Sudoku Written by Brian Hamilton and Karan Hamilton.
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Every row, column and box of a sudoku holds the digits 1 to 9 once, and 1 + 2 + … + 9 = 45. A killer sudoku prints no digits, only cage totals, so that 45 turns into a clue of its own. Total the cages lying wholly within one house, subtract from 45, and the difference is the sum of the house’s cells whose cages cross its edge. Those cells are the innies and outies, and the counting is the rule of 45, or the 45 rule.
We show it on one real Expert board from the generator behind our Killer Sudoku game, replayed by our own solver. It works the way a person does, easiest technique first, and gets stuck three times on this board; the rule of 45 frees it each time, with one cell, with two, and with two rows counted together. Then we counted how often each of our levels needs the rule, and where it turns up. If cages and totals are new to you, start with our rules of Killer Sudoku.
What is the rule of 45 in killer sudoku?
A complete house adds up to 45, two houses to 90 and three to 135. Pick a house and sort the cages that touch it into two kinds: those lying wholly inside, and those crossing its edge. Then:
- Innies are the cells inside the house that belong to crossing cages. Together they make 45 minus the totals of the whole cages.
- Outies are the cells outside the house that belong to crossing cages. Together they make the totals of every cage touching the house, minus 45.
Either count can leave a single cell, which then has its digit, or a few cells with a known sum, which you treat like a cage of your own. The rule needs cage totals to work on. A classic sudoku has nothing to subtract, so there the rule only says what you knew already: the last empty cell of a house takes the missing digit.
How do innies and outies work?
Here is our board nine steps in. Its four one-cell cages are filled, cage arithmetic has narrowed everything it can, and there is no single, no locked digit and no pair to find. Look at the middle-left box. Three cages lie wholly inside it, with totals of 8, 19 and 10. The box’s ninth cell, R6C3, belongs to a 14-cage that carries on into row 7.
The three whole cages use 37 of the box’s 45, so R6C3 must be 8. Nothing about R6C3 itself pointed that way: its own cage and its column still allowed any digit but 2. That is what makes the rule worth learning. Cage arithmetic reads a cell’s own cage; the rule of 45 reads the cages around it, and so reaches cells nothing else can.
An outie is the same count from the other side. Total every cage that touches the house, whole or crossing, and the result overshoots 45 by exactly the sum of the cells sticking out. The two-row example below is an outie.
What if two or three cells are left over?
Then you have found a cage the puzzle didn’t print. Three steps after the 8, the bottom-left box leaves two loose cells. Its whole cages total 15, 18 and 2 (a one-cell cage counts like any other), so R7C3 and R9C3 add up to 45 − 35 = 10.
Treat the two cells exactly like a printed two-cell cage of 10. Its combinations are 1 + 9, 2 + 8, 3 + 7 and 4 + 6; 5 + 5 is out, because the cells share column 3. Loose cells from one house always share that house, so they never repeat a digit. Counted over two or three houses, they may share nothing at all, and then a repeat is allowed.
Leftovers like this are the usual case. Across our 300 Hard boards, our solver counted a single house 346 times: 63 counts left one cell, 127 left two and 156 left three. It takes the first count it finds that removes anything, though, and a one-cell count was available somewhere at 235 of those 346 moments, so it pays to look for one before working through a bigger set.
How does the rule of 45 work over two or three rows, columns or boxes?
Count more than one house and the target becomes 90 for two, 135 for three. It helps when every single house has too many loose cells, which happens when long cages wander across several houses. Our solver tries two or three neighbouring rows or columns, and two neighbouring boxes. Three boxes side by side are the same 27 cells as three rows, so that case is covered too.
Five steps after the pair, our board needs exactly that. No single row, column or box gives anything any more, but rows 8 and 9 taken together do.
The two rows hold five whole cages, totalling 75, and one 20-cage that reaches up into row 7 with one cell. All six cages come to 95, five more than two rows can hold, so the cell outside, R7C9, is 5. Until that moment R7C9 still had all nine candidates, and this was the only count over several houses that removed anything.
Our levels need the bigger counts less often than the single-house kind: on 13 of 300 Hard boards and 32 of 300 Expert, never on Easy or Medium. Two houses were enough for 66 of the 90 multi-house counts our solver made on those boards, and only 16 of the 90 left a single cell, so expect a two- or three-cell sum to come out of them.
How often do you need the rule of 45?
We ran our solver over 300 boards per level, dealt by the same generator as the game, then ran it again with the rule of 45 switched off. A board needs the rule if the solver finishes it with the rule and not without. Easy never does: singles and cage arithmetic finish every Easy board. Medium needs it now and then, Hard on more than a quarter of boards and Expert on two in five.
| Medium | Hard | Expert | |
|---|---|---|---|
| Boards that need the rule of 45 | 12 | 88 | 124 |
| … over two or three houses | 0 | 13 | 32 |
| Single-house counts made on them | 34 | 346 | 444 |
| Digits placed before the first count (median) | 4 | 6 | 6 |
| Counts where some box would have worked | 34 | 345 | 433 |
| … some row | 18 | 92 | 73 |
| … some column | 20 | 124 | 80 |
| … a one-cell innie or outie | 25 | 235 | 251 |
300 boards per level from the generator behind our Killer Sudoku game, fixed seeds; Easy needs the rule on none. Our solver tries singles, cage arithmetic, a digit confined to one house, pairs and triples, then the rule of 45 on one house (at most three loose cells), then over two or three. At each single-house count we listed every house whose innies or outies would have removed a candidate. Measured October 2026.
Our solver counts houses only when nothing easier works, and on every one of the 224 boards where it did, switching the rule off left the board unfinished. It comes early, too. On Hard and Expert the first count came with a median of six digits on the board, barely past the one-cell cages: the rule of 45 is an opening move here, not an endgame trick.
Where should you look for innies and outies first?
In the boxes. At nearly every moment our solver needed the rule, some box offered a count that worked: 345 times out of 346 on Hard, 433 out of 444 on Expert. On those two levels a row offered one at only 16% to 27% of those moments, and a column at 18% to 36%.
That isn’t because cages follow box lines. Our generator grows each cage from a random cell into random neighbours and pays the boxes no attention. It is shape. A cage that turns a corner can still fit inside a 3×3 box, but never inside a row. On our Hard boards, 58% of the cages with two or more cells lie wholly inside one box, against 28% inside one row and 29% inside one column. More whole cages means fewer loose cells, and a house with three or fewer loose cells is one you can count.
Go round the nine boxes. For each, add the whole cages, one-cell ones included, and count the cells left over. Three or fewer: do the arithmetic. If no box gives anything, try the rows and columns, then two neighbouring rows or columns together. Write each result down as a small cage, such as R7C3 + R9C3 = 10, and use it at once.
Why doesn’t my rule of 45 count add up?
When the arithmetic gives a digit the cell can’t take, or a sum no set of cells could make, one of these is usually behind it:
- A crossing cage counted as whole. Trace the edge of the house before adding. A cage counts as inside only if every one of its cells is.
- A one-cell cage left out. Every one of our levels deals them. It is a whole cage whose total is its digit, like the 2 in R8C3, which counts in both the bottom-left box and rows 8 and 9.
- Innies and outies in one sum. Innies are 45 minus the whole cages; outies are all the touching cages minus 45. Both hold at once, but keep them as two equations.
- A two-cell sum read as one digit. 45 − 35 = 10 over two cells is a cage of 10, not a 10.
- A repeat ruled out that isn’t. Over two or three houses, two loose cells in different rows, columns, boxes and cages can hold the same digit.
If every count checks out and the puzzle still contradicts itself, it may have been copied wrong. Draw its cages into our Killer Sudoku solver, which reports whether that layout has one answer, several or none.
Stuck even after the rule of 45?
Go back to the easy techniques. A digit from the rule changes the cages around it: the 8 in R6C3 left its 14-cage needing 6 from two cells, and that set up the pair. Our solver returns to singles and cage arithmetic after every count, and on the Hard boards that need the rule it counted about four times each, 346 counts on 88 boards.
Between counts it uses one more killer move, a digit that every way of filling a cage needs, kept within one row, column or box. The rest are plain sudoku techniques: locked candidates, then naked pairs, hidden pairs and their triple cousins. Our Killer Sudoku tips and strategies guide puts the whole order together, level by level.
Beyond that, 5 of our 300 Hard boards and 34 of 300 Expert went past what our solver can do, though each has exactly one answer. It counts at most three loose cells and never puts innies and outies into one equation. Experienced solvers do both, with innie–outie differences and longer chains of cage reasoning. For steady practice, the daily Killer Sudoku works up from Easy early in the week to Expert at the weekend.
Killer Sudoku
Each board has been proved to have one answer. The rule of 45 decides about two Hard boards in seven, and two Expert boards in five.
▶ Play Killer Sudoku