How Killer Sudoku Works

The rules, in full

Fill the nine-by-nine grid so that every row, every column and every three-by-three box contains each of the digits 1 to 9 exactly once. That is the sudoku part and nothing about it changes.

The grid is also divided by dotted outlines into cages, and every cell belongs to exactly one cage. The small number printed in the top-left cell of a cage is the total its digits must add up to, and no digit may appear twice inside the same cage. That is the whole of it. There is nothing else to know before you start.

The killer sudoku rules above are complete as they stand. There is no variant in play on this site, no extra constraint hidden in the layout, and nothing that appears only on the harder boards. What changes between an easy puzzle and an expert one is how much the cages tell you, never what the cages mean.

A cage is not a box

This is the commonest early confusion, and it is worth clearing up before the first move. Cages ignore the grid’s own structure completely. One can sit inside a single box, straddle two of them, run along part of a row, bend round a corner in an L, or take a shape with no name at all. The one structural guarantee is that a cage is connected: every cell in it touches another cell of the same cage along an edge, never only at a corner.

A cage is not a region with rights of its own. It has no requirement to contain particular digits and no relationship to the boxes it crosses. It is a statement about a set of cells, and the set was chosen when the puzzle was built.

The no-repeat rule is sometimes redundant and sometimes the whole point

Think this through once, because it governs how much any given cage is telling you. If a cage lies entirely inside one row, ordinary sudoku already forbids a repeat there, so the cage rule adds nothing at all. The same goes for a cage sitting wholly inside one box.

But a cage spanning two rows and two boxes contains cells that share no unit whatsoever, and for those cells ordinary sudoku is perfectly content to see the same digit twice. The cage rule is the only thing standing in the way. That is why a total of 8 across two cells means 3 and 5, 2 and 6, or 1 and 7, and never 4 and 4, wherever on the grid those two cells happen to sit.

Every cell belongs to exactly one cage

That is a rule rather than an observation, and two things follow from it that are worth having before the first move. The first is that the cages tile the grid: they overlap nowhere and they leave nothing out, so there is no cell anywhere on the board that is not part of some printed total. That property is what makes the whole-region arithmetic on the hard page possible at all, and it fails immediately on any board where a cell has been left uncaged.

The second is that a cage may hold a single cell, and that is the one place on a killer board where a total is simply a digit – one square, one number, nothing to work out. Gentler puzzles carry a great many of them and the hardest ones almost none, which is most of what easy means here.

How to play killer sudoku from a cold start

Read every cage before writing anything down. Fill in the ones with only a single possible combination. Pencil the possibilities into the rest. Then work the rows, columns and boxes in the ordinary way, using the fact that each of them totals 45 whenever a group of cages happens to lie neatly inside one. You are finished when the grid is full and every cage adds to its printed total, and in practice you only need to check one of those two things: get every row, column and box right and the cages must come out right as well, because there is only ever one way to fill a board of this kind.