Cage Combinations

What Is It?

A cage tells you its total and how many cells it spans — and that already rules out digits. List every way to make the sum with distinct digits; any digit that appears in NO surviving combination cannot go anywhere in the cage.

Reading a cage as a list of digit sets

Two numbers start the work: how many cells the cage covers, and what those cells must total. Because a cage never repeats a digit, only particular sets of distinct digits reach that total. A 3-cell cage totalling 20 can be filled by 3+8+9, 4+7+9, 5+6+9 or 5+7+8, and by nothing else. Those 4 sets are the cage's combination list, and the union of the digits they use — 3, 4, 5, 6, 7, 8 and 9 — is the whole inventory of what the cage may hold. Digits 1 and 2 are out of all 3 cells before any other constraint is consulted.

The list is never static. A digit already sitting inside the cage cannot appear a second time, and a digit blocked in every cage cell by its row, column or box makes any combination containing it unfillable. The Killer engine on sudokuchallenge rebuilds the surviving list each time it examines the cage, so one cage can yield a fresh elimination several times across a solve rather than once at the start.

Turning the list into eliminations

The routine has 4 steps. Read the cage's size and total. Write out every set of distinct digits of that size reaching that total. Strike out any set that cannot be laid into the cage's cells given what each cell can still be — the digits have to go somewhere, one per cell. Then take the union of the survivors and delete every digit outside it from every cell of the cage.

Two different eliminations come out of that. The plain one is a digit that appears in no surviving set at all: it leaves the entire cage in one stroke. The finer one is a digit that does belong to a surviving set yet cannot sit in one particular cell, because the set's other digits would then have nowhere to go. The engine matches a combination's digits to cells one-to-one before accepting the combination, and again before accepting a digit in a given cell.

Two-cell cages make the finer case easy to state, since every digit there has exactly one partner. If the cage totals 8, a 3 in one cell obliges a 5 in the other; if that other cell cannot hold a 5, the 3 goes as well.

Worked example: 8 across two cells

The board in this lesson opens with a 2-cell cage at R1C5 and R1C6 totalling 8, both cells empty and no digits placed anywhere. Two different digits reaching 8 gives 1+7, 2+6 and 3+5. The tempting fourth option, 4+4, is unavailable, because a cage cannot repeat a digit.

The union of those 3 sets is 1, 2, 3, 5, 6 and 7, so 4, 8 and 9 come out of both cells. Each removal carries its own arithmetic. A 4 would need a second 4 beside it. An 8 reaches the total on its own and leaves nothing for a partner. A 9 overshoots 8 before a partner is even chosen. Both cells fall from 9 candidates to 6.

Now let the solve run on, and suppose a 7 appears elsewhere in column 5 so that R1C5 can no longer be 7. All 3 combinations still survive — 1+7 remains available as R1C5 = 1 with R1C6 = 7 — so no digit leaves the cage outright. The placement question, though, has become one-sided. R1C6 = 1 would force R1C5 = 7, which is now impossible, so the 1 comes out of R1C6 and stays in R1C5. The cage settles with R1C5 holding 1, 2, 3, 5 or 6 and R1C6 holding 2, 3, 5, 6 or 7 — an asymmetry the union alone would never reveal.

Where the technique pays, by cage size

Cage size decides how much a combination list is worth. Every 2-cell total from 3–17 rules at least one digit out, which makes small cages the reliable place to start. The weakest are 9, 10 and 11, each barring one digit: a 9 has no partner in a 9-cage, a 5 would need a second 5 in a 10-cage, and a 1 would need a 10 in an 11-cage.

From 3 cells upward the yield collapses in the middle of the range. A 3-cell cage totalling anywhere from 12–18 spreads its combinations across all 9 digits and rules nothing out. The same holds for 4-cell cages at 15–25, 5-cell cages at 19–31, 6-cell cages at 24–36 and 7-cell cages at 30–40. Reading a large mid-range cage combination by combination returns nothing.

The payoff sits at the ends of each range. A 3-cell cage totalling 22 has only 5+8+9 and 6+7+9, striking 1, 2, 3 and 4 out of all 3 cells at once; a 4-cell cage totalling 12 has only 1+2+3+6 and 1+2+4+5, striking out 7, 8 and 9.

Common ways to misapply it

The most frequent slip is forgetting that combinations use distinct digits, which invents entries such as 4+4 for a 2-cell 8 or 5+5 for a 2-cell 10 and inflates the union. The opposite slip is reading a digit that appears in most combinations as guaranteed. In the 3-cell 20 cage, 9 appears in 3 of the 4 combinations, but 5+7+8 contains none, so no 9 is promised anywhere.

A combination list names digits, never cells. Knowing a 3-cell cage holds 5+8+9 or 6+7+9 says nothing about which cell takes the 9 until row, column and box constraints settle it. The union is an upper bound rather than a candidate list, and the ordinary Sudoku exclusions still apply on top of it.

Working from a stale list is the practical error. Every digit placed inside the cage, and every digit placed where a cage cell can see it, may retire a combination and open a fresh elimination. A cage whose size and total alone admit a single combination is reported as its own separate step, so this technique carries the cages that keep two or more.

Cages with two or more combinations that still rule digits out (2–4 cells)

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 rule out the most digits?

The totals nearest a cage's minimum and maximum rule out the most. A 2-cell cage totalling 5 admits only 1+4 and 2+3, barring 5, 6, 7, 8 and 9. A 3-cell cage totalling 22 admits only 5+8+9 and 6+7+9, barring 1, 2, 3 and 4. Mid-range totals bar nothing.

How many combinations can a killer cage have?

No cage has more than 12 combinations. A 4-cell cage totalling 20 and a 5-cell cage totalling 25 each reach that maximum. 3-cell cages peak at 8 combinations (totals 14, 15 and 16), 2-cell cages at 4 (totals 9, 10 and 11), and 7-cell cages at 4 (totals 34, 35 and 36).

Why does a cage combination remove a digit from one cell but not its neighbour?

Because the digit still belongs to a live combination yet cannot be placed in that one cell. In a 2-cell cage totalling 8, if one cell has already lost its 7, the combination 1+7 survives in only one arrangement, so the other cell loses its 1 while the first cell keeps it.

Which cage totals give no eliminations at all?

Mid-range totals give none. A 3-cell cage totalling 12–18 spreads its combinations across all 9 digits, so enumeration removes nothing, and the same is true of 4-cell cages at 15–25, 5-cell cages at 19–31 and 6-cell cages at 24–36. Every 2-cell total removes something.

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