Cage Combinations
The foundational technique for Killer Sudoku. Learn which cages have unique digit sets and how to use this constraint to start solving.
What Are Cage Combinations?
Every cage in Killer Sudoku must be filled with digits that (1) sum to the target, and (2) are all different from each other. For many cage sizes and sums, there is only one possible set of digits that satisfies both constraints.
These "unique combination" cages give you immediate information: you know exactly which digits belong in the cage, even if you don't yet know their order. This constrains candidates and often unlocks the puzzle.
Key insight: Extreme sums (very low or very high) for a given cage size almost always have unique digit sets. Start your solving here.
2-Cell Cages: The Easiest Entry Point
2-cell cages are the simplest to analyze. Here are the unique combinations:
Sum = 3
{1, 2}
Only way to make 3 with two different digits (1+2=3)
Sum = 4
{1, 3}
1+3=4 (not 2+2, which violates no-repeat)
Sum = 16
{7, 9}
7+9=16 (8+8 is impossible; no other pair sums to 16)
Sum = 17
{8, 9}
Only way to make 17 with two different digits (8+9=17)
All other 2-cell sums (5, 6, 7, ..., 15) have multiple possible digit pairs. For example, sum 5 could be {1,4} or {2,3}.
3-Cell Cages: Unique Extremes
3-cell cages have unique combinations at the low and high ends:
Sum = 6
{1, 2, 3}
Minimum sum for 3 cells: 1+2+3=6
Sum = 7
{1, 2, 4}
Next smallest: 1+2+4=7 (no other triple works)
Sum = 23
{6, 8, 9}
Next largest below maximum: 6+8+9=23
Sum = 24
{7, 8, 9}
Maximum sum for 3 cells: 7+8+9=24
For sums like 8, 9, ..., 22, multiple combinations exist. For example, sum 10 could be {1,2,7}, {1,3,6}, {1,4,5}, or {2,3,5}.
4-Cell Cages: Minimum and Maximum
4-cell cages have unique combinations at the extremes:
Sum = 10
{1, 2, 3, 4}
Minimum sum: 1+2+3+4=10
Sum = 11
{1, 2, 3, 5}
Next smallest: 1+2+3+5=11
Sum = 29
{5, 7, 8, 9}
Next largest: 5+7+8+9=29
Sum = 30
{6, 7, 8, 9}
Maximum sum: 6+7+8+9=30
Example: Starting a Puzzle
Let's apply cage combinations to a small section of a puzzle:
Partial grid with cage sums:
Cages: 2-cell sum 3, 3-cell sum 11, 2-cell sum 17, 2-cell sum 6
Step 1: Identify Unique Combinations
2-cell sum 3: must be {1, 2}
2-cell sum 17: must be {8, 9}
2-cell sum 6: could be {1,5} or {2,4} (not unique)
3-cell sum 11: not a unique combination (could be {1,2,8}, {1,3,7}, {1,4,6}, {2,3,6}, or {2,4,5})
Step 2: Use Sudoku Constraints
Now we know: top-left cage contains 1 and 2 (order unknown), left-middle cage contains 8 and 9 (order unknown).
These cells are in the same column. Standard Sudoku rules say the column cannot repeat digits. This is already satisfied (no overlap between {1,2} and {8,9}).
If these cages are in the same 3×3 box, we can now eliminate 1, 2, 8, 9 as candidates for other cells in that box.
Step 3: Narrow Down Non-Unique Cages
The 2-cell sum 6 cage: could be {1,5} or {2,4}.
If this cage shares a row/column/box with the sum-3 cage (which uses 1 and 2), we can eliminate any overlap. If they're in the same box, {1,5} becomes impossible (1 is already used), so it must be {2,4}. But wait—2 is also in the sum-3 cage!
This shows how unique combinations propagate: knowing one cage forces another cage's digits through elimination.
Why This Technique Is Foundational
Cage combinations are your entry point into every Killer Sudoku puzzle. Unlike classic Sudoku, which gives you some digits to start, Killer starts empty. Unique-combination cages are your first "givens."
Scanning Strategy:
Scan for 2-cell cages with sums 3, 4, 16, 17
Look for 3-cell cages with sums 6, 7, 23, 24
Check 4-cell cages with sums 10, 11, 29, 30
Mark the digit sets for these cages as candidates
Use Sudoku rules to propagate the constraints
Once you've locked down the unique cages, other cages become easier to solve because you've eliminated many candidates from their rows, columns, and boxes.
Common Mistakes to Avoid
❌ Forgetting the no-repeat rule
A 2-cell sum-6 cage cannot be {3,3}. All digits in a cage must be different. This eliminates many naive arithmetic solutions.
❌ Assuming non-unique cages are useless
Even if a cage has multiple possible digit sets, knowing the options is valuable. Combined with Sudoku constraints, you can often eliminate all but one.
❌ Ignoring digit order
Knowing a cage is {1,2} tells you the digits, but not which cell gets which digit. Use row/column/box rules to determine order.
Quick Reference: Unique Combinations
2-cell cages:
3-cell cages:
4-cell cages: