How to solve
a Killer sudoku?

The rest of the article details each step, with figures measured on our grids and a real example.

Our Killer grids are rated by the techniques needed to finish them: at Easy, the cage combinations are enough; at Medium, you need the rule of 45 on a row, a column or a box; at Hard, on several at once. That is the rating rule. What is more instructive is the shape of the grids at each level:

Two things stand out. First, the higher the level, the fewer the cages and the bigger they get. A big cage has many possible combinations: on its own, it says almost nothing. Second, the last row: in 4 Medium grids out of 5, you cannot place a single digit until you have applied the rule of 45. The rule of 45 is not an endgame tool: it is often the way into the grid.

A two-cell cage that adds up to 3 must contain 1 and 2: no other pair of different digits makes 3. These safe cages are your first foothold. For cages of 2 to 6 cells (the ones in our grids), there are only four per size, and they all follow the same rule: the two smallest and the two largest possible sums.

A safe two-cell cage gives you a pair: you know which digits go there, not yet in which order. But if both cells are in the same row, column or box, those two digits already disappear from the rest of that row, column or box.

Be careful, though: these cages get rarer as the level goes up. There are 3.5 per grid on average at Easy, 1.5 at Hard, and none at all in nearly one Hard grid in five. The most frequent sums are precisely the least telling: for two cells, they are 9, 10 and 11, which each have four combinations (for 10: 1+9, 2+8, 3+7, 4+6).

A cage can have several combinations and still force a digit. Three cells adding up to 8: that is 1+2+5 or 1+3+4. The 1 is in both, so it must be in the cage, even if we don't know where yet.

Now, a digit appears only once per box. If that cage lies entirely inside one box, the box's 1 is in the cage, and the other cells of the box lose their 1. The same goes if the cage lies within a row or a column. It is the most discreet deduction in Killer, and one of the most used: our engine relies on it about twenty times per grid, from Easy level onwards.

The digits 1 to 9 add up to 45. So every row, every column and every box adds up to 45. When some cages are entirely enclosed in a box, you know their total, and whatever is missing to reach 45 is in the remaining cells.

The example

Here is one of our Medium grids, at the very start of the game. It has two safe cages: 16 in two cells (7+9, top middle) and 4 in two cells (1+3, at the bottom). They give pairs, but not a single digit. Like 4 Medium grids out of 5, this one only opens up with the rule of 45.

Top-right box: 45 − 38 = 7 for the remaining cell, then 13 − 7 = 6 for the cell that sticks out.

Look at the top-right box. Three cages are entirely enclosed in it (in blue): 18, 7 and 13. They add up to 18 + 7 + 13 = 38. Only one cell of the box lies outside those cages: R3C9 (row 3, column 9). So it is worth 45 − 38 = 7.

And that is not all. This cell belongs to another 13 cage, which sticks out of the box by one cell downwards (in orange). 13 − 7 = 6: R4C9 is a 6. Two digits placed on an empty grid, without pencilling in a single candidate.

The most common case: an invisible cage

A ready-made digit, as in the example, is the best surprise. But it is rare: in more than nine cases out of ten, the rule of 45 gives the sum of several cells. On another of our Medium grids, two cells remain in the top-left box, and the cages entirely inside add up to 42: those two cells make 3. It isn't written anywhere, but it is a cage like any other (an invisible cage), and 3 in two cells must be 1+2.

Think about the cells that stick out, too. We naturally look at what remains inside; yet for our engine, nearly two times out of three, the calculation goes through cells that spill over the area.

At Hard level, cages are bigger and often straddle two boxes: on a single row or a single box, nothing adds up cleanly any more. The same rule then applies to a wider area:

• two adjacent rows or columns add up to 90, three to 135;
• two boxes side by side add up to 90.

For example, if the cages lying entirely within rows 4 and 5 add up to 81, the remaining cells in those two rows make 9. All our Hard grids need it, about five times per grid. The reflex: when a box gives nothing, look at the whole band of three boxes, or at the two rows along the edge.

A classic sudoku player arrives with a toolkit: locked candidates, pairs, X-Wing. In Killer, these are used much less than you might think:

• locked candidates help in a little under one Medium grid in two, and in almost every Hard grid;
• pairs show up in one Hard grid in three, triples in fewer than one in ten;
• the X-Wing: across all our Killer grids, our engine has never needed it.

The difficulty of a Killer lies in the sums. When you get stuck, look for a new area to apply the rule of 45 before hunting for a classic sudoku pattern.

1. An invisible cage can repeat a digit. A real cage never contains the same digit twice. A sum found with the rule of 45 can: two cells that are neither in the same row, nor the same column, nor the same box can hold the same digit. Two such cells adding up to 10 might be 5+5. Before using the combination table, check that its cells are all in the same row, column or box.
2. An addition mistake. Everything that follows from a wrong sum is wrong, and the error only shows up much later. Redo the addition the other way round before placing the digit.
3. Forgetting the cells that stick out. When a single cell sticks out of a box, it is a gift: the rule of 45 gives you its digit directly.
4. Guessing. It is never necessary: every one of our Killer grids has a unique solution and can be solved by logic alone.

Where do you start a Killer sudoku?

With the cages that have only one combination (two cells adding up to 3 must be 1+2), then with the rule of 45, box by box. At Medium or Hard level, the rule of 45 places the first digit in more than 4 grids out of 5.

What is the rule of 45 in Killer sudoku?

Every row, column and box contains the digits 1 to 9, which add up to 45. Add up the cages lying entirely inside a box: the difference from 45 gives the sum of the remaining cells. If some cages stick out, whatever exceeds 45 lies in the cells that stick out. The rule also works across two or three rows at once (90, 135).

Do you need to learn the combinations by heart?

No. Just remember that, for each cage size, only the two smallest and the two largest sums have a single combination. For the others, Sudoku-Land shows every possible combination of the selected cage below the grid.

On the Killer Sudoku page, the three levels follow this method step by step: Easy for combinations, Medium for the rule of 45 on a box, Hard for the rule of 45 across several rows. When you select a cell, the line below the grid shows the possible combinations of its cage, given the digits already placed, and highlights the last one in gold when only one is left.

If you get stuck, the Hint button opens a step-by-step solver that explains each Killer deduction (combinations, required digits, rule of 45…) on a small grid with the cages drawn in. Reading the steps is free; only “Apply this move” uses up a hint.

Killer is also one of the six faces of the Variant Cube, and the Variants page compares it with the site's other variants.