Killer Sudoku Cage Combinations: The Complete Table

Why the combinations carry the opening

The rules are quick to state: the normal Sudoku rules apply, and in addition the digits in each cage add up to the number shown and may not repeat within the cage. The full version is on our Killer Sudoku page.

Everything else follows from that second rule. Because no digit may repeat, each cage size and each sum allows only a limited number of digit groups. A two cell cage with a sum of 3 can only be 1 and 2, never anything else. A two cell cage with 10, on the other hand, can be 1 and 9, 2 and 8, 3 and 7 or 4 and 6, but never 5 and 5. If you know these groups, you write in candidates in the first few minutes that other players still have to work out.

This matters even more in our puzzles, because our Killer Sudokus start without a single given digit. Each one has 31 cages: 16 with two, 11 with three and 4 with four cells. So just over half of all cages are two cell cages, and theirs is the shortest table of all.

The sums with only one combination

One rule makes memorising easy. For every cage size there are exactly four sums with only one possible combination: the two smallest and the two largest. For two cells they are 3, 4, 16 and 17. For three cells 6, 7, 23 and 24. For four cells 10, 11, 29 and 30. For five cells 15, 16, 34 and 35.

The reason is simple. The smallest possible sum takes the smallest digits, and there is only one way to do that. The next sum up swaps the largest of those digits for the next one, and there is only one way to do that as well. From the third sum on, there are two ways to raise a digit, and the uniqueness is gone. The same holds in mirror image at the top end.

In the table these sums are shown in bold. Knowing them gives you a way in to almost every puzzle: a two cell cage with 17 is always 8 and 9. If it lies in a row, the 8 and the 9 of that row are settled, and every other cell in the row loses both candidates.

The complete table

Each row gives a sum and every digit group that can make it in a cage of that size. The digits of a group are written without separators, so 129 means 1, 2 and 9. The order inside the cage is open; the table only tells you which digits they are.

Two cells

Three cells

Four cells

Five cells

Five cell cages do not appear in our puzzles, but they do in many other collections. The table is included here for completeness.

What ambiguous sums still tell you

Most cages do not have a unique sum, yet the table tells you more than a plain list. Three observations help in almost every puzzle.

First, fixed digits. A three cell cage with a sum of 8 can be 125 or 134. Which one stays open, but the 1 is in both. In the same way, a three cell cage with 22 always contains the 9, a four cell cage with 12 always contains the 1 and the 2, and a four cell cage with 28 always contains the 8 and the 9. You can write these fixed digits in straight away and remove them from the rest of the box if the cage sits inside one box.

Second, forbidden digits. In a two cell cage with an even sum, half of that sum is always missing, because it would have to appear twice. A two cell cage with 10 never holds a 5, one with 12 never holds a 6. That sounds trivial, but it removes a candidate in every other two cell cage.

Third, overlaps. If two cages share a row, they limit each other. If both need a 9, only one of them can have it, and the other falls back on its remaining combinations. That often turns two ambiguous cages into one unique one.

How to use the table while you play

Nobody needs to learn the whole table by heart. A fixed routine at the start of each puzzle works well. First, find every cage with a unique sum and write its digits in as candidates. Then go through the two cell cages and remove what cannot fit, such as the 5 in a cage with 10. Only then is it worth looking at the larger cages and their fixed digits.

After a few puzzles you will know the most common rows without looking them up. For the rare ones, it is enough to keep this page open until they stick as well.

The rule of 45

The second big help in Killer Sudoku is not a table but a sum. Every row, column and box contains the digits 1 to 9 exactly once and therefore always adds up to 45. If all cages in a box lie completely inside it except for one cell, that cell is the difference between 45 and the sum of the cages. If, the other way round, a cage sticks out of the box by one cell, that cell is the excess over 45.

The rule also works across several units: two rows add up to 90, three rows to 135. With 31 cages per puzzle there is almost always a box or a row where only one cell sticks out. Our own Killer Sudoku solver uses the rule of 45 as one of its techniques, right after cage combinations and fixed digits. If you use both together, you get through most of our puzzles without ever needing one of the hard Sudoku solving techniques.

The combinations are the same as in Kakuro, by the way, because there too no digit may repeat within a sum. Learn the table for one puzzle and you have it for the other. An overview of all types is on the puzzle types page.

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