Killer Sudoku Rules: How to Play, Strategies and Variations

What is Killer Sudoku?

Killer Sudoku keeps every rule of classic Sudoku: a 9×9 grid in which each row, column and 3×3 box contains the digits 1 to 9 exactly once. The twist is that the grid usually starts completely empty. Instead of givens, the grid is divided into cages, groups of cells outlined with dotted lines, each with a small number in its corner. The digits inside a cage must add up to that number.

Those totals are all the information you get, and they turn out to be plenty. A two-cell cage marked 3 can only hold 1 and 2. A three-cell cage marked 24 can only hold 7, 8 and 9. From footholds like these you work outward, using the same elimination logic as regular Sudoku.

The result feels like a cross between Sudoku and Kakuro. The arithmetic is simple addition, and after a few puzzles you will know the useful sums by heart. Despite the name, gentle Killer puzzles are no harder than a medium Sudoku, while the toughest ones rank among the most demanding pencil puzzles around.

A short history

Sum-based Sudoku puzzles were already established in Japan before the Western Sudoku boom, where they were known as samunamupure, a Japanese rendering of "sum number place". Number Place was the original English name for Sudoku, so the name simply means Sudoku with sums.

English-speaking readers met the puzzle in 2005, when The Times of London began printing Killer Sudoku alongside the regular Sudoku it had introduced a few months earlier. The dramatic name stuck, and Killer Sudoku quickly became one of the most widely published Sudoku variants in newspapers, puzzle magazines and apps.

Its appeal is that it adds exactly one idea to a puzzle millions of people already knew, and that idea (addition inside cages) combines with the Sudoku rules in surprisingly rich ways. You can read more background in Wikipedia's Killer Sudoku article.

The rules of Killer Sudoku

A standard Killer Sudoku follows five rules. The first three are ordinary Sudoku; the last two are what make it Killer:

1. 1Rows contain 1 to 9Each row must contain every digit from 1 to 9 exactly once.
2. 2Columns contain 1 to 9Each column must contain every digit from 1 to 9 exactly once.
3. 3Boxes contain 1 to 9Each 3×3 box, outlined by the thick lines, must contain every digit from 1 to 9 exactly once.
4. 4Cages add up to their totalThe digits in each dotted cage must sum to the number printed in the cage's corner. Every cell belongs to exactly one cage.
5. 5No repeats inside a cageA digit may not appear twice in the same cage. This matters most for cages that stretch across two boxes, where the Sudoku rules alone would allow a repeat.

Because every cell sits in exactly one cage, all the cage totals together add up to 405, which is nine rows of 45. That is a handy check that you have read the totals correctly.

A worked example

Here is a mini 4×4 Killer Sudoku that uses the digits 1 to 4, with 2×2 boxes instead of 3×3. Start at the top: the two-cell cage marked 3 can only be 1 and 2, so the 7 cage beside it must hold the remaining pair, 3 and 4. Now look at the bottom row. Three different digits from 1 to 4 that total 9 can only be 2, 3 and 4, so the bottom-right cell must be the 1. That settles the 4 cage above it: 4 minus 1 leaves 3.

From there, ordinary Sudoku logic finishes the grid. A full-size puzzle works exactly the same way, just with more cages and digits up to 9.

How to play: a step-by-step method

Killer grids start empty, so the opening is different from regular Sudoku. This routine gives you a reliable way in:

1. Find the forced cages. Scan for totals that allow only one set of digits, such as 3 or 17 in two cells, or 6 or 24 in three. Pencil those digits into the cage.
2. Apply the Rule of 45 to every row, column and box. Each unit totals 45. Add up the cages that sit entirely inside a unit and subtract from 45 to find the total of whatever is left. Often that leftover is a single cell, which you can fill immediately.
3. Pencil in candidates for small cages. For two- and three-cell cages, list the possible combinations and mark the candidates. A 10 in two cells, for example, can be 1+9, 2+8, 3+7 or 4+6, but never 5+5.
4. Use Sudoku elimination. Every placed digit removes that digit from its row, column, box and cage. Candidates in a cage that sits inside one row also clear that row, just like a naked pair.
5. Revisit the cages. As cells fill, cages with few options become cages with one. Recheck sums whenever a cage gets a new digit.
6. Finish with regular Sudoku. Once a good share of the grid is filled, the puzzle usually turns into a standard Sudoku and the cages become a checking tool.

Strategies strong solvers use

These techniques do most of the work in Killer Sudoku. The Rule of 45 alone solves a surprising share of every puzzle.

Keep a combination list nearby at first. After a dozen puzzles the common sums become instinctive and your solving speeds up dramatically.

Cage combinations worth memorizing

For each cage size, the smallest and largest totals can be made only one way. These are your guaranteed starting points:

• Two cells. 3 is always 1+2, 4 is always 1+3, 16 is always 7+9 and 17 is always 8+9.
• Three cells. 6 is always 1+2+3, 7 is always 1+2+4, 23 is always 6+8+9 and 24 is always 7+8+9.
• Four cells. 10 is always 1+2+3+4, 11 is always 1+2+3+5, 29 is always 5+7+8+9 and 30 is always 6+7+8+9.
• Five cells. 15 is always 1+2+3+4+5, 16 is always 1+2+3+4+6, 34 is always 4+6+7+8+9 and 35 is always 5+6+7+8+9.
• Nearly unique totals. Some totals have only two options, which is still very useful: 5 in two cells is 1+4 or 2+3, and 15 in two cells is 6+9 or 7+8.

Popular variations

Killer Sudoku mixes well with other rules. You may come across:

• Killer Sudoku with givens. Easier puzzles sometimes add a few starting digits to help beginners in.
• Killer Sudoku X. The two long diagonals must also contain 1 to 9, which gives the Rule of 45 extra lines to work with.
• Greater-than Killer. Inequality signs between cells, as in Futoshiki, add ordering clues alongside the sums.
• KenKen and Calcudoku. A close relative in which cages can use subtraction, multiplication or division, and there are no 3×3 boxes. See our KenKen guide.
• Little Killer Sudoku. Instead of cages, small arrows outside the grid give the total of a diagonal line of cells. It is a popular companion to other variants such as Thermometer Sudoku.

Common mistakes

The most frequent error is forgetting the no-repeat rule inside cages. A three-cell cage of 12 can be 1+2+9, 1+3+8, 1+4+7, 1+5+6, 2+3+7, 2+4+6 or 3+4+5, but never 4+4+4, even if the cells sit in different rows and boxes.

Second, many solvers ignore the Rule of 45 because it feels like extra work. On anything beyond an easy grid it is the main engine of the solve, so make a habit of checking every box and every row early on.

Third, arithmetic slips are costly. A cage total misread as 18 instead of 16 can go unnoticed for a long time. When a contradiction appears, reread the totals before assuming your logic is wrong.

Finally, do not abandon standard Sudoku technique. Hidden singles, pairs and pointing still apply in every row, column and box, and in the late stages they usually finish the puzzle faster than more cage arithmetic.

Frequently asked questions