Killer Sudoku techniques: the moves that don't exist in classic Sudoku

Every row, every column, and every 3x3 box in any sudoku contains the digits 1 through 9 once. They add to 45. That's the only fact this technique needs.

In a classic sudoku the fact is useless because you never get sums. In killer, every cage gives you a sum. So if you can find a row where every cage either fits entirely inside it, sits entirely outside it, or leaks by exactly one cell, you can subtract your way to the value of that leaking cell.

Pick a row. Walk every cage with at least one cell in it. Add the sums of cages entirely inside the row. Note any cage that crosses the row's boundary. If exactly one cell of the row belongs to a leaking cage, that cell's value is 45 minus the inside total minus whatever the rest of the leaking cage already accounts for outside the row.

The same trick scales. Two adjacent rows sum to 90, three sum to 135. When the cage layout is hostile at the single-row level it's often friendly across a pair, and you get back two-cell sums you couldn't see otherwise.

These are the cells the Rule of 45 exposes when only one cell sticks out from the cages of a house.

An innie sits inside the house you're analyzing while the rest of its cage sticks out. An outie is the opposite: most of its cage is in the house, but one cell pokes out. The naming is bookkeeping. What matters is the math.

For an innie, the inside contribution from the leaking cage is unknown but the rest of the cells in the house come from cages whose sums are fully inside. The innie equals 45 minus the sum of all complete inside cages, minus the part of the leaking cage that you can already compute.

When two cells leak instead of one, the trick still works. You don't get a single-cell answer; you get a virtual two-cell cage with a known sum. From there it's combinations.

The Rule of 45 scales linearly. Two adjacent rows sum to 90. Three, 135. Any rectangular block of N full houses sums to N times 45.

This matters when the cage layout is hostile to one row but friendly to a pair of rows. Sometimes every cage either lives inside the pair or pokes out cleanly with the sticking-out parts pinned to small numbers. You compute the leak for the pair, not the row, and you get back single-cell or two-cell sums you couldn't see at the single-house level.

A useful default scan: once for clean single-house leaks, once for clean two-house leaks. By the time you've done both you've usually found enough to make the rest of the killer-specific work redundant.

A locked combination is a cage whose size and sum together force exactly one digit set. A two-cell cage summing to 3 must be {1, 2}. A four-cell cage summing to 30 must be {6, 7, 8, 9}. There are no other options.

These are gold. The cage doesn't tell you which cell holds which digit, but it pins the digit set. That set then participates in every Sudoku-style constraint elsewhere: naked subsets, locked candidates, line-box reductions. A locked four-cell cage at the bottom of a 3x3 box claims four of the box's nine digits before you've placed a single number.

The complete list is short. You don't need to memorize it; you need to recognize it on the board. The cheat sheet on the cages reference page covers every locked cage size and sum.

Locked combinations are forced by size and sum alone. Hidden combinations are forced by everything else.

Take a three-cell cage summing to 12. There are seven combinations: {1,2,9}, {1,3,8}, {1,4,7}, {1,5,6}, {2,3,7}, {2,4,6}, {3,4,5}. If even one of those combinations is impossible because the cage's box already contains a 1, or a row through the cage rules out 9, you cross it off. Sometimes you cross off six and the seventh is your answer.

Hidden combinations are how ordinary cages turn into useful ones. The work is bookkeeping. A scrap of paper with the surviving combinations under each cage is worth more than any clever trick on the board.

A killer pair is two cells, possibly in different cages, whose candidate sets are identical and consist of the same two digits. The classical naked-pair logic applies: those two digits are claimed by those two cells and can be eliminated from any house they share with other cells.

Killer pairs show up most often inside a single cage when the cage's surviving combinations all share the same two digits in the same two cells. They also show up across cages when two cages partially overlap a house and their possible digits force the same restriction.

The triple version works the same way with three cells and three digits. Beyond triples the pattern keeps generalizing, but the practical payoff drops fast: finding a killer quad takes more work than running the rest of the techniques on the page would.

Cage splitting takes a cage that crosses a house boundary and treats the part inside the house as a virtual mini-cage with a derived sum.

Consider a five-cell cage summing to 25 with three cells in a 3x3 box and two cells outside it. If the two outside cells are pinned to {7, 8} by some other constraint, the three inside cells must sum to 25 minus 15, which is 10. Now you have a virtual three-cell, sum-10 cage. The combinations table gives you the four candidate sets: {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}.

The split rarely solves cells outright. What it does is eliminate two or three candidates per cell, which is the difference between a stuck puzzle and a moving one.

A pseudo-cage is a virtual cage built from partial cages plus the Rule of 45.

Suppose row 1 contains pieces of three cages: a two-cell cage entirely in the row, a four-cell cage with one cell in the row and three in row 2, and a three-cell cage entirely in the row. The cells of row 1 sum to 45. The two cages entirely in the row contribute their full sums. The four-cell cage contributes its full sum minus whatever its three cells in row 2 amount to.

This is the same idea as innies and outies, but instead of solving for a single cell you're constructing a hypothetical cage from leftover row-1 cells. That hypothetical cage has a known sum (45 minus the others) and a known size (whatever cells aren't claimed). Now you can apply combinations to the hypothetical cage. Pseudo-cages are how the Rule of 45 produces useful information when the leaks aren't just one or two cells.

Sometimes a cage has several surviving combinations and they all share a digit. Take a three-cell cage summing to 8: the possibilities are {1, 2, 5} and {1, 3, 4}. Both contain 1. Whatever else the cage holds, one of its cells must be a 1.

That digit can be eliminated from any cell outside the cage that shares a row, column, or box with one of the cage's cells (because the 1 must live somewhere inside the cage and can't live in the houses the cage already covers).

Consistent numbers are weak by themselves but they compose well with hidden combination work. They often unlock the next round of eliminations.

Big cages are awful to enumerate. A seven-cell cage has too many combinations to track. The trick is to enumerate its complement instead.

If a seven-cell cage sits inside a 3x3 box, the two cells of the box not in the cage form an implicit two-cell cage whose sum is 45 minus the big cage's sum. A seven-cell cage summing to 38 leaves a two-cell complement summing to 7, which has just three combinations: {1, 6}, {2, 5}, {3, 4}. Whichever the complement turns out to be, the big cage holds the other seven digits.

The same idea applies to six-cell and eight-cell cages within a single house. You don't need a dedicated combinations table for them; you need the small-cage tables you already have.