Some cages can only be filled one way — spot them and the digits are handed to you.
Scan for cages sitting at the extremes of their range: 3, 4, 16 or 17 in 2 cells, 6, 7, 23 or 24 in 3 cells, 10, 11, 29 or 30 in 4 cells. Each of those totals has exactly 1 digit set, so every other digit can be struck from the cage immediately.
A cage tells you two things for free: how many cells it has, and what they must add up to. Usually that leaves several possible digit sets — but sometimes it leaves exactly one, and those cells are pinned to that set before you place a single digit. The trick is knowing where to look: for any cage size, only the two LOWEST and the two HIGHEST reachable totals are forced. Two cells: 3 (1+2) and 4 (1+3) at the bottom, 16 (7+9) and 17 (8+9) at the top. Three cells: 6, 7, 23, 24. Four cells: 10, 11, 29, 30. Five cells: 15, 16, 34, 35. Learn to scan for those numbers — it is the first thing an experienced Killer solver does on a new board.
The number of digit sets fitting a given cage size and total peaks in the middle of the range and falls away at both ends. A 3-cell cage can total 6 to 24, and across that range the count of valid sets runs 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1. Only the 4 totals at the ends — 6, 7, 23 and 24 — leave a single set standing.
The arithmetic is short. Three different digits cannot total less than 1 + 2 + 3 = 6, so a cage of 6 is exactly 1, 2, 3. Reaching 7 means gaining 1 over that minimum, and the only way to do so without repeating a digit is to lift the top member from 3 to 4, giving 1, 2, 4. The same argument runs down from the ceiling: 7 + 8 + 9 = 24 is the most 3 digits can make, and shedding exactly 1 means pulling the 7 to 6, giving 6, 8, 9. From the third total inward the surplus can be spread 2 ways, and uniqueness collapses.
That pattern holds at every cage size the technique covers. Sizes 2–7 each have precisely 4 forced totals — the minimum, the minimum plus 1, 1 below the maximum, and the maximum — 24 size-and-total pairs in all, well worth committing to memory.
These cages are found by reading printed clue numbers, not by working with candidates, so the scan costs nothing and belongs at the start of a solve. Sweep the grid once for 2-cell cages totalling 3, 4, 16 or 17, again for 3-cell cages at 6, 7, 23 or 24, then for 4-cell cages at 10, 11, 29 or 30.
Geometry decides how much each find is worth. A forced cage whose cells all sit inside one box, row or column doubles as a naked subset in that unit: 3 cells holding 6, 8 and 9 between them strip those digits from every other cell of the unit. A cage of the same total straddling 2 boxes fixes its own cells but radiates nothing outward.
Sudoku Challenge's step solver applies the technique in that spirit. It considers cages of 2–7 cells with no digit yet placed, checks whether the size and total admit exactly 1 combination, and if so removes every digit outside that combination from the cage's cells. The move is an elimination rather than a placement, and sits immediately below naked and hidden singles on the ladder.
The lesson board above carries exactly one forced cage, and it is a small one: r1c9 and r2c9, printed total 17. With 2 cells the range is 3–17, so 17 is the ceiling — 8 + 9, the largest 2 different digits can make. Nothing else reaches it, so those cells hold 8 and 9 in some order. The board's other 32 multi-cell cages all sit away from their extremes, so this is the whole harvest.
The immediate payoff is inside the cage. Digits 2, 3, 4, 5, 6 and 7 come out of both cells, and 1 was never a candidate there because r2c8 already holds the 1 for box 3 — so the step records 12 removals, 6 from each cell. Note what has not been established: nothing yet says which of 8 and 9 belongs to r1c9 rather than r2c9. The set is fixed, the arrangement is not.
The larger payoff is outward. Both cells lie in column 9 and both lie in box 3, so the cage doubles as a naked pair in each unit, consuming 8 and 9 in both. Those digits leave every other cell of the 2 units — a union of 13 cells, 12 of them still empty, for 24 further removals.
Now chain it. r3c8 and r3c9 form a 2-cell cage totalling 13, a total that on its own allows 4 + 9, 5 + 8 or 6 + 7. Both cells sit in box 3, which has just surrendered its 8 and 9, killing the first 2 options — so that pair must be 6 and 7. A Rule of 45 check then squeezes the box: the cages lying wholly inside box 3 total 8 + 17 + 1 + 13 = 39, leaving r2c7 and r3c7 to make 45 − 39 = 6. Those 2 cells are part of the 14-cage running r2c7, r3c7, r4c7, so r4c7 = 14 − 6 = 8 — a placement, grown from a single printed number.
The commonest misapplication is expecting a placement. A unique combination cage never places a digit on its own; it restricts. A placement follows only when a row, column, box or neighbouring cage narrows the arrangement, so treat the find as groundwork.
The second is reaching for it after the cage has been partly filled. Once a digit is written in, the size-and-total argument is superseded by the stronger question of which combinations remain placeable alongside the digit already sitting there — the residual cage-combination technique. Sudoku Challenge's solver skips any cage with a placed cell for that reason.
The third is over-hunting. A 3-cell cage totalling 15 has 8 possible sets and a 4-cell cage totalling 20 has 12; enumerating those by hand early in a solve yields nothing. Confine the scan to the 4 extreme totals at each size and move on.
Cages of 8 cells are a degenerate case worth knowing. Every total from 36 to 44 admits exactly 1 set, because 8 distinct digits are 1–9 with one missing and that missing digit is 45 minus the total. Such cages are rare, and the deduction belongs to the Rule of 45 anyway.
| Cage size | Total range | Lowest | Second lowest | Second highest | Highest |
|---|---|---|---|---|---|
| 2 cells | 3–17 | 3 = 1+2 | 4 = 1+3 | 16 = 7+9 | 17 = 8+9 |
| 3 cells | 6–24 | 6 = 1+2+3 | 7 = 1+2+4 | 23 = 6+8+9 | 24 = 7+8+9 |
| 4 cells | 10–30 | 10 = 1+2+3+4 | 11 = 1+2+3+5 | 29 = 5+7+8+9 | 30 = 6+7+8+9 |
| 5 cells | 15–35 | 15 = 1+2+3+4+5 | 16 = 1+2+3+4+6 | 34 = 4+6+7+8+9 | 35 = 5+6+7+8+9 |
| 6 cells | 21–39 | 21 = 1+2+3+4+5+6 | 22 = 1+2+3+4+5+7 | 38 = 3+5+6+7+8+9 | 39 = 4+5+6+7+8+9 |
| 7 cells | 28–42 | 28 = 1+2+3+4+5+6+7 | 29 = 1+2+3+4+5+6+8 | 41 = 2+4+5+6+7+8+9 | 42 = 3+4+5+6+7+8+9 |
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.
For every cage size from 2 to 7, exactly 4 totals are forced: the 2 lowest and the 2 highest that size can reach. Two cells: 3, 4, 16, 17. Three cells: 6, 7, 23, 24. Four cells: 10, 11, 29, 30. Five cells: 15, 16, 34, 35. Six cells: 21, 22, 38, 39. Seven cells: 28, 29, 41, 42.
No — a unique combination cage fixes the set of digits, not their arrangement. A 3-cell cage totalling 23 must hold 6, 8 and 9, but any of the 3 could sit in any of the 3 cells. Placement arrives only when a row, column, box or neighbouring cage rules out the alternatives.
Because the surplus above the minimum can be spread across the digits several ways. A 3-cell cage totalling 6 has 1 set, one totalling 15 has 8 sets, and one totalling 24 has 1 again. Only at the extremes is there a single route to the total, which is why the scan targets the ends of each range.
Run the forced-cage scan first as a one-off pass, since it needs no candidate work at all, then solve singles. Sudoku Challenge's step solver orders it the other way — naked singles, hidden singles, then unique combination cages — because once candidates exist, singles are cheaper to detect than a cage sweep.
Explore every Classic and Killer technique in the complete Learning Hub, or read how to play Sudoku to get started.