Killer sudoku cage combinations

A cage sum is a clue about a set of digits. Most solvers learn the two famous ones — a two-cell 3 is 1 and 2, a two-cell 17 is 8 and 9 — and then guess at the rest. The tables below are the whole set, enumerated rather than remembered, so you can look up any cage on the board instead of re-deriving it in the margin.

What the tables assume

Three assumptions hold everywhere on this page, and they are the ones that make the arithmetic come out the way it does.

The digit set is fixed by the board. A 9×9 killer sudoku uses the digits 1 to 9. A 6×6 uses 1 to 6. Those are different puzzles with different answers, so the 6×6 tables live in their own section at the bottom and the two sets never mix.
Digits are distinct inside a cage. Every combination below uses each digit at most once. That is what makes a cage of k cells have a smallest and largest possible total in the first place.
These are combinations, not permutations. 1 2 6 and 6 2 1 are one entry, written in ascending order. Which cell holds which digit is a separate question that the rest of the grid answers.

Cages are written as digit strings throughout: 126 means the set {1, 2, 6}.

Sum ranges on a 9×9 board

The smallest total a cage of k cells can hold is 1+2+…+k. The largest is 9+8+…+(10−k). Everything in between is reachable, so each cage size has an unbroken band of possible sums. A printed sum outside its cage’s band is a misprint, not a hard puzzle.

Cage size	Lowest sum	Highest sum	Possible sums	Combinations in total
2 cells	3	17	15	36
3 cells	6	24	19	84
4 cells	10	30	21	126
5 cells	15	35	21	126
6 cells	21	39	19	84
7 cells	28	42	15	36
8 cells	36	44	9	9
9 cells	45	45	1	1

Two things fall straight out of that table. A nine-cell cage is always the full digit set, so it totals 45 and tells you nothing new. And the counts mirror: 36 and 36, 84 and 84, 126 and 126. That is the complement rule, and it is worth the paragraph below.

The complement rule. A cage of k cells summing to S has exactly the same number of combinations as a cage of 9−k cells summing to 45−S, and each combination of one is the set of digits the other leaves out. A two-cell 3 is 12; a seven-cell 42 is 3456789. So the whole seven-cell table is the two-cell table read backwards, and the eight-cell table is just “45 minus the one digit you left out” — an eight-cell 36 leaves out the 9, an eight-cell 44 leaves out the 1.

Locked sums: one combination only

This is the table to memorise. A locked sum has exactly one combination, so the moment you see it you know the cage’s digits — before you know their order, before you know anything else on the board. Locked cages are where nearly every killer sudoku opens.

Cage size	Sums with exactly one combination
2 cells	3 = 12, 4 = 13, 16 = 79, 17 = 89
3 cells	6 = 123, 7 = 124, 23 = 689, 24 = 789
4 cells	10 = 1234, 11 = 1235, 29 = 5789, 30 = 6789
5 cells	15 = 12345, 16 = 12346, 34 = 46789, 35 = 56789
6 cells	21 = 123456, 22 = 123457, 38 = 356789, 39 = 456789
7 cells	28 = 1234567, 29 = 1234568, 41 = 2456789, 42 = 3456789

The pattern is the same at every size: the two lowest sums and the two highest sums are locked, and nothing else is. That is because the lowest sum has to be the smallest digits, the next sum up can only be reached by pushing the top digit one step, and the same argument runs from the other end. Eight-cell cages are locked at every sum, since the sum names the missing digit.

Almost as useful are the sums with exactly two combinations, because one eliminated digit finishes them. Cages of two to five cells are listed here because that is the range a printed killer sudoku actually uses; by the complement rule the larger cages have them too, at six cells 23 and 37, and at seven cells 30, 31, 39 and 40.

Cage size	Sums with exactly two combinations
2 cells	5 = 14/23, 6 = 15/24, 14 = 59/68, 15 = 69/78
3 cells	8 = 125/134, 22 = 589/679
4 cells	12 = 1236/1245, 28 = 4789/5689
5 cells	17 = 12347/12356, 33 = 36789/45789

Every two-cell cage

Fifteen sums, thirty-six combinations, and you will use this table more than all the others put together. The last column is the set of digits that cannot appear in a cage of that sum at all, which is often the faster deduction: a two-cell 11 rules the 1 out of both its cells no matter what else happens.

Sum	Combinations	How many	Digits ruled out
3	12	1	3456789
4	13	1	2456789
5	14 23	2	56789
6	15 24	2	36789
7	16 25 34	3	789
8	17 26 35	3	489
9	18 27 36 45	4	9
10	19 28 37 46	4	5
11	29 38 47 56	4	1
12	39 48 57	3	126
13	49 58 67	3	123
14	59 68	2	12347
15	69 78	2	12345
16	79	1	1234568
17	89	1	1234567

Three oddities in that column are worth a second look. A two-cell 9 can never contain a 9, because the other cell would have to be a 0. A two-cell 10 can never contain a 5, because the other cell would repeat it. And a two-cell 12 can never contain a 6, for the same reason.

Every three-cell cage

Nineteen sums, eighty-four combinations. The middle of this table is wide open — a three-cell 14 or 15 has eight combinations and constrains almost nothing on its own — so the value sits at the two ends.

Sum	Combinations	How many	Digits ruled out
6	123	1	456789
7	124	1	356789
8	125 134	2	6789
9	126 135 234	3	789
10	127 136 145 235	4	89
11	128 137 146 236 245	5	9
12	129 138 147 156 237 246 345	7	—
13	139 148 157 238 247 256 346	7	—
14	149 158 167 239 248 257 347 356	8	—
15	159 168 249 258 267 348 357 456	8	—
16	169 178 259 268 349 358 367 457	8	—
17	179 269 278 359 368 458 467	7	—
18	189 279 369 378 459 468 567	7	—
19	289 379 469 478 568	5	1
20	389 479 569 578	4	12
21	489 579 678	3	123
22	589 679	2	1234
23	689	1	123457
24	789	1	123456

A three-cell 8 always contains the 1, and a three-cell 22 always contains the 9. Those “always includes” facts are the other half of the table, and at four and five cells they get much stronger.

Four- and five-cell cages: work the ends

There are 126 combinations at each of these sizes, and listing all of them buys you nothing — a four-cell 20 has twelve combinations and is a clue about almost nothing. What pays is the band at either end, where the count drops to three or fewer and the “always includes” column starts doing real work.

Cage	Sum	Combinations	How many	Always includes	Digits ruled out
4 cells	10	1234	1	1234	56789
4 cells	11	1235	1	1235	46789
4 cells	12	1236 1245	2	12	789
4 cells	13	1237 1246 1345	3	1	89
4 cells	14	(5 combinations)	5	—	9
4 cells	26	(5 combinations)	5	—	1
4 cells	27	3789 4689 5679	3	9	12
4 cells	28	4789 5689	2	89	123
4 cells	29	5789	1	5789	12346
4 cells	30	6789	1	6789	12345
5 cells	15	12345	1	12345	6789
5 cells	16	12346	1	12346	5789
5 cells	17	12347 12356	2	123	89
5 cells	18	12348 12357 12456	3	12	9
5 cells	19	(5 combinations)	5	1	—
5 cells	31	(5 combinations)	5	9	—
5 cells	32	26789 35789 45689	3	89	1
5 cells	33	36789 45789	2	789	12
5 cells	34	46789	1	46789	1235
5 cells	35	56789	1	56789	1234

The five-cell 19 is the sleeper in that list. It has five combinations, so it looks unhelpful, and every one of them contains the 1. If that cage sits inside a single box, you have just placed that box’s 1 into five known cells — which is frequently enough to finish a different cage entirely. The five-cell 31 does the same thing for the 9.

The 45 rule

Every row, every column and every box on a 9×9 board holds each digit from 1 to 9 exactly once. So each of them totals 45. That single number turns cage sums into a second set of clues.

If a row, column or box is covered exactly by whole cages, their sums add to 45 — a useful arithmetic check on a board you are about to publish. More often the coverage is nearly exact, and that is where it pays: if every cage in a box lies inside it except for one stray cell, that cell holds 45 minus the sum of the whole cages. If two cells stray out of the box, you get their total instead of a digit, which is still a clue a two-cell table can finish.

On a 6×6 board the same argument gives 21, since 1+2+3+4+5+6 = 21.

Worked example one: a box that resolves itself

A 3×3 box holds nine cells. Suppose four cages sit entirely inside it — a two-cell cage printing 16, a three-cell cage printing 7, a two-cell cage printing 8, and a two-cell cage printing 14. Two plus three plus two plus two is nine cells, and 16 + 7 + 8 + 14 = 45, so the cages cover the box exactly.

The 16 is locked. Two cells totalling 16 can only be 79. Those two cells hold the box’s 7 and its 9.
The 7 is locked. Three cells totalling 7 can only be 124. Those three hold the box’s 1, 2 and 4.
That accounts for five digits. The box’s remaining four cells must hold the digits nobody has claimed: 3, 5, 6 and 8. They add to 22, which matches the 8 and the 14 still to be placed.
The 8 resolves. A two-cell 8 is 17, 26 or 35. The 1 and the 7 are already spoken for, and so is the 2. Only 35 survives.
The 14 resolves. A two-cell 14 is 59 or 68. The 9 is in the 16 cage. Only 68 survives.

Every cage in that box now has a known digit set, and not one digit was placed by a sudoku row-or-column argument. The sums did all of it. Which cell of the 79 cage holds the 7 is settled later, by the rows and columns running through the box.

Worked example two: one digit elsewhere kills two combinations

Now a column, with a three-cell cage printing 21 and a two-cell cage printing 4 inside it, plus a given 8 somewhere else in the same column.

The 4 is locked. Two cells totalling 4 can only be 13. The column’s 1 and 3 are in those two cells, and nowhere else in the column.
The 21 has three candidates. From the three-cell table: 489, 579, 678.
The 8 is already used in this column, so the cage cannot hold another one. That kills 489 and 678 in one stroke.
The cage is 579. Five digits are now fixed for the column — 1, 3, 5, 7, 9 — so its four remaining cells hold 2, 4, 6 and 8, one of which is the given you started from.

This is the everyday killer move, and it is why it pays to pencil whole combinations into a cage corner rather than candidate digits into each cell. Crossing off a set removes digits from every cell of the cage at once.

A second habit worth building from the tables above: read the “digits ruled out” column before you read the combinations. A three-cell 21 rules out the 1, the 2 and the 3 for all three of its cells the instant you see the number, and that is often more immediately useful than knowing its three candidate sets.

6×6 killer sudoku is a different table

A 6×6 board uses the digits 1 to 6, in rows, columns and boxes of six. The sums look familiar and mean different things: a two-cell 10 is locked to 46 here, where on a 9×9 it has four combinations. Read these tables only for 6×6 boards.

Cage size	Lowest sum	Highest sum	Possible sums	Combinations in total
2 cells	3	11	9	15
3 cells	6	15	10	20
4 cells	10	18	9	15
5 cells	15	20	6	6
6 cells	21	21	1	1

Cage size	Sums with exactly one combination
2 cells	3 = 12, 4 = 13, 10 = 46, 11 = 56
3 cells	6 = 123, 7 = 124, 14 = 356, 15 = 456
4 cells	10 = 1234, 11 = 1235, 17 = 2456, 18 = 3456
5 cells	15 = 12345, 16 = 12346, 17 = 12356, 18 = 12456, 19 = 13456, 20 = 23456

Every five-cell cage on a 6×6 is locked, because five of six digits is the whole set minus one — a five-cell 15 leaves out the 6, a five-cell 20 leaves out the 1. That makes a five-cell cage on a small board one of the strongest clues in killer sudoku.

Here is the full two-cell and three-cell set for 1 to 6.

Cage	Sum	Combinations	How many	Digits ruled out
2 cells	3	12	1	3456
2 cells	4	13	1	2456
2 cells	5	14 23	2	56
2 cells	6	15 24	2	36
2 cells	7	16 25 34	3	—
2 cells	8	26 35	2	14
2 cells	9	36 45	2	12
2 cells	10	46	1	1235
2 cells	11	56	1	1234
3 cells	6	123	1	456
3 cells	7	124	1	356
3 cells	8	125 134	2	6
3 cells	9	126 135 234	3	—
3 cells	10	136 145 235	3	—
3 cells	11	146 236 245	3	—
3 cells	12	156 246 345	3	—
3 cells	13	256 346	2	1
3 cells	14	356	1	124
3 cells	15	456	1	123

Where the boards come from

If you want to make killer sudoku rather than solve it, Killer Sudoku builds 6×6 and 9×9 boards with cages of two to five cells, which means the two-, three-, four- and five-cell tables above cover every cage you will ever print with it. Each cage is a connected group of cells, every cell belongs to exactly one cage, digits are distinct inside each cage, and each printed sum is computed from the finished grid. Every board has exactly one solution, and every puzzle comes with its matching answer key.

Difficulty runs across seven bands — Very Easy, Easy, Normal, Hard, Very Hard, Tough and Extreme — and the higher bands print the cage sums as the entire clue set, so the tables on this page are the whole of the solver’s starting material. The lower bands also reveal some digits on the grid. Cages print as dashed outlines set inside the cells with the sum in the corner of the cage’s top-left cell, and they can be filled with colour when the outlines need to read at small sizes. Pages export as PNG, JPG, SVG or PDF, puzzle and key.

The classic given-digit version is Sudoku 9×9, where the clues are printed digits rather than arithmetic. For assembling either into a book, that job belongs to Puzzle Book Studio.

The whole point

Four tables carry most of killer sudoku. The locked sums tell you a cage’s digits outright. The “digits ruled out” columns strike candidates off three cells at a time. The 45 rule turns a nearly-covered box into one more clue. And the ends of the four- and five-cell ranges tell you which digit a cage must contain even when you cannot say which cell holds it. Everything above was enumerated rather than recalled, so a cage you look up here is a cage you can write into the grid.