells can eliminate 8 and 9 from their other candidates.
Every 2-Cell Cage Combination (The Most Common Cage)
Two-cell cages appear more often than any other size in a standard Killer Sudoku grid. It's worth knowing this table cold — or at least keeping it open.
The table below covers every possible 2-cell cage sum (3 through 17). "Options" is the number of valid combinations for that sum.
| Sum | Valid combinations | Options |
|---|---|---|
| 3 | {1, 2} | 1 |
| 4 | {1, 3} | 1 |
| 5 | {1, 4} · {2, 3} | 2 |
| 6 | {1, 5} · {2, 4} | 2 |
| 7 | {1, 6} · {2, 5} · {3, 4} | 3 |
| 8 | {1, 7} · {2, 6} · {3, 5} | 3 |
| 9 | {1, 8} · {2, 7} · {3, 6} · {4, 5} | 4 |
| 10 | {1, 9} · {2, 8} · {3, 7} · {4, 6} | 4 |
| 11 | {2, 9} · {3, 8} · {4, 7} · {5, 6} | 4 |
| 12 | {3, 9} · {4, 8} · {5, 7} | 3 |
| 13 | {4, 9} · {5, 8} · {6, 7} | 3 |
| 14 | {5, 9} · {6, 8} | 2 |
| 15 | {6, 9} · {7, 8} | 2 |
| 16 | {7, 9} | 1 |
| 17 | {8, 9} | 1 |
Notice the symmetry: sums 3 and 17 each have one option, sums 9–11 peak at four options, then the count mirrors back down. The midpoint sums (9, 10, 11) are the most ambiguous — four possible pairs each. Extreme sums (3, 4, 16, 17) are the easiest because they leave no room for choice.
Quick check (sum 10): {1, 9}, {2, 8}, {3, 7}, {4, 6} — four options. Note that {5, 5} is not listed: the no-repeat rule within a cage eliminates it even though 5+5 = 10.
How Many Numbers Can a Cage Hold? (Min and Max Sums)
Before you pencil in candidates, it's worth knowing whether a cage sum is even legal for that cage size. The table below shows the minimum and maximum possible sum for each cell count.
| Cells | Min sum | Max sum |
|---|---|---|
| 1 | 1 | 9 |
| 2 | 3 | 17 |
| 3 | 6 | 24 |
| 4 | 10 | 30 |
| 5 | 15 | 35 |
| 6 | 21 | 39 |
| 7 | 28 | 42 |
| 8 | 36 | 44 |
| 9 | 45 | 45 |
The min sum for N cells = 1+2+…+N (the N smallest distinct digits). The max = 9+8+…+(10−N) (the N largest). A 9-cell cage must contain every digit 1–9 and therefore always sums to exactly 45.
Practical use: if you see a 2-cell cage with sum 2 or 18, the puzzle has a typo — those are outside the valid range. And if you're looking at a 3-cell cage with sum 25, that's also impossible (max for 3 cells is 24). Spotting impossible sums early saves you from chasing dead ends.
How to Actually Use These Tables
The combination tables are only useful if you have a method for applying them. Here's the order that works:
- Scan for unique cages — lock in the digits immediately (no guessing)
- Fill definite digits — write confirmed candidates in pencil
- Cross-check row / column / box — eliminate candidates that clash with locked-in digits
- Use the 45 rule on near-full boxes — subtract known cage totals from 45 to find leftover cells
- Repeat — each locked digit feeds new eliminations
Step 1 — Unique cages first. Circle every cage from the unique-combinations table and fill those digits in pencil immediately — free information, no deduction needed.
Step 2 — Cross-check. Each locked cage eliminates its digits from every crossing row, column, and box. A chain of unique cages can collapse a large chunk of the puzzle before any real deduction begins.
Step 3 — Narrow non-unique cages. For a 2-cell cage summing to 9, the table gives four options. If the row already has a 2 and 3 locked in, cross off {2, 7} and {3, 6} immediately — you're down to {1, 8} or {4, 5}.
Step 4 — The 45 rule (brief version). Every row, column, and 3×3 box sums to 45. Subtract known cage totals from 45 to find leftover cells. For a full walkthrough, the how to play Killer Sudoku guide covers it step by step.
Step 5 — Never guess. A wrong digit corrupts its cage, row, column, and box at once. Work only from confirmed combinations.
Want to sharpen the add-to-a-target instinct these cages need? Try Make 10 ↓
Is Killer Sudoku Good for Your Brain? (Honest Answer)
Killer Sudoku keeps you practicing addition, logical elimination, and pattern recognition — all at once, which is what makes it feel engaging rather than tedious. The "just one more cage" pull is real.
On bigger cognitive claims: playing logic puzzles keeps you sharp at logic puzzles, and there's genuine enjoyment in that. Whether it translates to broad memory or reasoning gains is more limited and task-specific than a lot of puzzle marketing suggests. For an honest look at what the research does and doesn't say, brain games for seniors covers that territory directly.
If you enjoy the combination-thinking in Killer Sudoku, Kakuro uses the same skill — but in a crossword-shaped grid instead of a 9×9 box. The cages become "runs" and the no-repeat rule still applies. How to solve Kakuro has the run combination tables and beginner walkthrough.
Just for fun — not medical advice.
Frequently Asked Questions
What two numbers add up to 16 in Killer Sudoku?
Only {7, 9}. That's the sole valid 2-cell combination for a sum of 16 — {8, 8} repeats, and any pair using a digit above 9 or below 1 is out of range. This makes a 2-cell cage summing to 16 one of the easiest free moves on the board.
What is the 45 rule in Killer Sudoku?
Every row, column, and 3×3 box contains the digits 1 through 9 exactly once, so each always sums to 45. When a set of cages almost completely fills a box, you can subtract their totals from 45 to calculate the value of any leftover cell — without needing to know that cell's own cage combination. The how to play Killer Sudoku guide walks through this with a worked example.
Which cages have only one possible combination?
See the unique-combinations table above. The pattern: extreme low sums (like a 2-cell cage of 3) and extreme high sums (like a 3-cell cage of 24) tend to be unique because the no-repeat rule leaves almost no room. Mid-range sums have multiple options and require cross-referencing.
Do numbers repeat inside a Killer Sudoku cage?
No — never. Even if a repeated digit would reach the correct sum, it's not allowed within the same cage. This is why a 2-cell cage summing to 4 must be {1, 3}, not {2, 2}. The no-repeat rule is what keeps combination lists short and makes the puzzle solvable by logic alone.
Do I have to be good at math to play Killer Sudoku?
No. The arithmetic is single-digit addition — checking whether two or three small numbers reach a target. The real skill is logical elimination, the same as in regular Sudoku. Knowing that a cage summing to 17 must be {8, 9} is the kind of number comfort that helps — not speed arithmetic.
Play now: Make 10 is open in your browser — place number tiles so a row or column sums to ten, and they clear. Same add-to-a-target instinct as a Killer Sudoku cage, in a faster format. No account, no download.
More from the Make10s blog: how to play Killer Sudoku · how to solve Kakuro · brain games for seniors · all posts
Source: SudokuWiki.org — Killer Combinations (cage rules & combination structure). Combination values verified against the Make10s internal table.