Killer Sudoku: The Rules and How to Solve It

What makes it Killer

A Killer Sudoku is a normal nine-by-nine board with the usual rule: each row, column, and three-by-three box holds every value once. On top of that, the grid is divided into dotted regions called cages, and each cage shows a small target number. The values inside a cage must add up to that target, and no value may repeat within a cage. Most Killer puzzles begin with no given numbers at all, which surprises newcomers. The cages and their sums are the only clues you get, and that is precisely what keeps the puzzle feeling fresh even to solvers who have finished thousands of ordinary grids.

The rule of 45

Here is the single most useful trick in the whole variant. Every row, column, and box contains the values 1 through 9 exactly once, and those nine values always add up to 45. So if the cages inside a row very nearly fill it, you can subtract their sums from 45 to find the value of the leftover cell. Suppose a row is covered by cages totaling 40, with just one cell left uncaged at r1c9. That lone cell must hold 45 minus 40, which is 5. This one idea, sometimes called the 45 rule or the innie-outie method, unlocks a huge number of early moves that would otherwise be invisible.

Think in combinations

Small cages are your best friends, because their sums have very few possible fillings. A two-cell cage summing to 3 can only be 1 and 2, in some order. A two-cell cage summing to 17 can only be 8 and 9. A three-cell cage summing to 6 can only be 1, 2, and 3. Learning these locked combinations lets you pencil candidates into a cage almost instantly, long before the surrounding cells are solved. The extremes are the most valuable, since the smallest and largest sums allow the fewest arrangements and therefore pin down the most information in a single glance.

Combinations worth memorizing

• Two cells summing to 3: only 1 and 2.
• Two cells summing to 4: only 1 and 3.
• Two cells summing to 16: only 7 and 9.
• Two cells summing to 17: only 8 and 9.
• Three cells summing to 6: only 1, 2, and 3.
• Three cells summing to 24: only 7, 8, and 9.

A short worked example

Imagine a two-cell cage at r1c1 and r1c2 with a target of 17. From the table, those cells must be 8 and 9 in some order, so you can pencil both candidates into each. Now suppose r1c1 also sits in a box that already contains an 8 elsewhere. That removes 8 from r1c1, forcing it to 9, which in turn forces r1c2 to 8. Two cells solved from a single cage and one outside clue. This is the rhythm of Killer: the cage math narrows things down to a couple of options, then ordinary Sudoku logic arrives from the row, column, or box to finish the job.

Then it is just Sudoku, so keep practising

Once the cage arithmetic gives you a foothold, the classic techniques take over completely. Singles, pairs, and plain scanning all work exactly as they do in normal Sudoku, because the row, column, and box rules are still fully in force. Killer simply front-loads the puzzle with an extra source of clues, the sums, and then hands the board back to your usual logic. If you can already solve a standard grid, you are most of the way to solving a Killer one; you have only added a layer of light arithmetic at the start. So explore the variants if you enjoy them, but keep your core skills sharp with a standard board too, since the sharper your scanning and singles become, the easier every Killer cage will be to crack.

Working with three and four-cell cages

The small two-cell cages are where you begin, but the larger cages are where a Killer puzzle is really won, and they reward a little extra thought. A three-cell cage has more possible fillings than a two-cell one, yet the extremes still lock down hard: a three-cell cage summing to 7 can only be 1, 2, and 4, and one summing to 23 can only be 6, 8, and 9. With four-cell cages the same idea holds at the edges, where very low or very high totals leave only a handful of arrangements. A useful trick is to note which values a large cage cannot contain rather than which it must, since ruling values out is often faster than pinning them down. Combine that with the rule of 45 across a whole box, and even a cage that looks wide open will usually surrender one or two certainties. From there the ordinary Sudoku logic takes hold, and the rest of the region tends to unravel steadily, one forced value pulling the next into place.