Killer Sudoku
Forced Cells in Cages
A forced cell is a cell whose digit is settled by its cage. List the sets of different digits that reach the cage's sum, strike every set whose digits cannot each find a cell, and see what each cell can still take: when one cell is left with a single digit, place it, even though its pencil marks still show several. It applies to a cage with two or more empty cells; with one empty cell, simple subtraction gives the answer.
What you need to see on the grid
Look for forced cells in cages with few combinations — small sums, large sums, few cells — whose row, column or box already holds some of the digits involved. Four terms carry the method:
- Combination
- A set of different digits that fills the cage's empty cells and reaches what is left of its sum. Order does not matter and no digit repeats inside a cage, so 8 in three cells has exactly two: {1, 2, 5} and {1, 3, 4}.
- Remaining sum
- The cage's sum minus the digits already placed in it. Those digits also drop out of every combination: a cage never holds the same digit twice.
- Crossing
- A combination counts only if its digits can be handed out one per empty cell, each cell receiving a digit it still lists. A single digit that no cell can take is enough to rule it out.
- Forced cell
- A cell that gets the same digit in every fill of the cage that survives the crossing. It is not a Naked Single: its pencil marks can still show several digits. And unlike Cage Combinations, which only trims candidates, it places a digit.
How to apply it, step by step
One example runs through all four steps: a three-cell cage of sum 8 in row 5, early in 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.
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Step 1Read the cage and list its combinations
Start from a cage that still has two or more empty cells. Note its sum, how many cells are empty and which digits are already inside it. This cage covers R5C4, R5C5 and R5C6, totals 8 and holds no digit yet, so three different digits must add up to 8.
Write out every way to do it. There are exactly two: 1 + 2 + 5 and 1 + 3 + 4. Nothing larger fits, because any set containing 6 or more already adds up to at least 1 + 2 + 6 = 9. If you are not sure of a list, check it against the cage sum chart rather than your memory.
Only the cage matters here: three empty cells, sum 8. Its combinations are {1, 2, 5} and {1, 3, 4}. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 8 R6 R7 R8 R9 - Cage, sum in its corner
- Not needed for this step
Filled in: the cage. The other 78 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 2Strike out the combinations the grid rules out
A combination survives only if each of its digits can sit in some cell of the cage. Look at the units the cage lies in. Row 5 already holds a 3, at R5C1, and all three cage cells are in row 5 — none of them lists 3. The combination {1, 3, 4} needs a 3 somewhere in the cage, so it is impossible.
Only {1, 2, 5} is left: the cage holds exactly 1, 2 and 5. Nothing is placed yet — so far you know which digits the cage uses, not where each one goes.
Row 5 in full, with the cage. The 3 at R5C1 keeps 3 out of all three cage cells, so {1, 3, 4} cannot be the cage's combination. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 3 8 6 R6 R7 R8 R9 - Digit already on the grid
- Row 5: its 3 is already placed
- Cage cell: 3 is not a candidate
- Cage, sum in its corner
- Not needed for this step
Filled in: row 5; 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 3Place the cell that has one digit left
Now hand out 1, 2 and 5, cell by cell. R5C4 lists 4, 5, 7, 8 and 9. Column 4 already holds 1 (R3C4) and 2 (R7C4), so of the three digits it can only take 5. Its other candidates go too: 7, 8 and 9 belong to no combination of 8, and 4 only belonged to {1, 3, 4}, which step 2 struck out. R5C4 = 5.
This is not a Naked Single: R5C4 still shows five pencil marks, and only the cage brings them down to one. You do not need the other cells settled either. R5C5 and R5C6 can each still be 1 or 2, so they stay open — narrowing them down is the work of Cage Combinations, a separate technique.
Technique: Forced cell in a cage
Row 5 and column 4 in full, with the cage. Column 4 already holds 1 and 2, so R5C4 can only be the 5 of {1, 2, 5}. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 1 R4 R5 3 85 6 R6 R7 2 R8 R9 - Digit already on the grid
- Column 4: 1 and 2 already placed
- The cell the cage forces
- Digit placed
- Cage, sum in its corner
- Not needed for this step
Filled in: row 5; column 4; the cage. The other 64 cells are left blank: they are not needed for this step. Placed: R5C4 = 5.
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Step 4Collect what the new digit unlocks
A placed digit clears its row, column and box at once, so look around R5C4. In box 6, 5 had only two places left: R5C7 and R6C7. R4C8 and R6C8 see the 5 at R7C8, and R4C9, R5C9 and R6C9 see the 5 at R2C9. The new 5 in row 5 now takes R5C7 away as well, so R6C7 is the only cell for 5 in the box: a Hidden Single.
The cage has moved on too: R5C5 and R5C6 must now add up to 3, which only 1 + 2 can do. Check the units around a forced cell before you look elsewhere.
Technique: Hidden Single
Box 6 and row 5 after the placement, plus the 5s at R2C9 and R7C8. Only R6C7 can still hold 5 in the box. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 5 R3 R4 3 R5 3 5 6 R6 5 R7 5 R8 R9 - Digit already on the grid
- Box 6: one place left for 5
- The cell that must be 5
- Candidate 5, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 5; box 6; cells R2C9 and R7C8. The other 64 cells are left blank: they are not needed for this step. Placed: R6C7 = 5.
Common mistakes
Listing combinations from memory
Forget one combination and a cell can look forced when it is not. Remember 9 in three cells as only {1, 2, 6} and {2, 3, 4}, and you have lost {1, 3, 5}: a cell listing 5 and 6 would look like a certain 6. Count the sets carefully, or check them against the cage sum chart, before you place anything.
Checking only the cell you are looking at
R5C4 lists 4, and 4 belongs to {1, 3, 4}, so a quick look gives the cell two options and the move is missed. A combination only survives if the other cells can take the rest of it — here that means a 3, which row 5 already holds. Cross every combination with every empty cell of the cage, not just the one in front of you.
Confusing "in the cage" with "in this cell"
Both combinations of 8 contain a 1, so the cage is sure to hold one — but that does not say where. R5C5 and R5C6 can both still be 1. A digit shared by every combination becomes a placement only when exactly one empty cell of the cage can take it; until then it tells you what the cage holds, not which cell holds it.
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 forces a cell, the hint in the app points to that cell, names the cage and its remaining sum, and explains why nothing else fits — the same argument as step 3 above.