Killer Sudoku
Cage Combinations
A cage combination is a set of different digits that fills a Killer cage to its sum. List the combinations for the cage's size and sum, drop every one the grid rules out — one that lacks a digit already in the cage, or needs a digit none of its empty cells can take — and remove every candidate that no surviving combination can put in its cell. The cage sum chart gives you the list; the grid decides which entries are still alive.
What you need to see on the grid
Use it on any cage whose sum leaves few combinations — the smallest and largest sums first — and come back to a cage whenever it gains a digit or its row, column or box fills up. Four terms carry the method:
- Combination
- A set of different digits, one per cell of the cage, that adds up to its sum. Order does not matter: 5 6 7 9 is one combination however its digits end up arranged inside the cage.
- Unique sums
- Sums with only one possible set for their size, such as 7 in three cells (1 2 4) or 30 in four (6 7 8 9). The cage's digits are known at once and only their order is left to find.
- Always and never digits
- A digit that is in every combination is certainly in the cage; a digit that is in none can never be. Both can be read off the chart before you even look at the grid.
- Surviving combination
- A combination that still fits the grid: it contains every digit already placed in the cage, and each digit it still needs is a candidate in at least one empty cell of the cage. Only these count when you remove candidates.
How to apply it, step by step
One cage runs through all five steps, part-way through a Killer puzzle. Every diagram shows the whole grid: the cells a step needs are filled in, the others are left blank and hatched, so you know you can ignore them. Only the cage we work on is outlined; the puzzle's other cages play no part in these steps.
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Step 1Read the cage's size and sum, then list its combinations
Start with what the cage itself tells you: how many cells it covers and what they add up to. This one covers four cells, R5C7, R5C8, R6C7 and R6C8, and its sum is 27.
Look up 27 in four cells on the Killer cage sum chart (linked at the end of this guide). It lists three combinations: 3 7 8 9, 4 6 8 9 and 5 6 7 9. Those are the only sets of four different digits that make 27; which one is right, and in what order, is for the grid to decide. One cell, R5C7, already holds a 6 — keep that in mind, it comes back in step 3.
Only the cage is drawn: four cells, sum 27, with a 6 already in R5C7. Pencil marks are left out — the chart does not need them. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 276 R6 R7 R8 R9 - Digit already on the grid
- Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 77 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Note the digits every combination shares, and those none uses
Compare the three combinations. 9 is in all of them, so whichever one is right, the cage contains a 9 — and since R5C7 is a 6, that 9 is in one of the three empty cells. 1 and 2 are in none of them, so no cell of this cage can ever be a 1 or a 2.
Now look at the pencil marks: all three empty cells still list 1 and 2. Those candidates are already doomed; note them, and step 4 removes them together with whatever step 3 finds. When the chart gives a unique sum, this comparison finishes the job on its own: every digit outside the one combination leaves the cage.
The same cage with its pencil marks. 9 is in every combination for 27; 1 and 2, in none of them, are still listed in all three empty cells. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 276 R6 R7 R8 R9 - Digit already on the grid
- Empty cell of the cage
- Candidates 1, 2 and 9, the digits studied
- Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 77 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Cross each combination with the grid
The chart only knows the size and the sum; the grid knows more. Each combination must pass two tests: it contains every digit already placed in the cage, and every other digit it needs is still a candidate in at least one empty cell of the cage.
3 7 8 9 has no 6, but R5C7 already holds one: it dies. 4 6 8 9 has the 6 but also needs an 8, and box 6 already has its 8 in R4C7, so none of the three empty cells can take 8: it dies too. 5 6 7 9 passes both tests, so the empty cells hold 5, 7 and 9, in some order.
This is the part a printed chart cannot do. On the chart alone, 4 still looks possible here; against the grid, it is gone.
Technique: Cage combination elimination
Box 6 in full, around the cage. The 6 in R5C7 rules out 3 7 8 9; the 8 in R4C7 keeps 8 out of every empty cage cell, which rules out 4 6 8 9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 8 R5 276 R6 3 R7 R8 R9 - Digit already on the grid
- Box 6, which already holds 3, 6 and 8
- Empty cell of the cage
- Cage, sum in its corner
- Not needed for this step
Filled in: box 6; the cage. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 4Remove every candidate no surviving combination can use
Only 5 6 7 9 is left, and its 6 is already placed, so each empty cell keeps just the 5, 7 and 9 it lists. Everything else goes from R5C8, R6C7 and R6C8: 1 and 2, which are in no combination (step 2), and 4, whose only combination died in step 3.
The exact rule works cell by cell: a candidate stays if at least one complete fill of the cage — different digits, the right total, each taken from its own cell's pencil marks — puts it in that cell. Here the three empty cells must make 27 − 6 = 21, and only 5 + 7 + 9 does it. Each of 5, 7 and 9 can still go in each cell, so nothing else is removed and no cell is decided yet.
Technique: Cage combination elimination
The cage alone is enough to check this: with the 6 placed, the three empty cells must make 21 from their own pencil marks, and only 5 + 7 + 9 does. 1, 2 and 4 leave all three cells. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 276 R6 R7 R8 R9 - Digit already on the grid
- Empty cell of the cage
- Candidate removed
- Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 77 cells are left blank: they are not needed for this step. Removed: 1 from R5C8, R6C7 and R6C8; 2 from R5C8, R6C7 and R6C8; 4 from R5C8, R6C7 and R6C8.
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Step 5Collect what the elimination unlocks
Candidates that leave a cage also leave its rows, columns and boxes. Before step 4, 2 had four places in box 6: R5C8, R5C9, R6C7 and R6C8 (R4C8 and R4C9 cannot take it, because row 4 already has a 2 in R4C2). Three of those have just lost their 2, so R5C9 is the only place left for 2 in box 6: a Hidden Single.
After every cage elimination, take each digit you removed and check the rows, columns and boxes the cage touches — that is where the cage pays you back.
Technique: Hidden Single
Box 6 after the cage elimination, and the 2 in R4C2 that keeps 2 out of row 4. 2 now fits only R5C9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 2 8 R5 276 2 R6 3 R7 R8 R9 - Digit already on the grid
- Box 6: one place left for 2
- The cell that must be 2
- Candidate 2, the digit studied
- Digit placed
- Cage, sum in its corner
- Not needed for this step
Filled in: box 6; the cage; cell R4C2. The other 71 cells are left blank: they are not needed for this step. Placed: R5C9 = 2.
Common mistakes
Forgetting the digits already in the cage
Once a cage holds a digit, either keep only the combinations that contain it, or look up the empty cells alone with the sum minus the placed digits — here, 21 in three cells without a 6. Reading the four-cell list but ignoring the 6 keeps 3 7 8 9 on it. Box 6 happens to block 3 and 8 here anyway; on another grid they would be candidates you never remove.
Throwing out every digit of a dead combination
When 4 6 8 9 dies, only 4 leaves the cage: 6 and 9 are still in 5 6 7 9. A candidate goes only when no surviving combination can put it in that cell, so check each digit against every combination that is left, not against the one that just died.
Clearing an "always" digit outside a cage that crosses a border
A digit in every combination is certainly in the cage, so it can also leave other cells — but only cells that see every cage cell where it could go. This cage lies inside box 6, so its 9 is box 6's 9. When a cage straddles two boxes, its 9 may sit on either side, and a cell of one box can lose 9 only if it also sees the cage cells in the other box.
Every cage combination, for every size and sum: Killer Sudoku Cage Sum Chart — all combinations
Learn the move, not just the digit
In a real Killer game, when a cage combination is what opens the next cell, the hint in the app shows that cell and, in its details, names cage combination elimination as the move that unlocked it — the same move as step 5 above.