Solving hard killer sudoku

Hard killer is the tier where the 45-rule stops being a fancy technique and becomes the move that fires every puzzle, often more than once. Easy and medium killer mostly resolve on cage-sum arithmetic and basic singles — the 45-rule is available but rarely necessary. At hard, the puzzles are designed so that without the 45-rule, you stall. Combined with a few cage-specific patterns and the geometric thinking that comes from looking at cage shapes rather than just cage sums, the 45-rule is the technique that turns a hard killer puzzle from impossible to satisfying. The tier's bargain is straightforward: arithmetic isn't enough; geometry is required.

The 45-rule as everyday tool

The 45-rule is the foundational killer technique. The digits 1 through 9 sum to 45, and every row, column, and box on the grid must contain those digits exactly once — so every row, column, and box sums to 45. If a row contains five complete cages totalling 38, the remaining cage in that row must total 7. That's already enough to narrow the cage's possible digit sets dramatically, and it doesn't require solving any individual cell. The 45-rule fires hardest on rows and boxes; column-level 45s tend to be easier to spot but rarer to be useful, because cages cross row and box boundaries more often than they cross column boundaries on the average killer grid.

Innies and outies are what the 45-rule reveals. An innie is a cell that sticks into a region from a cage that mostly lives outside the region. An outie is the reverse — a cell that sticks out of the region from a cage that mostly lives inside it. The 45-rule applied to a region with one innie tells you that innie's value directly: the total of complete cages in the region, minus 45, gives the innie's required sum, and if the innie is one cell, that's the cell's value. Innies and outies often deliver the cleanest single-cell placements at hard, and they get faster to spot once you stop reading the cage outlines as colour and start reading them as shapes.

The cage-set patterns worth memorising

Killer pairs and triples are the cage-constraint version of the classic naked pair. A 2-cell cage summing to 17 can only contain 9 — there's no other set of two distinct digits that sum to 17. A 3-cell cage summing to 6 can only contain 3. A 2-cell cage summing to 4 can only contain 3 (since duplicates aren't allowed in a cage). These eliminate every other digit from those cells, and from every other cell in the cage's row, column, and box that those digits could otherwise occupy.

The combinations worth memorising at hard: 3 = 2, 4 = 3, 16 = 9, 17 = 9 for two-cell cages. For three-cell cages: 6 = 3, 7 = 4, 23 = 9, 24 = 9. Spotting these in a fresh grid takes a few seconds and frequently delivers two or three eliminations on the first pass.

Cage geometry

At hard, the shape of the cages relative to each other starts mattering as much as their sums. When two cages share a row, column, or box, the digits in one cannot also appear in the same row, column, or box as the digits in the other. This is just the standard sudoku constraint applied through cage outlines, but it's easier to forget when the cages are visually separated. Walking the cage shapes once before placing any digit — checking which cages share rows, columns, or boxes — is a habit worth building. Our piece on meeting killer sudoku covers the rules in more depth if any of this is unfamiliar.

When hard killer stalls

Hard killer stalls usually come from one of two failure modes.

Single-axis 45-rule. Most solvers learn the 45-rule on rows first, and many never apply it systematically to boxes. But box-level 45s often deliver cleaner innie/outie eliminations because boxes have more cage-edges than rows do — cages enter and exit boxes more frequently than they enter and exit rows. When a hard puzzle stalls, applying the 45-rule to a box you haven't checked is often the next move. Same for columns, though they're typically less productive.

Cage-sum tunnel vision. Spending too long on the "what digits sum to N" arithmetic at the expense of the cage's row/column/box position. Hard killer puzzles are joint constraints — cage sum AND grid position — and when one fails to break a stall, switching to the other usually does. If a 3-cell cage summing to 14 has too many possible digit combinations to narrow on the sum alone, look at where those cells sit on the grid; some combinations get eliminated by what's already in the surrounding rows and columns.

Two habits to drop

Pencil-marking only the digit candidates without tracking cage-sum candidates. The full pencil mark for a killer cell isn't just "which digits could go here" — it's also "which cage-sum combinations are still possible." If a 4-cell cage summing to 22 has been narrowed from 7 or 8 or 9 (etc.) down to two combinations, that's information worth tracking somewhere on the grid. Most solvers keep this in their head; the fast solvers note it next to the cage sum.

Treating cages as decorative. The cage outlines aren't just visual grouping — they're constraints that fire constantly through the solve. When a cage gets fully resolved, its outline can be visually mentally "turned off" since it no longer adds information. When a cage is partially resolved, the remaining cells' digit options are dramatically constrained. Reading the cage state actively, rather than treating cages as static decoration, is the difference between a 30-minute hard killer solve and a 60-minute one.

The 45-rule becomes second nature. You'll start applying it without consciously deciding to, and the box-level applications will feel as natural as the row-level ones. The next tier, expert killer, introduces layered 45-rules (innies and outies of innies and outies) and the same advanced classic techniques (X-wings, unique rectangles, forcing chains) but with cage constraints making them appear earlier than they do in classic. Most solvers spend several months at hard before expert feels accessible, and that's correct pacing — the cage geometry takes time to become intuitive.