Killer Sudoku Cage Combinations Chart (All Sums, 2-9 Cells)

In killer sudoku a cage's digits cannot repeat, so every cage sum has a fixed, finite list of possible digit sets, and knowing those lists turns arithmetic into instant pencil marks. The charts below give every combination for two-cell and three-cell cages, the full combination counts for four-cell cages, and the sums at every size that allow only one possible set.

The single most valuable thing in this article is not the big tables. It is the short list of unique-combination sums, the cages that have exactly one possible digit set. Those are free information the moment you see them, and experienced solvers can recite them.

The two rules the whole chart rests on

Digits never repeat inside a cage. This is the standard killer rule and it is what makes the combination lists finite. A cage of four cells summing to 20 draws four different digits from 1 to 9.

Every cage of nine cells sums to 45. So does every row, column and box. That fact powers the innies and outies technique further down.

A minority of publishers use a looser variant where a cage may repeat a digit if the repeated cells sit in different rows, columns and boxes. Every chart here assumes the standard no-repeat rule. If your puzzle allows repeats, the counts below are lower bounds and the unique-combination shortcuts do not hold.

Minimum and maximum sums by cage size

Before reaching for a combination list, check whether the sum is near an extreme. Extreme sums carry the most information.

The minimum is the sum of the lowest digits, 1 through n. The maximum is the sum of the highest n digits. A nine-cell cage is always the full digit set, which is why it tells you nothing on its own but is often used to fence off a box.

The mirror rule that halves your memorization

For a cage of n cells, the combinations for sum S are the exact mirror image of the combinations for sum 10n minus S, with each digit d replaced by 10 minus d.

Two-cell cages: 20 minus S. Sum 3 is {1,2}, so sum 17 is {9,8}. Three-cell cages: 30 minus S. Sum 7 is {1,2,4}, so sum 23 is {9,8,6}. Four-cell cages: 40 minus S.

Learn the low half of any chart and you have the high half for free. This is also a fast way to check yourself when you list combinations by hand.

Two-cell cage combinations: the complete chart

Every possible pair, all fifteen sums.

The four unique pairs are 3, 4, 16 and 17. Mark those cages immediately. The two-cell 5, 6, 14 and 15 cages are nearly as good, since two combinations means only three or four candidate digits survive in each cell.

Also note what is missing. A two-cell cage summing to 9 can never contain a 9, a two-cell 10 can never contain a 5, and a two-cell 11 can never contain a 1. Reading the chart for absent digits is often more useful than reading it for present ones.

Three-cell cage combinations: the complete chart

Unique three-cell sums: 6, 7, 23 and 24.

The two-combination sums are almost as strong. A three-cell 8 is either 125 or 134, so it always contains a 1, and it never contains 6, 7, 8 or 9. A three-cell 22 is 589 or 679, so it always contains a 9. Hunting for the digit common to every combination is one of the quickest wins in the game.

Four-cell cages

There are 126 possible four-cell sets, so a full list is unwieldy. Here are the tight ends, where the information is concentrated.

Unique four-cell sums: 10, 11, 29 and 30.

For the middle sums, the count is what matters, because it tells you whether the cage is worth analyzing at all.

The counts mirror for sums 21 to 30, so a four-cell 21 also has 11 combinations and a four-cell 25 has 6. A four-cell cage summing to 18, 19, 20, 21 or 22 gives you essentially nothing on its own. Do not waste time on it until neighboring cages have narrowed the digits.

Five to nine cells: the unique sums

Large cages are usually easier than they look, because the extremes are so restricted.

Every eight-cell cage has exactly one combination. An eight-cell cage is the full digit set minus one digit, so its sum is 45 minus the missing digit. Subtract the sum from 45 and you know precisely which digit is absent. A 39 cage is missing the 6, a 44 cage is missing the 1.

Seven-cell cages work the same way one level down: the sum is 45 minus a two-digit pair, so a seven-cell cage summing to 30 is missing either {6,9} or {7,8}, and therefore definitely contains 1, 2, 3, 4 and 5.

The 45 rule, innies and outies

The combination chart handles cages. The 45 rule handles the space between them, and in hard killer puzzles it opens more cells than any lookup.

Every row, column and box totals 45. So if you can account for all but a few cells of a house with complete cages, the leftover cells are forced.

Innies. Suppose box 5 is covered by cages that lie entirely within it, summing to 38, and one cell of box 5 is not part of any of them. That cell must be 45 minus 38 = 7. One subtraction, one digit placed.

Outies. Suppose the cages that overlap column 3 sum to 51, and all of them lie inside column 3 except a single cell that pokes out into column 4. That stray cell is 51 minus 45 = 6.

Multi-house versions. Two adjacent boxes total 90, three total 135, and a full band of three boxes totals 135 as well. Apply the same subtraction across a band and you can often nail an innie in a region where no single cage is helpful. Two innies rather than one gives you a sum for a pair, which then goes straight back to the two-cell chart above.

Putting the chart to work

A short sequence shows how these fit together. Say a three-cell cage summing to 7 occupies r1c1, r1c2 and r2c1, all inside box 1.

The chart gives one combination: 1, 2 and 4. Those three digits are now locked into box 1, so 1, 2 and 4 can be erased from every other cell of box 1, exactly like a naked triple. They are also partly locked into row 1 and column 1, though not fully, since the cage straddles both.

Now suppose an adjacent two-cell cage summing to 16 sits at r1c3 and r2c3. The chart gives 7 and 9 only. Row 1 now has 1, 2 or 4 in its first two cells and a 7 or 9 at r1c3, so 7 and 9 are gone from r1c4 through r1c9 only if both cage cells were in row 1, which they are not. Be careful here: locking only applies to the house a cage sits entirely inside.

That caution is the most common source of wrong eliminations in killer sudoku. A unique combination locks its digits into every house that fully contains the cage, and no others.

If you want to drill these lookups, working through fresh grids beats rereading the table. Our killer sudoku generator prints cages at a range of difficulties, and the easier settings tend to use more small cages, which is exactly where the two-cell and three-cell charts pay off fastest.

Three habits that speed up killer solving

Annotate cages before you place anything. Write the possible digit set next to every cage with three or fewer combinations. That pass takes two minutes and usually reveals several forced cells.

Look for cages that share a house. Two cages inside the same box, one summing to 6 in three cells and one to 17 in two cells, mean digits 1, 2, 3, 8 and 9 fill five of that box's nine cells. The other four cells hold 4, 5, 6 and 7 in some order.

Check the extremes first every time you get stuck. Sums at or near the minimum and maximum for their cage size are the only ones that carry hard information without further deduction. Scan for those before you start branching.

Frequently Asked Questions

Can a digit repeat inside a killer sudoku cage? Not under the standard rules. Digits within a cage are always different, which is what makes fixed combination lists possible. A small number of publishers use a variant that allows repeats when the cells sit in different rows, columns and boxes, so check the instructions on an unfamiliar puzzle.

Which killer sudoku cage sums have only one possible combination? Two-cell cages of 3, 4, 16 and 17. Three-cell cages of 6, 7, 23 and 24. Four-cell cages of 10, 11, 29 and 30. Five-cell cages of 15, 16, 34 and 35. Six-cell cages of 21, 22, 38 and 39. Seven-cell cages of 28, 29, 41 and 42. Every eight-cell cage, and the nine-cell 45.

What is the 45 rule in killer sudoku? Every row, column and box sums to 45. If complete cages account for all but one cell of a house, that leftover cell equals 45 minus the cage total. If cages cover a house plus one extra cell outside it, that extra cell equals the cage total minus 45.

Do I need to memorize the whole combination chart? No. Memorize the unique sums for two-cell and three-cell cages, plus the fact that eight-cell cages are 45 minus the missing digit. Everything else can be looked up or worked out from the minimum and maximum sums, and the mirror rule means you only ever have to learn half of any list.