What Is the 45 Rule in Killer Sudoku? Innies and Outies Explained

If there's one technique that turns a stuck killer sudoku into a solved one, it's the 45 rule. It sounds like math homework, but it's really just a counting trick: since every row, column, and 3x3 box contains the digits 1 through 9 exactly once, each of those regions always adds up to 45. Once you internalize that, you can pull single digits straight out of a blank-looking grid with nothing but addition and subtraction. This guide explains the 45 rule, then shows you the two moves it powers — innies and outies — that strong solvers rely on every game.

New to the puzzle? Skim the killer sudoku rules first, or follow the step-by-step beginner solve. Then come back for the technique that levels you up.

Why every region sums to 45

The logic is short. The digits 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 add up to 45. A killer sudoku still obeys all the normal sudoku rules, so every row holds each of those nine digits once, every column does, and every 3x3 box does. That means the total of any complete row, column, or box is always exactly 45 — no exceptions, on every puzzle ever made.

That fixed total is leverage. The cages tell you the sums of small groups of cells. If you can account for almost all the cells in a region using cage sums, whatever's left over has to make up the difference to 45.

The basic move: a single missing cell

Here's the simplest use. Take a 3x3 box. Suppose four cages sit entirely inside it, summing to 17, 9, 11, and 4. That's 17 + 9 + 11 + 4 = 41, covering eight of the nine cells. The ninth cell isn't part of any of those cages — maybe it belongs to a cage that mostly lives in the next box over.

Since the box must total 45, that lone cell is 45 − 41 = 4. You placed a digit without touching a single candidate list. On a hard killer sudoku with no givens, this is often your very first solved cell.

Innies: a cell that pokes in

An innie is a cell inside a region that belongs to a cage extending outside it. You can find its value with the same subtraction.

Picture a box where all the cages are contained except one cage that has just a single cell inside the box (the rest of that cage spills into the neighboring box). Add up the fully-contained cages — say they total 38. The box totals 45, so the cells outside those cages must total 7. If only one cell is left unaccounted for inside the box, that innie is 45 − 38 = 7. The name comes from the cell being the part of an outside cage that reaches in.

Outies: a cell that pokes out

An outie is the mirror image: a cell just outside a region that one of the region's cages reaches out to grab.

Suppose every cage in a box is fully inside it except one cage that has eight of its cells in the box and a single cell poking out into the row below. Add the box's cage sums. The amount above 45 is exactly the value of that one outside cell. If the cages total 50 and the box can only hold 45, then the extra 5 lives in the outie — that single external cell is 5.

Innies and outies are two sides of the same coin: you're comparing the sum of a set of cages against the 45 a clean region must hold, and the leftover lands in one cell.

Going multi-region: the big leagues

The real power shows up when you combine regions. Three stacked boxes form a band that must total 3 × 45 = 135. Three columns form a stack worth 135 too. By adding cage sums across a whole band and accounting for the cages that cross its borders, you can isolate a single innie or outie that no single-box analysis would ever reveal.

This is the engine behind expert and Einstein puzzles, and it's covered further in the advanced techniques guide. Don't reach for it until single-region innies and outies feel natural — but when you're stuck on a tough grid, a band calculation is often the only thing that breaks it.

A quick checklist for using the 45 rule

When you're stuck, run through this:

1. Scan every box for one that's nearly covered by fully-contained cages. Subtract their total from 45.
2. Do the same for rows and columns — they total 45 too, and beginners forget that.
3. Look for cages that cross a region boundary; those create innies and outies.
4. If single regions are exhausted, add cage sums across a band of three boxes (135) and find the crossing cell.

Where this fits in your overall approach

The 45 rule isn't the whole game — you still need locked cage combinations, candidate elimination, and the standard sudoku techniques. But it's the deduction that produces digits when nothing else will, which is why it sits near the top of the complete killer sudoku strategy. Treat it as your go-to move whenever the grid goes quiet.

Want to try it live? Open a hard killer sudoku, find a box that's almost cage-complete, and subtract from 45. The first time a digit just falls out, you'll never solve without it again.

Frequently asked questions

What is the 45 rule in killer sudoku?

The 45 rule states that every row, column, and 3x3 box in a killer sudoku contains the digits 1 through 9 once, so each one always sums to 45. By adding the cage sums inside a region and subtracting from 45, you can deduce the value of a single uncovered cell.

What are innies and outies in killer sudoku?

An innie is a cell inside a region that belongs to a cage reaching in from outside; an outie is a cell just outside a region that one of its cages reaches out to. Both are found by comparing the region's cage sums against the 45 it must total — the difference reveals the cell's value.

Does the 45 rule work for rows and columns too?

Yes. Every complete row and every complete column contains 1 through 9 once, so both total 45 just like boxes do. Many solvers only apply the rule to boxes and miss easy deductions in rows and columns.

Can I add the 45 rule across multiple boxes?

Yes. Three stacked boxes total 135 (3 × 45), and so do three columns. Adding cage sums across a whole band and accounting for crossing cages lets you isolate cells that single-region analysis can't reach. It's a core expert technique.

Is the 45 rule the same as cage combinations?

No. Cage combinations tell you which digits can fill a cage based on its sum. The 45 rule uses the fixed total of a whole region to deduce a single cell. They work together — combinations narrow candidates, the 45 rule produces exact digits.