Killer Sudoku cage combinations — chart and calculator

Cage combination calculator

Pick the cage size and the target sum. If the surrounding rows, columns, or box have already ruled digits out of the cage, tap them under Exclude digits and the list narrows instantly. Everything runs in your browser — nothing is sent anywhere.

Two reading tips. First, when the calculator shows exactly one combination, every digit in it is locked into the cage — you can pencil-mark all of them immediately. Second, look for digits that appear in every listed combination: a three-cell 22 has two combinations (5+8+9 and 6+7+9), but both contain a 9, so the 9 is certain even though the full set is not. Shared digits are deductions in disguise.

Reference chart: 2-cell cages (sums 3–17)

Two-cell cages are the workhorses of killer solving — never more than four combinations, and four sums with exactly one. Highlighted rows have a single combination: memorize those first.

Notice the symmetry: sum 3 mirrors sum 17, sum 4 mirrors 16, and so on around the midpoint. That's because replacing every digit d with 10−d turns a combination for sum s into one for 20−s. The same mirror trick works for every cage size (around 30 for three cells, 40 for four), so you only ever need to memorize half of any chart.

Also note that a two-cell cage can never sum to 2 or to anything over 17 — and no two-cell sum can use the digit that is half of it (a 10-cage never contains a 5, a 12-cage never contains a 6), since that would force a repeat.

Reference chart: 3-cell cages (sums 6–24)

Three-cell cages range from sum 6 (1+2+3) to 24 (7+8+9). Highlighted rows again mark the single-combination sums — 6, 7, 23, and 24.

Beyond the unique sums, watch for forced digits in the near-unique rows: every 22-cage contains a 9 and every 8-cage contains a 1, even though neither sum is unique. Not every low-count row obliges — a 20-cage forces nothing, since 5+7+8 avoids the 9 the other three combinations share. Whether a cage forces a digit is exactly the kind of second-order question the calculator answers faster than a static chart: exclude a digit and see whether any combination survives.

Cage sums with only one combination

These are the cages that crack a puzzle open, because the digit set is fully determined the moment you see the sum. There are four per cage size, always the two smallest and two largest possible sums:

• 2 cells: 3 = 1+2 · 4 = 1+3 · 16 = 7+9 · 17 = 8+9
• 3 cells: 6 = 1+2+3 · 7 = 1+2+4 · 23 = 6+8+9 · 24 = 7+8+9
• 4 cells: 10 = 1+2+3+4 · 11 = 1+2+3+5 · 29 = 5+7+8+9 · 30 = 6+7+8+9
• 5 cells: 15 = 1+2+3+4+5 · 16 = 1+2+3+4+6 · 34 = 4+6+7+8+9 · 35 = 5+6+7+8+9

The pattern continues (6 cells: 21, 22, 38, 39; 7 cells: 28, 29, 41, 42; 8 cells: any sum, since eight distinct digits are just "45 minus the missing one"). A unique-combination cage confined to a single box or line also fires eliminations outward: a 30-cage inside one box strips 6, 7, 8, and 9 from every other cell of that box.

The 45 rule, innies, and outies

The charts above cover printed cages — but the strongest killer solvers manufacture their own. Since the digits 1–9 sum to 45, every row, column, and 3×3 box totals exactly 45. Add up the cages covering a box: if they total 48 with one cell poking out (an outie), that cell is 48 − 45 = 3. If cages inside a box total 41 and leave one cell uncovered (an innie), that cell is 45 − 41 = 4.

When the innie or outie is two or three cells instead of one, you've effectively created a new cage with a sum you computed yourself — and if those cells share a row, column, or box, the no-repeat rule applies too. That's when you come back to this page: a self-made two-cell "cage" summing 16 is 7+9 with certainty, and a three-cell innie of 7 is 1+2+4, straight off the chart. The 45 rule generates the cage; the combination list converts it into digits. For the full technique with a worked example, see our killer sudoku guide.

How to actually use combinations at the board

A chart is only as useful as the habits around it. The workflow that works:

• Open with the unique sums. Scan the whole grid for the sixteen unique 2–5-cell sums before entering anything, and pencil-mark their digit sets. On easier puzzles this alone places your first digits.
• Look for digits common to all combinations. A two-cell 14 (5+9, 6+8) locks nothing, but a two-cell 15 (6+9, 7+8) locks nothing either — while a three-cell 22 locks a 9 and a three-cell 8 locks a 1. Common digits are certainties; pencil-mark them as such.
• Feed eliminations back in. Whenever a row, column, or box rules a digit out of a cage, re-derive the cage's combinations with that digit excluded — this is precisely what the calculator's toggle buttons are for. Cages collapse fast once one or two digits are banned.
• Don't memorize the big tables. Nobody memorizes the 4-cell chart (it peaks at twelve combinations for sums 18–22). Memorize the unique sums and the mirror symmetry; look the rest up.

If you're playing in Sudoku by Joseph, the app's pencil marks are the natural place to store what the chart tells you — and since killer is in the free tier with no lives or mistake counters, a mis-marked cage costs you nothing but the correction.

Other options worth knowing

Other combination references worth knowing: SudokuWiki's killer combinations page lists every cage size in full static tables and pairs well with its strategy articles. BrainBashers has a compact printable chart if you solve on paper. DailyKillerSudoku.com shows combinations contextually inside its browser player. The gap this page fills is the interactive part: filtering by excluded digits — the operation you actually perform mid-solve — is something a static table can't do.

Frequently asked questions