Killer Sudoku Solver

A Killer Sudoku solver fills a 9×9 grid that has no starting digits at all — only dashed cages, each with a sum. The rules of ordinary Sudoku still apply, plus two more: the digits in a cage must add to its total, and no digit repeats inside a cage. You solve it with cage combinations, the rule of 45, and the standard Sudoku techniques underneath. Our free Sudoku solver handles the classic grid; this guide gives you the method for the Killer variant.

Below: the rules, the combination tables you actually need, the rule of 45, the solving order, and answers to the questions people ask about Killer Sudoku.

What is Killer Sudoku?

Killer Sudoku (also called Sumdoku, Sum Sudoku or Addoku) combines Sudoku with the arithmetic of Kakuro. The grid is the familiar 9×9 with nine 3×3 boxes, but it usually starts completely empty. Instead of givens, it has cages: groups of cells outlined by a dashed border, each carrying a small number in its top-left corner — the cage sum.

Three rules apply at once:

• Every row, column and 3×3 box contains the digits 1 to 9 exactly once — standard Sudoku.
• The digits inside a cage add up to the cage sum.
• No digit repeats within a cage, even if the cage spans two boxes.

That third rule is what makes cages useful. A two-cell cage summing to 4 could be 1+3 or 2+2 in ordinary arithmetic; in Killer Sudoku only 1+3 is legal.

The rule of 45

This is the technique that makes Killer Sudoku solvable at all, and it is remarkably simple.

The digits 1 to 9 add up to 45. Therefore every row, every column and every 3×3 box sums to exactly 45.

So if a box is covered by cages totalling 38, and one extra cell of that box belongs to a cage sticking out elsewhere, that cell must be 45 − 38 = 7. You have placed a digit without any other reasoning.

The same works in reverse: if cages inside a box total 52 and one cell of one cage sticks outside the box, that outside cell holds 52 − 45 = 7.

Cage combinations you must know

Some cage sums have only one possible set of digits. Those are free information — learn them and you get several cells at a glance.

Beyond these, the useful skill is knowing the minimum and maximum a cage can hold: an n-cell cage cannot be smaller than 1+2+...+n, nor larger than 9+8+...+(10−n). A four-cell cage summing to 12 is therefore very tightly constrained, because the minimum is 10.

How to solve Killer Sudoku step by step

1. Mark the forced cages. Go through the grid writing the digit set for every cage that has only one combination — the table above. These are certainties.
2. Apply the rule of 45 to every box, row and column. Look for units where cages almost exactly cover the unit, leaving one cell in or out. Each of those gives you a digit directly.
3. Use the no-repeat rule inside cages. A three-cell cage summing to 8 cannot contain 1+3+4 and 2+2+4; only sets with distinct digits survive.
4. Convert cage sets into candidates. Once a cage's digit set is known, those digits are the only candidates in its cells, and they are eliminated from any unit the cage fully occupies.
5. Switch to ordinary Sudoku technique. Naked singles, hidden singles, naked pairs, pointing pairs — the whole standard ladder now applies to your candidates.
6. Alternate. Every digit you place changes cage arithmetic; every cage you resolve changes candidates. Killer Sudoku is solved by going back and forth between the two.

Killer-specific tactics

Innies and outies

An innie is a cell inside a unit that is not covered by the cages contained in it; an outie is a cell of a cage that sticks outside the unit. Both are solved by the rule of 45, and hunting for them systematically is the core skill of Killer Sudoku.

Cages within a single box

If a cage lies entirely inside one 3×3 box, its digits are consumed by that box — they can be removed from every other cell in it, exactly like a naked set.

The 45 rule on bands

Three stacked boxes total 135. If cages covering them add to 128, the seven missing points sit in whatever cells are uncovered. On large sparse grids this is often the only way in.

How a Killer Sudoku solver works internally

A program solving Killer Sudoku uses the same engine as a classic solver — constraint propagation plus backtracking — with the cage sums added as extra constraints.

• For each cage it precomputes every legal set of distinct digits reaching the sum.
• Whenever a digit is placed, sets containing it in the wrong position are discarded and the cage's candidate pool shrinks.
• Rule-of-45 deductions are applied as a global constraint over rows, columns, boxes and their unions.
• When propagation stalls, the solver branches on the most constrained cell, exactly as in classic Sudoku.

The extra arithmetic makes Killer grids easier for a computer than for a human: cage sums prune the search tree hard, while for a person they add a whole second layer of bookkeeping.

Cage combination tables in full

Beyond the unique-combination cages, most sums have a handful of options. Knowing the count is often enough: a cage with three possibilities is worth working on, a cage with fifteen is not.

Two-cell cages

A two-cell cage can sum from 3 to 17. Sums 3, 4, 16 and 17 are unique. Sums 5 (1+4, 2+3), 6 (1+5, 2+4), 14 (5+9, 6+8) and 15 (6+9, 7+8) have exactly two options, which makes them nearly as valuable. A two-cell cage summing to 10 has four options and tells you little on its own — but it does tell you the cage contains no 5, since 5+5 is illegal.

Three-cell cages

Range 6 to 24. Unique at 6 (1+2+3), 7 (1+2+4), 23 (6+8+9) and 24 (7+8+9). Sums 8, 9, 22 and 21 have two or three options each. The middle sums, around 15, have the most and should be left until the grid constrains them.

Larger cages

A cage of five or more cells is usually more useful for what it excludes than for what it contains. A five-cell cage summing to 16 is only one point above its minimum of 15, so it must be 1+2+3+4+6 — the only way to add a single point to the minimum set. The same trick works from the maximum downwards.

A worked opening

Take a grid where the top-left 3×3 box is covered by three cages summing to 15, 22 and 9, plus one cell belonging to a cage that extends to the right.

1. The three contained cages total 46. The box must total 45.
2. The extra cell therefore contributes 45 − 46 = −1 — impossible, so one of those cages must extend outside the box instead. Re-read the outlines.
3. Say the 22-cage has one cell in the neighbouring box. The cells inside total 45 minus the covered part, and that single outside cell equals 46 − 45 = 1.
4. You have placed a 1 without touching a single Sudoku rule.

That is the whole method in miniature: count what the cages cover, compare to 45, and read off the difference. Repeat it on every row, column and box before doing anything else.

Killer Sudoku difficulty levels

Killer Sudoku times run longer than classic Sudoku at every level, because each deduction requires arithmetic as well as logic. That is normal and not a sign you are doing it wrong.

Killer Sudoku variants you will meet

Publishers apply the cage idea in several ways, and each changes the tactics slightly.

• Standard Killer. Cages cover the whole grid, no digits given. The rule of 45 does the opening work.
• Killer with givens. A few digits are printed alongside the cages. Much easier, because ordinary Sudoku scanning starts immediately.
• Non-consecutive Killer. Adjacent cells may not hold consecutive digits. A strong extra constraint that prunes cage combinations fast.
• Killer X. Both main diagonals must also contain 1 to 9, which gives two more units for the rule of 45.
• Greater-than Killer. Cages carry inequality signs instead of, or alongside, sums.
• Zero Killer. Some cages show no sum at all; you deduce their total from the surrounding 45 arithmetic.

Whatever the variant, the method is unchanged: resolve forced cages, apply 45 to every complete unit, convert cage sets to candidates, then solve as ordinary Sudoku.

Reading a cage the way an expert does

A cage carries four separate pieces of information, and beginners usually read only the first.

1. Its total. The obvious one.
2. Its possible digit sets. How many combinations reach that total with distinct digits.
3. What it excludes. A three-cell cage summing to 7 can only be 1+2+4, so it contains no 3, no 5 and nothing above 4. Exclusions are often worth more than inclusions.
4. Which units it sits in. A cage entirely inside one box behaves like a naked set for that box. A cage crossing two boxes links their arithmetic through the rule of 45.

Train yourself to note all four each time you look at a cage. It is slower for the first few puzzles and much faster afterwards.

Cage sums that carry hidden certainties

These near-limit facts are where most Killer breakthroughs actually come from — not from the famous unique-combination cages, which are rarer than beginners expect.

A full solving order for a blank Killer grid

1. Mark unique cages. Write the forced digit sets from the combination table.
2. Mark near-limit cages. Any cage within two or three of its minimum or maximum collapses to one or two sets.
3. Rule of 45 on every box. Look for boxes almost covered by whole cages; the innie or outie gives a digit outright.
4. Rule of 45 on rows and columns. Same logic, often overlooked because cages sit visually inside boxes.
5. Rule of 45 on bands and stacks. Three boxes total 135. This unlocks sparse regions nothing else touches.
6. Cages as naked sets. A cage fully inside a unit removes its digits from the rest of that unit.
7. Switch to standard Sudoku. Hidden singles, naked pairs, pointing pairs, box-line reduction — all now apply to your candidates.
8. Alternate. Each placement changes cage arithmetic; each resolved cage changes candidates. Keep cycling between the two.

Why Killer Sudoku feels harder than it is

The intimidation comes from the blank grid, not from the difficulty. A standard Sudoku hands you 25 digits and asks for 56 more. A Killer hands you nothing — but it hands you roughly thirty cage constraints instead, and those are collectively far more informative than 25 scattered digits.

The arithmetic never exceeds adding single digits. What Killer really demands is bookkeeping: tracking which digit sets each cage can still hold, and updating them every time something is placed. Solvers who find it impossible are almost always trying to hold that information in their head instead of writing it down.

Write the possible sets next to each cage on your first pass. From that moment it becomes an ordinary Sudoku with extra help.

Common mistakes

• Allowing a repeat inside a cage. The single most frequent error. 2+2 never appears in a four-sum cage.
• Forgetting cages cross boxes. The no-repeat rule applies to the cage, not to the box.
• Ignoring minimum and maximum. A five-cell cage cannot sum below 15 or above 35, which often collapses the options instantly.
• Not re-checking 45 after each placement. Every digit placed changes the arithmetic of every unit it touches.
• Treating it as arithmetic only. The classic Sudoku techniques still do most of the work in the second half of the grid.

Frequently asked questions

Solve your grid

Learn the single-combination cages, then hunt innies and outies with the rule of 45 — that pair of habits opens almost every Killer Sudoku. And for a classic 9×9 grid, our free Sudoku solver gives you the answer instantly.

Next, read: How to Solve Sudoku — The Complete Guide · Advanced Sudoku Techniques · Free Sudoku Solver