Every set of digits that can fill a killer sudoku cage, for every cage size and every total, in one table you can read on your phone or print. The table is made by the same code as our game's Combos button, so it is complete and exact. Below it: how to use it, and today's puzzle to try it on.
By Brasslight Games ·
How to read the table
A killer sudoku cage holds different digits from 1 to 9, and they add up to the small total in its corner. For any cage size and total there is a fixed list of digit sets that can make it, and the table below gives every one. Each row is one total. The sets are written without spaces, so 124 means the digits 1, 2 and 4 in some order: the table tells you which digits, never where they go.
Two short columns save you some thinking. Always lists the digits that are in every set for that total, so the cage must use them. Never lists the digits that are in no set, so the cage can't use them. The shaded rows have only one set, which makes them the best place to start.
In all there are 120 possible totals for cages of two to nine squares, made by 502 different sets of digits. A cage of two squares can total anything from 3 to 17, three squares from 6 to 24, four from 10 to 30, five from 15 to 35, six from 21 to 39, seven from 28 to 42, eight from 36 to 44, and a nine-square cage always totals 45. A one-square cage is simply its own digit.
The totals to learn first
Of the 120 totals, 34 have only one possible set. You'll meet the same few again and again, so they are worth knowing by heart:
There is a pattern that makes them easy to remember: for every cage size, the two smallest totals and the two largest have only one set. The smallest uses the lowest digits (three squares adding up to 6 must be 1, 2 and 3), and the next one up swaps the top digit for the next one (three squares adding up to 7 must be 1, 2 and 4). The largest totals work the same way from the top: 24 in three squares is 7, 8 and 9, and 23 is 6, 8 and 9. Cages of eight squares are the exception that makes it simpler still: every total from 36 to 44 has one set, because the cage is just 1 to 9 with one digit missing, and the missing digit is 45 minus the total.
Above: today's Easy Killer Sudoku, free. Find a cage from the 'only one set' list and start there; tap Combos in the game to see the sets for any cage. Open Easy Killer Sudoku on its own page →
Using always and never
Most cages have more than one set, but the sets usually agree on something. Three squares adding up to 8 can be 1, 2 and 5 or 1, 3 and 4: either way the cage contains a 1. If that cage sits inside one box, no other square in the box can be a 1. Four squares adding up to 12 are 1, 2, 3 and 6 or 1, 2, 4 and 5, so both 1 and 2 are certain.
The never column works the other way round. Two squares adding up to 10 can be 1 and 9, 2 and 8, 3 and 7 or 4 and 6, but never 5, because a cage can't repeat a digit. Three squares adding up to 19 can be five different sets, and not one of them uses a 1. Write that down as a pencil mark and it often finishes a row for you.
Then combine the cage with its neighbors. Say two squares adding up to 4 (only 1 and 3) share a row with a two-square cage of 9, which could be 1 and 8, 2 and 7, 3 and 6 or 4 and 5. The 1 and the 3 are taken, so the 9 must be 2 and 7 or 4 and 5. Each set you rule out narrows the cages around it.
The complement trick
You don't need to memorize the big cages separately, because they mirror the small ones. The digits 1 to 9 add up to 45, so the digits a cage leaves out are a set of their own. A cage of seven squares with a total of 41 leaves out two digits that add up to 4, and the only two digits that do are 1 and 3: so the cage is 2, 4, 5, 6, 7, 8 and 9. A cage of six squares with a total of 38 leaves out three digits that add up to 7, which can only be 1, 2 and 4.
So a big cage is a small cage in reverse: subtract its total from 45, count the squares it leaves out, and read the small cage's row. This is also why the table looks the same from both ends.
The rule of 45
The same number gives the most useful trick in killer sudoku. Every row, every column and every 3 by 3 box holds 1 to 9 once, so each adds up to 45. Add the cages that sit wholly inside a box: if they come to 38, the squares left over make up the other 7. When only one square is left over, you have its digit. When a cage pokes out of the box by one square, the difference between the cage totals and 45 gives you that square too.
Across two rows the total is 90, across three it is 135. Medium puzzles on Suvi need the rule of 45 somewhere; Hard ones need it across several rows or columns at once, which is where a cheat sheet saves the most time.
The full table
Every cage size from two squares to nine, every total, every set of digits. Shaded rows have only one set. On a phone, swipe a wide table sideways to see its last column.
On Suvi you never need the table at your elbow while you play. In Killer Sudoku, tap a square and then Combos: the game lists every set of digits that fits that cage, the same sets as this table. There is a new puzzle every day at Easy, Medium and Hard, the same for everyone, each checked by our solver to have exactly one solution you can reach without guessing.
Easy needs only cage combinations and the usual sudoku singles, so it is the place to practice the totals above. Hard needs the rule of 45 across rows or columns. If plain sudoku is your game, our sudoku techniques guide covers the moves that killer sudoku also uses once the cages are filled in, and today's Sudoku is free at four levels.
Questions
Can a killer sudoku cage repeat a digit?
No. The digits in a cage are always different, even when a cage bends across two boxes. That is why two squares adding up to 10 can never be 5 and 5, and why some totals have only one possible set.
How many cage combinations are there?
For cages of two to nine squares there are 120 possible totals and 502 different sets of digits between them. 34 of those totals have only one set.
Which cages should I look at first?
The ones with only one set: the two smallest and the two largest totals for each cage size, such as 3 or 4 in two squares, 6 or 7 in three, and 23 or 24 in three. Then any cage whose sets all share a digit.
Is this table the same as the game's Combos list?
Yes. Both come from the same code in our Killer Sudoku engine, so the table always matches what the game shows for a cage before you have filled anything in.
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