Killer Sudoku Combinations Cheat Sheet: All Cage Sums

What Are Killer Sudoku Cage Combinations?

Killer sudoku layers a second rule on top of classic sudoku: the grid is divided into “cages”—groups of two or more cells outlined with a dotted border, each labeled with a target sum. Every cage must contain digits that add up to that sum, and just like a row, column, or 3x3 box, no digit can repeat within a cage.

That single constraint is what makes killer sudoku combinations the fastest way into a puzzle. Instead of guessing, you can work out exactly which digits are mathematically allowed to sit in a cage before you even look at the rest of the grid. This killer sudoku combinations cheat sheet lists every possible combination for 2-cell, 3-cell, 4-cell, and 5-cell cages so you can check candidates in seconds instead of doing mental arithmetic mid-solve.

The 45 Rule: Your First Tool

Before you touch the combination tables, learn the 45 rule. Every row, column, and 3x3 box in a standard sudoku grid contains the digits 1 through 9 exactly once, and 1+2+3+4+5+6+7+8+9 = 45. That means the digits in any complete row, column, or box must always sum to 45—no exceptions.

This becomes a solving weapon whenever a set of cages lines up almost perfectly with a row, column, or box, but one cell sticks out or one cell is missing. Subtract the known cage totals from 45 and the leftover single cell is solved instantly.

For example, imagine a box is covered by three cages summing to 14, 12, and 15, plus one extra cell that belongs to a cage outside the box. The three cages account for 14 + 12 + 15 = 41 of the box's cells. Since the whole box must total 45, that leftover cell is worth 45 − 41 = 4. No candidates, no trial and error—just subtraction.

2-Cell Cage Combinations

Two-cell cages are the building blocks of most killer sudoku puzzles. Here is every possible combination for sums 3 through 17, with the unique combinations (sums that only have one possible pair of digits) called out.

• 3 — 1+2 (unique combination)
• 4 — 1+3 (unique combination)
• 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 (unique combination)
• 17 — 8+9 (unique combination)

Sums of 3, 4, 16, and 17 are the ones worth memorizing first. The moment you spot a 2-cell cage totaling one of those four numbers, you already know both digits—you just need to work out which cell gets which value.

3-Cell Cage Combinations

Three-cell cages have more possible combinations in the middle of the range, but the extreme sums still lock down to a single answer. Here is the full list for sums 6 through 24.

• 6 — 1+2+3 (unique combination)
• 7 — 1+2+4 (unique combination)
• 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
• 13 — 1+3+9, 1+4+8, 1+5+7, 2+3+8, 2+4+7, 2+5+6, 3+4+6
• 14 — 1+4+9, 1+5+8, 1+6+7, 2+3+9, 2+4+8, 2+5+7, 3+4+7, 3+5+6
• 15 — 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6
• 16 — 1+6+9, 1+7+8, 2+5+9, 2+6+8, 3+4+9, 3+5+8, 3+6+7, 4+5+7
• 17 — 1+7+9, 2+6+9, 2+7+8, 3+5+9, 3+6+8, 4+5+8, 4+6+7
• 18 — 1+8+9, 2+7+9, 3+6+9, 3+7+8, 4+5+9, 4+6+8, 5+6+7
• 19 — 2+8+9, 3+7+9, 4+6+9, 4+7+8, 5+6+8
• 20 — 3+8+9, 4+7+9, 5+6+9, 5+7+8
• 21 — 4+8+9, 5+7+9, 6+7+8
• 22 — 5+8+9, 6+7+9
• 23 — 6+8+9 (unique combination)
• 24 — 7+8+9 (unique combination)

Just like the 2-cell table, focus your memory on the edges: sums of 6, 7, 23, and 24 each resolve to exactly one combination of digits.

4-Cell and 5-Cell Cages: The Ones Worth Memorizing

Four- and five-cell cages have too many possible combinations in the middle range to memorize practically. The good news is that the same pattern holds: the lowest and highest possible sums for each cage size still only have one valid combination, and those are the ones worth learning by heart.

4-cell cage unique combinations:

• 10 — 1+2+3+4 (unique combination)
• 11 — 1+2+3+5 (unique combination)
• 29 — 5+7+8+9 (unique combination)
• 30 — 6+7+8+9 (unique combination)

5-cell cage unique combinations:

• 15 — 1+2+3+4+5 (unique combination)
• 16 — 1+2+3+4+6 (unique combination)
• 34 — 4+6+7+8+9 (unique combination)
• 35 — 5+6+7+8+9 (unique combination)

For every sum in between, don't try to memorize a table—lean on the 45 rule and elimination instead. Cross off digits that are already locked into the same row, column, or box, and the remaining candidates narrow quickly even without a full combination list.

How to Use This Cheat Sheet While Solving

Knowing the combinations is only half the job. Here is the order of operations top killer sudoku solvers actually use during a puzzle:

1. Scan for unique-combination cages first. Any 2-cell cage summing to 3, 4, 16, or 17, any 3-cell cage summing to 6, 7, 23, or 24, and the 4-cell and 5-cell extremes above give you locked-in digits before you've solved anything else.
2. Apply the 45 rule to define single cells. Whenever cages nearly cover a full row, column, or box, subtract their totals from 45 to solve the leftover cell directly.
3. Cross-reference combinations to eliminate candidates. When two overlapping cages share a possible digit in only one of their combination lists, you can rule that digit out of the other cells in the grid.
4. Layer in standard sudoku technique. Once cage logic narrows your candidates, switch back to normal scanning, naked pairs, and other row/column/box techniques to finish the puzzle. Practice this combined approach on our hard sudoku generator, or build your own custom difficulty puzzles with the sudoku puzzle maker.

Practice Makes Permanent

The combinations above will start to feel automatic after a handful of puzzles, but only if you actually put them to work. Print a blank sudoku grid, sketch your own killer cages onto it by hand, and try solving with nothing but this cheat sheet and the 45 rule. Hand-drawing a few cages is one of the fastest ways to make the unique combinations stick in memory.

Once cage combinations feel natural, round out your solving toolkit with our sudoku solving strategies guide, which covers the classic scanning and elimination techniques that carry you through the rest of any killer sudoku grid.