Expert puzzles occasionally reach a point where every basic technique is exhausted. The patterns below are the usual next step. None of them require guessing; all of them are ordinary elimination applied across two or more regions at once.
X-wing
Pick a digit, say 5. Find two rows in which 5 has exactly two candidate cells each, and where those cells sit in the same two columns. The four cells form a rectangle.
Here is the logic. In each row, the 5 is in one of two cells. Whichever way it falls, one 5 lands in each of the two columns. Therefore no other cell in those two columns can be a 5 — remove it everywhere else on those columns.
It works identically starting from columns and eliminating along rows. Whenever you are stuck, picking each digit in turn and counting its candidates per row is a fast way to find one.
Swordfish
The same idea with three rows and three columns. Three rows in which a digit has two or three candidates each, all confined to the same three columns, means that digit occupies one cell in each of those columns — so it can be removed from the rest of them.
Swordfish is genuinely rare and fiddly to spot. It is worth knowing the shape exists, but it is not where your effort belongs until X-wings are automatic.
XY-wing
Three cells, each with exactly two candidates. A pivot cell holds {X,Y}. Two other cells, each able to see the pivot, hold {X,Z} and {Y,Z}. Whatever the pivot turns out to be, one of the other two must be Z. So any cell that can see both of those cells cannot be Z.
This one takes a moment to accept, but the reasoning is airtight: if the pivot is X, the {X,Z} cell must be Z; if the pivot is Y, the {Y,Z} cell must be Z. Either way a Z appears in one of them, so a cell seeing both is safe to clear.
Unique rectangle
A technique that leans on the puzzle having exactly one solution. If four cells forming a rectangle across two boxes all hold the same two candidates, the puzzle would have two solutions — you could swap the pair around and both would be valid. Since a proper puzzle has one solution, that configuration cannot happen, which lets you eliminate.
It is a legitimate technique for puzzles guaranteed unique, which includes every puzzle on this site. Purists dislike it because it reasons about the puzzle rather than the grid.
Check the basics again first
Nine times out of ten, a grid that seems to need an X-wing actually has a hidden single or a locked candidate that was missed. Do a clean re-sweep of singles, pairs and pointing before reaching for anything exotic.
When these are worth the effort
Honestly, not often. Published puzzles up to hard almost never need anything beyond pairs, triples and locked candidates. The expert setting here occasionally produces grids where an X-wing is the cleanest path, but even there a patient re-scan usually finds something simpler.
Learn X-wing because it appears often enough to matter and because its logic teaches you to think across regions. Treat swordfish, XY-wing and unique rectangles as curiosities to recognise rather than tools to hunt for.
A practical stuck-point routine
- Re-verify every candidate list. A single stale mark can make a solvable grid look impossible.
- Sweep for naked and hidden singles from scratch.
- Check pointing and claiming in all nine boxes.
- Look for naked and hidden pairs in all twenty-seven regions.
- Only then, pick each digit in turn and count candidates per row and per column, looking for an X-wing.
If you reach step five regularly, you are solving genuinely hard puzzles. If step one or two keeps producing results, the issue was bookkeeping rather than technique.