Solving expert killer sudoku

Expert killer is the tier where killer puzzles start looking like extreme classic puzzles with cage constraints layered on top. The 45-rule applies to itself — innies and outies of innies and outies — and the advanced classic techniques (X-wings, swordfish, unique rectangles, forcing chains) start firing earlier than they do in classic, because the cage constraints surface the patterns sooner. Most expert killer puzzles take 30 to 45 minutes. The longer ones run an hour. The pacing rewards solvers who commit to full notation and structured technique-by-technique scanning.

Layered 45-rule

The basic 45-rule — region totals to 45, subtract complete cages, deduce what's left — fires constantly at expert. The new layer at this tier is applying the rule recursively. Once an innie or outie's value is deduced from the 45-rule on one region, that deduction often unlocks a 45-rule application on a neighbouring region: the cell you just deduced is part of a cage in the next region over, and now that cage's running total shifts, opening a new innie/outie deduction.

The pattern looks abstract written down. In practice, it means walking the grid systematically: row 1's 45-rule gives you cell A; that lets you compute a partial cage sum that opens row 2's 45-rule; row 2 gives you cell B; cell B is the last unknown in a column-axis cage, opening column 3's 45-rule. The chain runs until it terminates — sometimes after two steps, sometimes after five.

Innies and outies at expert often involve multiple cells per region. A region with two innies doesn't tell you either cell directly, but it tells you their combined sum — which combined with a killer pair constraint inside the cage frequently nails down both cells. Reading multi-cell innies/outies as cumulative information is the new mental shift expert asks for.

Classic techniques fire earlier

Most expert killer puzzles need at least one classic-style advanced technique — an X-wing, a swordfish, a unique rectangle, or a forcing chain. The cage constraints often surface these patterns earlier than they would appear in classic at equivalent difficulty. A digit's positions inside a region are constrained not just by the row/column/box rules but by which cages can accept that digit, and that extra constraint frequently produces X-wing-shaped configurations after only two or three placements.

The piece on why the X-wing keeps tripping people up is worth reading at expert killer if the X-wing recognition pattern hasn't fully clicked yet. The piece is written for classic but the recognition habit transfers directly. Same for unique rectangles and forcing chains — the techniques are unchanged; what changes is the frequency they appear and the additional cage-sum verification you can run alongside them.

Pencil marks at expert killer

Expert killer requires the most aggressive notation discipline of any tier covered in this series. Full digit candidates in every empty cell, full cage-sum-set candidates next to every cage sum, and ideally chain-marking notation when forcing chains are used. The notation looks intimidating for the first few puzzles. After about ten expert killer puzzles, most solvers find the routine settles and the notation overhead drops — the eye learns to read the busy grid as structured information rather than visual noise.

Our piece on reading pencil marks like a shape covers the visual habit that makes full notation readable rather than overwhelming.

When expert killer stalls

Expert killer stalls in two characteristic ways, both rooted in tracking failures rather than missing techniques.

Nested innie/outie tracking errors. The layered 45-rule chains can be three or four deductions long, and a single arithmetic error breaks the rest of the chain. Most expert killer stalls trace back to a deduction made twenty minutes earlier that was off by one or two — the cell labelled wrong, the cage total miscomputed. The fix is to verify each 45-rule application on paper rather than in your head, and to mark the deduced values clearly enough that you can re-check them later. Audit-style backtracking is the move when a stall has lasted ten minutes; the error is almost never the technique you're trying to apply, it's a deduction earlier in the chain.

Cage-state drift. Over the 30 to 45 minutes of an expert killer solve, the cage-set candidates accumulate small errors. A cage that was narrowed to two possible digit sets has had one of those sets eliminated by a placement, but the notation wasn't updated. Walking every unresolved cage every fifteen or twenty minutes and verifying its remaining digit sets against the cells inside it surfaces these errors before they hide the next move.

Two habits to drop

Bashing. Expert killer rewards method, not endurance. When a stall has lasted ten minutes, the move that's there isn't going to come from staring harder. Re-pencil-mark, re-walk every region's 45-rule, technique-by-technique scan, take a break. Solvers who push through stalls usually compound errors rather than break them.

Skipping notation. The discipline that worked at hard — full digit candidates, killer-pair tracking, basic 45-rule arithmetic — extends at expert with cage-set notation and chain-marking. Solvers who try to hold expert killer in their head occasionally succeed on shorter puzzles but routinely fail on the longer ones. The notation isn't optional at this tier, and the time spent maintaining it is less than the time spent unwinding errors caused by skipping it.

Expert killer is the top of the killer ladder for most generators. The graduation from this tier, if there is one, is from solving expert in 45 minutes to solving it in 25 — speed rather than new techniques. Most solvers spend years at expert killer without exhausting it; the cage configurations vary enough that each puzzle still asks for something specific even after hundreds of solves. Worth coming back to a few easy killer puzzles occasionally too — the contrast resets the technique-tracking habit and the lower-tier rhythm is its own pleasure.