Cage Must-Contain

What Is It?

Sometimes every single way to fill a cage uses the same digit. That digit is guaranteed to live inside the cage — so if the cage sits within one row, column, or box, that unit’s copy of the digit is used up, and you can erase it from the rest of the unit.

Why a shared digit is a certainty

A cage of n cells with a given total admits only a finite list of digit sets, because the cells hold distinct digits from 1–9. Write that list out and look at what the sets have in common. If one digit survives in every set, then whichever filling turns out to be correct, that digit occupies one of the cage's cells. The cage does not say which cell — only that the digit is somewhere in there. That certainty, weak as it sounds, is what the whole technique trades on, and it costs nothing more than reading the cage's size and total off the board.

On its own a guaranteed digit changes nothing. It becomes useful the moment the cage sits entirely inside a single row, column or box. A unit holds each digit exactly once, so if the unit's copy of that digit is already spoken for by the cage, no cell that lies inside the unit but outside the cage can hold it. Every such cell loses the digit from its candidate list. The cage behaves as a container that has quietly claimed one of the unit's nine slots, and the claim is enforceable even though the slot has no fixed address. The board on this page carries a clean specimen: the three-cell cage totalling 8 at R2C6, R3C5 and R3C6 never leaves box 2, and 8 across three cells is only 1+2+5 or 1+3+4, so box 2's single 1 is inside that cage and the box's six other cells lose it.

Working through it at the board

The scan is mechanical. First, find cages that fit inside one unit — a horizontal cage that never leaves its row, a vertical cage that never leaves its column, or a compact cage that sits inside one 3×3 box. Cages of 2–6 cells are the practical range, and the Next Step engine on this site scans exactly that range. Second, enumerate the combinations the size and total allow. Third, intersect those combinations. Fourth, check whether the shared digit is still a candidate somewhere else in the unit; if it is not, the pattern is real but yields nothing, and you move on to the next cage. The 12-cage at R7C1, R8C1 and R9C1 on this board shows why the order matters: it passes the geometry test twice over — column 1 and box 7 both contain it — yet 12 across three cells has seven combinations that share no digit, so the scan ends there.

The combination list is not fixed for the whole solve. A combination only counts if its digits can each land in a distinct cage cell given what those cells still allow, so as candidates thin out, combinations drop away and the intersection grows. A cage that guaranteed nothing at the start can guarantee a digit twenty moves later. That is why the pattern is worth re-checking after any placement that touches a single-unit cage, and why a static lookup of totals is a starting point rather than the whole answer. The engine recomputes the surviving combinations every time it is asked for a step, which is why it will sometimes offer a must-contain on a cage you had already dismissed as barren.

Totals that force a digit before you start

Some totals guarantee a digit before any candidate work at all. Two-cell cages never do once more than one combination survives: a pair must be {d, S − d}, so two different pairs for the same total can share no digit. The only two-cell cages with a forced digit are those with a single combination — totals of 3, 4, 16 and 17 — and those belong to the unique-combination pattern rather than here. From three cells upward the picture changes, because a set can be varied in more than one place while one member stays fixed. The two-cell cage totalling 7 at R3C7 and R3C8 on this board is the type case: it sits inside both row 3 and box 3, and still forces nothing, because 1+6, 2+5 and 3+4 have no digit in common.

The table on this page lists every cage of 2–6 cells whose total leaves more than one combination and still forces a digit. The shape of it is that totals near a size's minimum force low digits, totals near its maximum force high digits, and the guarantee weakens as the total moves towards the middle. A 3-cell cage summing to 8 must hold a 1; a 3-cell cage summing to 22 must hold a 9; a 3-cell cage summing to 15 has eight combinations and forces nothing at all. The broad middle of each size's range never forces anything, which is why the table has a hole through its centre.

A worked example: 22 across three cells

Take the cage on this lesson's board that occupies R5C9, R6C8 and R6C9 and totals 22. All three cells lie in box 6, so the containment condition holds. Three distinct digits from 1–9 adding to 22 can only be 5 + 8 + 9 or 6 + 7 + 9; nothing else reaches 22 without repeating a digit, since 4 + 9 + 9 and 7 + 8 + 7 are illegal and 6 + 8 + 9 overshoots at 23. Both surviving sets contain a 9. Box 6 therefore has its 9 inside the cage, and the 9 can be struck from the box's other six cells: R4C7, R4C8, R4C9, R5C7, R5C8 and R6C7.

Notice what the step does not do. It does not say whether the 9 sits at R5C9, R6C8 or R6C9 — all three keep it as a candidate — and it does not resolve 5/8 against 6/7 for the other two cells. It buys six eliminations outside the cage and nothing inside it. Shift the total by one and the technique evaporates: a 3-cell cage summing to 21 allows 4 + 8 + 9, 5 + 7 + 9 and 6 + 7 + 8, which share no digit at all, so no elimination follows however neatly the cage sits inside its box.

The same board carries a second, larger case. The cage running R1C3–R1C8 totals 34 across six cells, and five combinations reach 34 with six distinct digits: 1 + 3 + 6 + 7 + 8 + 9, 1 + 4 + 5 + 7 + 8 + 9, 2 + 3 + 5 + 7 + 8 + 9, 2 + 4 + 5 + 6 + 8 + 9 and 3 + 4 + 5 + 6 + 7 + 9. Every one of them contains a 9. The cage lies wholly in row 1, so row 1's 9 lives inside it and the digit leaves R1C1, R1C2 and R1C9. Five surviving combinations is a wide, undetermined list, which shows the cage need not be anywhere near pinned down for the technique to pay.

Where it fails and how it is misapplied

The containment condition is strict, and it is the usual place the technique goes wrong. A cage that spills across two boxes, or bends out of its row into the row below, still guarantees its digit, but there is no single unit whose copy of that digit is thereby consumed, so nothing can be erased anywhere. Check the geometry before doing the arithmetic — it is the cheaper test by a wide margin. An L-shaped or otherwise irregular cage inside one box does qualify: the cage's shape is irrelevant, and only the set of units that fully contain it matters. This board carries exactly that failure: the six-cell cage totalling 34 at R6C6, R7C5, R7C6, R8C5, R8C6 and R9C5 guarantees a 9 just as firmly as row 1's 34-cage does, but R6C6 lies in box 5 while the other five sit in box 8, so no unit's 9 is consumed and not one candidate can be struck.

Two further confusions are worth naming. A cage with exactly one surviving combination guarantees every digit in that combination, which is a stronger and separately named pattern; treat must-contain as the case where the combination list is still plural. And a cage can guarantee more than one digit at once — a 4-cell cage totalling 28 allows only 4 + 7 + 8 + 9 and 5 + 6 + 8 + 9, so both 8 and 9 are trapped, and each supports its own sweep of the enclosing unit. Work the guaranteed digits one at a time, applying every elimination one licenses before moving to the next, or the second digit's eliminations are easy to overlook.

Cages with more than one possible combination that still force a digit, by size and total

How Killer Sudoku Cages Work

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.

Frequently Asked Questions

Which cage sums guarantee a digit in Killer Sudoku?

Totals near the extremes of a cage's range guarantee a digit: 3 cells summing to 8 or 22, 4 cells to 12, 13, 27 or 28, 5 cells to 17–19 or 31–33, and 6 cells to 23–26 or 34–37. Middle totals leave too many combinations to share anything.

Does Cage Must-Contain work if the cage crosses two boxes?

No, a cage that crosses two boxes or two rows produces no elimination. The guaranteed digit still sits somewhere inside the cage, but no single row, column or box has its copy of that digit consumed, so there is no unit from which the digit can be removed.

Can one cage guarantee more than one digit?

Yes, a cage can trap several digits at once. A 4-cell cage totalling 28 allows only 4+7+8+9 and 5+6+8+9, so 8 and 9 are both guaranteed; a 5-cell cage totalling 33 allows 3+6+7+8+9 and 4+5+7+8+9, guaranteeing 7, 8 and 9. Each trapped digit clears the enclosing unit separately.

Is Cage Must-Contain the same as a unique combination cage?

No. A unique-combination cage has exactly one possible digit set, so every digit in it is fixed to the cage. Cage Must-Contain applies when several combinations survive and they happen to overlap on one or more digits. The must-contain case is weaker, far more common, and easier to walk past.

Why do two-cell cages never produce a must-contain elimination?

Because a two-cell cage's combinations are pairs of the form {d, S − d}, and two different pairs with the same total can never share a digit. Two-cell cages force digits only when a single combination remains — totals of 3, 4, 16 and 17 — which is the unique-combination pattern rather than a must-contain.

Does the guaranteed digit tell me which cell it goes in?

No, the technique fixes the digit to the cage, not to a cell. The step removes the digit from no cage cell, and the whole payoff lands outside the cage, in the rest of the enclosing row, column or box. Placing it needs a further step, such as a hidden single or a subset.

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