Killer sudoku is sudoku with sums. The grid is the usual 9 × 9, and the usual rule still holds: every row, every column and every 3 × 3 box contains the digits 1 to 9 exactly once. On top of that, the grid is divided into cages, groups of squares outlined with dashed lines, each with a small total in its corner. The digits in a cage must add up to that total, and no digit may repeat inside a cage. Most killer sudokus start with no digits at all, so the totals are your only clues. That sounds harder than it is. A handful of cage combinations and one counting rule do most of the work. This guide covers the rules, the combinations worth memorising, the 45 rule with worked numbers and a set of tips you can try in our free killer sudoku game, which lists the possible digit sets for whichever cage you select.
Fill every square with a digit from 1 to 9.
Each row contains 1 to 9 exactly once.
Each column contains 1 to 9 exactly once.
Each 3 × 3 box contains 1 to 9 exactly once.
The digits in each cage add up to the number shown in the cage's corner.
A digit cannot appear twice in the same cage, even when the cage crosses from one box into another.
Every square belongs to exactly one cage, and the cage totals across the whole grid add up to 405, because each of the nine rows sums to 45 (1 + 2 + … + 9 = 45, and 9 × 45 = 405). That is a useful check if you are building or copying a puzzle.
Because digits cannot repeat in a cage, some totals can be made in only one way. These are the starting points of almost every solve, so they are worth learning by heart.
The pattern is easy to remember: the two smallest and the two largest totals for each cage size are unique. A 2-square cage can total anything from 3 to 17, a 3-square cage from 6 to 24, a 4-square cage from 10 to 30 and a 5-square cage from 15 to 35. A total outside those ranges means you have misread the cage.
| Cage size | Smallest totals | Largest totals |
|---|---|---|
| 2 squares | 3 = 1+2, 4 = 1+3 | 16 = 7+9, 17 = 8+9 |
| 3 squares | 6 = 1+2+3, 7 = 1+2+4 | 23 = 6+8+9, 24 = 7+8+9 |
| 4 squares | 10 = 1+2+3+4, 11 = 1+2+3+5 | 29 = 5+7+8+9, 30 = 6+7+8+9 |
| 5 squares | 15 = 1+2+3+4+5, 16 = 1+2+3+4+6 | 34 = 4+6+7+8+9, 35 = 5+6+7+8+9 |
Two-square cages are the most common in easy puzzles, and the full list is short enough to keep beside you while you learn.
Note what is missing: an even total can never use half of itself twice. A 2-square cage of 10 cannot be 5+5, and a cage of 8 cannot be 4+4. So 5 never appears in a 2-square 10, and 4 never appears in a 2-square 8. Those small exclusions often break a stuck row.
| Total | Possible pairs | Total | Possible pairs |
|---|---|---|---|
| 3 | 1+2 | 11 | 2+9, 3+8, 4+7, 5+6 |
| 4 | 1+3 | 12 | 3+9, 4+8, 5+7 |
| 5 | 1+4, 2+3 | 13 | 4+9, 5+8, 6+7 |
| 6 | 1+5, 2+4 | 14 | 5+9, 6+8 |
| 7 | 1+6, 2+5, 3+4 | 15 | 6+9, 7+8 |
| 8 | 1+7, 2+6, 3+5 | 16 | 7+9 |
| 9 | 1+8, 2+7, 3+6, 4+5 | 17 | 8+9 |
| 10 | 1+9, 2+8, 3+7, 4+6 |
Every row, column and box adds up to 45. When cages almost exactly fill a box, a row or a column, subtraction tells you the digits in the gap. Solvers call a square inside the area but outside its whole cages an innie, and a square of a cage that sticks out of the area an outie.
Look for the 45 rule as soon as you start. Scan each box, row and column for areas where cages fit neatly and only one or two squares are left over. Easy puzzles often give you three or four innies in the first minute.
| Situation | Arithmetic | Result |
|---|---|---|
| Cages lying wholly inside a box cover 8 of its 9 squares and total 38 | 45 − 38 = 7 | The one remaining square (an innie) is 7 |
| Cages covering one row and one extra square outside it total 52 | 52 − 45 = 7 | The square that sticks out (an outie) is 7 |
| Cages inside a box cover 7 squares and total 35 | 45 − 35 = 10 | The two remaining squares add to 10: 1+9, 2+8, 3+7 or 4+6, never 5+5 |
| Three full rows (a band of three boxes) | 3 × 45 = 135 | Use the same subtraction on a whole band or stack of boxes |
Fill single-square cages first. A 1-square cage is just a given digit.
Mark the unique cages next. A 3 in two squares is 1 and 2; you do not know the order yet, but no other square in that row, column or box can hold a 1 or 2 if the cage lies in one line or box.
Run the 45 rule on every box, row and column. Write the innie and outie totals in the margin even when they do not fix a single digit.
Use pencil marks for cage options. For a 3-square 8 (1+2+5 or 1+3+4), mark 1 in all three squares: 1 must be in that cage either way.
Find digits common to every combination. In the 3-square 8 above, 1 appears in both sets, so the cage definitely contains a 1. That removes 1 from the rest of any row, column or box the cage sits in.
Watch cages that stay in one row or box. Their digits act like a locked set for that unit, the same way a naked pair does in classic sudoku.
Use ordinary sudoku techniques too. Hidden singles, pairs and pointing pairs all still work once some digits are placed. Our sudoku tips for beginners covers them.
Check sums as you go. When a cage has one square left, the last digit is the total minus the others. It is also the fastest way to catch an earlier mistake.
Move up one cage size at a time. Easy puzzles use small cages; harder ones use 4- and 5-square cages with many combinations. Learn the unique ones for each size before you move up.
Kakuro uses exactly the same digit combinations as killer sudoku cages, so the Kakuro combinations chart doubles as a reference for 3-, 4- and 5-square cages.
| Puzzle | Grid | Clues | Repeats allowed in a sum? |
|---|---|---|---|
| Sudoku | 9 × 9 with boxes | Given digits | No sums |
| Killer sudoku | 9 × 9 with boxes | Cage totals, usually no given digits | No |
| Kakuro | Crossword-style runs | Run totals | No |
| Calcudoku | Square grid, no boxes | Cage results with +, −, × or ÷ | Yes, if the squares are not in the same row or column |
Fill the 9 × 9 grid so each row, column and 3 × 3 box contains 1 to 9 once. The grid is split into cages; the digits in each cage must add up to its total, and no digit can repeat inside a cage.
No. A digit can appear only once in a cage, even when the cage spans two boxes.
Every row, column and box adds up to 45. If cages inside a box total 38 and one square is left over, that square is 45 − 38 = 7. The same subtraction works on rows, columns and groups of boxes.
405, because the nine rows each sum to 45.
It starts harder because there are no given digits, but the unique cage combinations and the 45 rule give you a clear way in. Easy killer sudokus with small cages are a good first step.
Learn the unique cages, run the 45 rule on every box, row and column, and pencil in the digits common to every combination a cage allows. With those three habits, easy killer sudokus fall in minutes and hard ones become a matter of patience. Try a fresh grid at any of three levels in the free killer sudoku game.