Killer Sudoku

Rule of 45 across Multiple Regions

Two whole rows of a Sudoku always add up to 90 (three add up to 135), and the same goes for columns. Add the sums of every cage that touches those rows: the total exceeds 90 by exactly what the cage cells lying outside them hold — the outies. When one outie is left empty, that difference is its digit; when two are, it narrows them to the pairs that make it up.

What you need to see on the grid

Reach for it when the rule of 45 on a single row, column or box gives nothing because the cages keep spilling into the next row, yet two neighbouring rows or columns together are left by only one or two cages. Four terms carry the method:

Region: Whole rows or whole columns taken together, each holding 1 to 9 once: two of them total 90, three total 135. The arithmetic works for any houses that do not overlap; the hint in the app looks at two neighbouring rows or columns, and at the three rows or columns of one band of boxes.
Touching cage: A cage with at least one cell inside the region. Its whole sum goes into the count, including the part that lies outside. A cage that only borders the region without entering it does not count.
Outie: A cell of a touching cage that lies outside the region. Together the outies hold the touching cages' total minus 90 (or 135). Subtract any outie already filled; what is left belongs to the empty ones.
Innie: The mirror image: a cell inside the region whose cage leaves it. 90 minus the cages lying wholly inside is what the innies hold. It is the same count seen from the other side, and a good way to check your sums; on a single row, column or box it is the subject of the Innies & Outies guide.

How to apply it, step by step

One example runs through all five steps: rows 8 and 9 of a Killer puzzle, 16 digits in. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them. Steps 1 to 3 are pure arithmetic, so they show the cages without pencil marks.

Step 1Pick two neighbouring rows the cages barely leave

Start with one row and follow its cages. Row 8 is crossed by six cages, and every one of them carries on into row 9 — the 21-cage even reaches up into row 7. Not one cage lies wholly inside row 8, so the rule of 45 on that row alone has nothing to subtract, and row 9 is no better.

Now take rows 8 and 9 together. Five of the six cages close up inside the pair; only the 21-cage sticks out, by a single cell, R7C1. That is the shape to look for: whole rows or columns that the cages leave in one or two places. Columns work exactly the same way.

Row 8 and the six cages that cross it. Each one carries on into row 9, and the 21-cage up to R7C1 as well.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	21
R8		18			27		14	9	6
R9				9		3

Digit already on the grid
Row 8: no cage fits inside it
Cage, sum in its corner
Not needed for this step

Filled in: row 8; the 6 cages. The other 62 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 2Add up every cage that touches the two rows

List every cage with at least one cell in rows 8 or 9, and add their sums, each cage once and in full: 21 + 18 + 27 + 14 + 9 + 6 = 95. The 21-cage counts whole, even though one of its cells lies in row 7.

Digits already placed change nothing here: the 9 at R9C4 and the 3 at R9C6 are already part of their cages' sums. Work along both rows cell by cell and tick each cage as you meet it, so that none is missed and none is counted twice.

Rows 8 and 9 with the six cages that touch them: 21, 18, 27, 14, 9 and 6, which add up to 95.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	21
R8		18			27		14	9	6
R9				9		3

Digit already on the grid
Rows 8 and 9: together they hold 90
Cage, sum in its corner
Not needed for this step

Filled in: rows 8 and 9; the 6 cages. The other 62 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 3Subtract 90 to find what the outies hold

Rows 8 and 9 hold the digits 1 to 9 twice, so their 18 cells add up to 90. The six cages add up to 95. They cover all 18 cells plus the cells they have outside the rows, so those outside cells — the outies — hold 95 − 90 = 5 between them.

Here there is one outie, R7C1, and it is empty: it holds the 5 on its own. Check it from the inside if you like. The five cages that stay inside the rows total 18 + 27 + 14 + 9 + 6 = 74, so R8C1 and R9C1, the innies, hold 90 − 74 = 16, and the 21-cage leaves 21 − 16 = 5 for R7C1. Same answer.

The cages total 95, the two rows 90: the only outie, R7C1, holds the difference, 5.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	21
R8		18			27		14	9	6
R9				9		3

Digit already on the grid
Rows 8 and 9: together they hold 90
Outie: a cage cell outside rows 8–9
Cage, sum in its corner
Not needed for this step

Filled in: rows 8 and 9; the 6 cages. The other 62 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 4Place the outie, or narrow a pair of them

R7C1 lists 1, 5, 6, 7, 8 and 9: neither its row, column and box nor the 21-cage on its own can decide it. The count can. R7C1 = 5.

Two variations come up often. If some outies are already filled, subtract them first: had the 21-cage left the rows with two cells and one of them held 2, the empty one would be 5 − 2 = 3. If two outies are empty, they share the total: keep only the candidates that pair up to make it — as two different digits if the two cells share a row, column, box or cage. With three empty outies or more, the total alone seldom settles anything, and the hint in the app does not try.

Technique: Rule of 45 across regions

Rows 8 and 9 in full with their pencil marks, and the six cages. R7C1 still lists six digits; the count makes it 5.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	215
R8		18			27		14	9	6
R9				9		3

Digit already on the grid
Rows 8 and 9: together they hold 90
Outie: a cage cell outside rows 8–9
Digit placed
Cage, sum in its corner
Not needed for this step

Filled in: rows 8 and 9; the 6 cages. The other 62 cells are left blank: they are not needed for this step. Placed: R7C1 = 5.

Step 5Collect what the new digit unlocks

Go back to the cage the outie came from. With 5 at R7C1, the 21-cage needs 16 from R8C1 and R9C1, and two different digits make 16 in only one way: 7 + 9. Row 9 already has its 9, at R9C4, so R9C1 cannot take it. R9C1 is the 7, and R8C1 = 9.

That is the innie total of step 3 paying off: the count told you R8C1 + R9C1 = 16 before anything was placed. The 9 at R8C1 is a forced cell — the cage pins it while its pencil marks still show seven digits — and R9C1 = 7 follows by simple subtraction.

Technique: Forced cell in a cage

The 21-cage after R7C1 = 5, and the 9 at R9C4. R8C1 and R9C1 must make 16 = 7 + 9, and R9C1 cannot be 9, so R8C1 = 9.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	215
R8	9
R9				9

Digit already on the grid
The cell the cage forces
Digit placed
Cage, sum in its corner
Not needed for this step

Filled in: the cage; cell R9C4. The other 77 cells are left blank: they are not needed for this step. Placed: R8C1 = 9.

Common mistakes

Missing a cage that only grazes the rows

A cage that dips into the region by a single cell — say, one hanging down from row 7 — counts in full, just like a cage lying wholly inside, and it is the easiest one to overlook. Leave a cage out and the total falls short by its whole sum. Here, dropping the 21-cage gives 74, below 90, so at least you would notice; dropping a small cage in a busier grid gives a wrong total that looks perfectly believable. Walk every cell of the rows and tick its cage before you add.

Forgetting outies that are already filled

The difference above 90 covers every outie, filled or not. When three cages cross out of the rows and two of their outside cells already hold digits, the empty outie gets what is left after you subtract those digits — not the whole difference. List the outies first, then the ones still empty.

Assuming two outies must be different

Two empty outies sharing a total of 8 can be 4 and 4, unless something keeps them apart. They must differ only when they share a row, column, box or cage; outies on opposite sides of the region often share none of these. Striking 4 from both cells because "a pair cannot repeat" removes a digit that may be the answer.

Keep going

Innies & OutiesA row, column or box totals 45: take away the cages inside it, and the cells left over must make the rest.
Killer Sudoku & the Rule of 45The rules of Killer Sudoku, the first easy cage deductions, and the rule of 45 that turns a box's cages into a digit.

Every cage combination, for every size and sum: Killer Sudoku Cage Sum Chart — all combinations

Learn the move, not just the digit

In a real Killer game, when this count is what opens the next cell, the hint in the app points to that cell, lists what rules out each other digit, and names the rule of 45 across regions for the ones it removes — the move in step 4 above.

Play a Killer Sudoku now