A locked pair, triple, or quad inside a cage claims its digits, freeing the cage’s other cells.
Look through each cage for a group of cells whose combined candidate list is exactly as long as the group — 2 cells showing only 4 and 5, say. Those digits are locked to that group, so strike them from the cage's other cells, which often leaves one cell holding a single digit.
Naked subsets work inside cages just like in classic Sudoku. If N cells of a cage can only hold the same N digits, those digits are locked to that group — remove them from the cage’s remaining cells.
The deduction rests on one property: the digits inside a cage are all different. That makes a cage behave like a miniature house — one of 1–9 cells rather than 9. If 2 cells of a cage can hold nothing but 4 and 5, one of them is the 4 and the other is the 5, in some order. No third cell of that cage can be a 4 or a 5, because both digits are already spoken for. The number in the cage's corner plays no part in that conclusion.
The locked group can be 2 cells (a pair), 3 (a triple), 4 (a quad), or more in a large cage. What matters is the count: the chosen cells' combined candidate list must hold exactly as many digits as there are cells, and each cell must draw only from that list — one stray candidate breaks the pattern. The cells need not share a row, column, or box. Cage membership alone is the link, which is why this reaches eliminations that classic scanning cannot see.
The pattern rarely shows in raw pencil marks. The Killer step engine builds a cage-aware candidate set for every cell before it looks for subsets, and the same order of work applies by hand. Start with the ordinary row, column, and box eliminations. Then list every combination of distinct digits that makes the cage's total in the number of cells it has, discarding any that cannot be laid out — a combination survives only when each of its digits has a distinct cell still willing to take it. Keep the digits those survivors use, drop the rest, and drop any digit already placed in the cage.
Then scan cage by cage, smallest groups first. A cage needs at least 3 empty cells to pay: 2 to lock and 1 to strip. With 4 empty cells you can test pairs and triples; with 5, pairs, triples, and quads. The engine uses the same bound, trying sizes from 2 up to one less than the number of empty cells, and it reports the pattern only when a digit actually comes off a remaining cell.
The board above carries the pattern in its top-left cage: 4 cells totalling 29, at R1C1, R1C2, R2C1, and R2C2. Four different digits reach at most 6 + 7 + 8 + 9 = 30, so 29 sits one short of that ceiling, and exactly one combination makes it — 5 + 7 + 8 + 9. The cage can hold nothing outside those 4 digits, which is what the pencil marks show.
Ordinary row, column, and box work has already cut R1C1 and R2C1 to 5 and 7, while R1C2 and R2C2 still carry all 4 digits. R1C1 and R2C1 are the locked pair: between them they take the 5 and the 7, in some order. So strike both digits from the cage's other 2 cells, leaving R1C2 = {8, 9} and R2C2 = {8, 9}. The 2 boards below show the cage before and after that strike, and the table repeats it in full.
The remainder then carries its own total: the pair absorbs 5 + 7 = 12, so the 2 cells left must make 29 − 12 = 17, and 8 + 9 is the only way 2 different digits reach 17. This particular cage sits wholly inside one box and its pair shares a column, so an ordinary naked pair would clear the same 2 cells; the cage route is what the teacher shows because cage logic runs first, and because it still holds when a cage sprawls across units, where no house argument reaches. Had R1C2 held only 5, 7, and 8, stripping the pair would leave a single 8, and the engine would offer that placement instead of the eliminations.
A locked subset does more than delete candidates. Because the group takes exactly those digits, its total is fixed, so the rest of the cage inherits a known smaller sum. In a 5-cell cage totalling 30, suppose 3 cells are locked to {5, 8, 9}. They contribute 5 + 8 + 9 = 22, so the other 2 cells must total 30 − 22 = 8 from digits outside the trio, leaving 1 and 7 or 2 and 6. A 5-cell cage worth 30 has 6 combinations in all; the locked triple cuts them to 1 + 5 + 7 + 8 + 9 and 2 + 5 + 6 + 8 + 9.
The narrowest case collapses on the spot. In a 3-cell cage totalling 15, two cells reduced to {4, 5} force the third: there are 8 ways to make 15 with 3 different digits, and only 4 + 5 + 6 carries both a 4 and a 5, so the third cell is 6. The engine prefers this outcome, offering the placement instead of the eliminations once a remaining cell is down to one candidate, and it checks that digit against the cage's surviving combinations before showing it.
The most common error is treating a cage as a full house. A row holds all 9 digits, so a hidden subset argument works there directly. A cage holds only as many digits as it has cells, and you cannot claim a digit must live inside a cage until combination analysis shows it in every surviving combination. Naked subsets transfer to cages cleanly; hidden subsets need that extra proof first.
The second error is reaching outside the cage. These eliminations run to the cage's own cells and no further. When the locked cells do sit inside a single row, column, or box, an ordinary naked subset applies in that house as well — but that is a separate deduction resting on the house, and it vanishes the moment the group spreads across two units.
Two smaller traps deserve naming. A cage with only 2 empty cells yields nothing: a pair across both is the entire remainder, with nothing left to strip. And an over-narrowed candidate list manufactures phantom subsets — pencil one digit off a cell in error and two cells can look locked when they are not, after which the eliminations remove a true digit. Rebuild a cage's combination list whenever a placement changes what it can still hold.
| Cell | Cage-aware candidates | After the locked pair | Role |
|---|---|---|---|
| R1C1 | 5, 7 | 5, 7 | Locked pair — takes the 5 or the 7 |
| R2C1 | 5, 7 | 5, 7 | Locked pair — takes whichever the other leaves |
| R1C2 | 5, 7, 8, 9 | 8, 9 | 5 and 7 removed |
| R2C2 | 5, 7, 8, 9 | 8, 9 | 5 and 7 removed |
Killer Sudoku keeps every rule of Classic Sudoku — fill the 9×9 grid so each row, column, and 3×3 box holds the digits 1 to 9 exactly once — and adds cages: dashed groups of cells with a small target sum, where no digit may repeat inside a cage. Puzzles usually start with zero given digits, so every deduction begins from the cage sums. That single extra rule unlocks a whole family of arithmetic techniques.
Practise this in Killer mode on Sudoku Challenge, where the in-game Next Step button walks the exact deduction taught in this lesson.
A naked pair inside a cage is 2 cells of that cage whose candidates have shrunk to the same 2 digits. Those 2 digits belong to those 2 cells in some order, and because a cage never repeats a digit, both can be deleted from every other cell of the same cage.
Not on its own — the deduction reaches only the cage's other cells. If the locked cells all sit inside one row, column, or box, the same digits also form an ordinary naked subset in that house and may be cleared there, but that is a second deduction resting on the house rather than on the cage.
A unique-combination cage is settled by size and total alone: 3 cells totalling 23 can only hold 6, 8, and 9. A cage naked subset depends on the current candidate lists instead, firing whenever a few cells have narrowed to matching digits, whatever the cage total, and it prunes the cage's remaining cells.
No, the elimination rests on the no-repeat rule rather than on the total. The sum still matters twice over: combination analysis is usually what trims the candidate lists far enough for a subset to become visible, and once the subset is locked, the cage's remaining cells inherit a known smaller total.
At least 3. You need cells to form the locked group and at least 1 more to strip, so a pair pays from 3 empty cells upward, a triple from 4, and a quad from 5. A cage with only 2 empty cells offers nothing, because the pair is the entire remainder.
Most likely the candidate lists have not been filtered by the cage yet. Mark each cell with the digits its row, column, and box allow, keep only those appearing in a combination the cage can still be filled with, then drop any digit already placed in that cage. Subsets surface after that pass.
Explore every Classic and Killer technique in the complete Learning Hub, or read how to play Sudoku to get started.