Elimination, naked single, naked pair, pointing pairs — if you've worked through all of those and the puzzle is still stuck, it's time for the next layer. These four techniques break through the blockades you hit at the intermediate and expert levels, each from a different angle.
X-Wing and Swordfish are built on row-column symmetry. XY-Wing is a logical chain linking three cells. Forcing Chains aren't guessing — they're a matter of following both possibilities simultaneously until a single contradiction-free conclusion emerges. All four are pure logic, no guesswork involved.
To apply the techniques in this article, candidate notes are essential. You'll also need a solid grasp of basic elimination, naked singles and naked pairs. If those foundations aren't firmly in place yet, start with our guide to fundamental techniques.
X-Wing
The X-Wing operates at the intersection of two rows and two columns. The name comes from the shape of the letter "X": when the candidates for a single digit in two rows fall in exactly the same two columns, those four cells form an X — and that digit can be eliminated from every other cell in those two columns.
The logic is straightforward: that digit in row 2 will end up in either column 3 or column 7. The same digit in row 6 will also land in either column 3 or column 7. Whichever combination plays out, the remaining cells in columns 3 and 7 cannot hold that digit.
Visual Example — X-Wing
Step-by-Step Solution
Column-Based X-Wing
The X-Wing isn't limited to rows — the same logic applies to columns. If the candidates for a digit in two columns fall in exactly the same two rows, that digit is eliminated from all other cells in those two rows. The direction flips; the logic is identical.
Work through each digit separately. For the 7, scan every row: which one has 7 candidates in only two columns? Asking that question for every digit from 1 to 9 is far faster than hunting for an X-Wing directly. The first time you find one, something clicks before the puzzle is even finished — and the second time, that instinct arrives much sooner.
Swordfish
Swordfish is X-Wing scaled up to three rows. X-Wing covered two rows × two columns = four corners. Swordfish covers three rows × three columns = nine potential corners — but not all of them need to be filled. What matters is that all candidates for a digit across those three rows fit into no more than three columns.
"No more than" are the key words here. One of the three rows might have that digit in just a single column — that doesn't break the Swordfish. The condition is simply that when you pool all the candidates for that digit across the three rows, you end up with three distinct columns or fewer.
Visual Example — Swordfish
Step-by-Step Solution
X-Wing compares two rows — the brain can handle that visually. Swordfish requires holding three rows in mind simultaneously and computing the union of their columns. That extra load on working memory is why even experienced solvers miss the Swordfish sometimes. The practical fix: work one digit at a time, take notes, and scan the three rows one by one.
XY-Wing
XY-Wing shares nothing with X-Wing except the name — the underlying logic is completely different. Three cells, three two-candidate lists, and the visibility relationships between them — that's all there is to it.
Terminology: one pivot cell and two pincer cells. The pivot sees both pincers. The pincers don't need to see each other directly — but they share one common candidate. That shared candidate is eliminated from every cell seen by both pincers at the same time.
Structure and Logic
Pivot cell candidates: {X, Y}. First pincer: {X, Z}. Second pincer: {Y, Z}.
Why is Z eliminated? The pivot must be either X or Y. If the pivot is X → the first pincer must be Z. If the pivot is Y → the second pincer must be Z. Either way, it's guaranteed that one of the two pincers will hold Z. Therefore, no cell visible to both pincers can contain Z.
Visual Example — XY-Wing
Step-by-Step Solution
When More Than One Cell Is Affected
Sometimes an XY-Wing affects more than one cell — if both pincers see multiple cells simultaneously, Z is eliminated from all of them. This happens most often when one of the pincers sits on the boundary of a box.
A naked pair acts on cells that share the same unit (row, column or box). XY-Wing bridges cells from different units — without the pivot, the two pincers might not "see" each other at all. That's why XY-Wing reaches larger sections of the grid and eliminates candidates in places a naked pair simply can't touch.
Forcing Chains
Forcing chains aren't guessing — they're about following both possibilities simultaneously and showing that both lead to the same result. "If this cell is A → this happens → conclusion: Z. If it's B → a different path → but still: Z." When both paths open the same door, Z is confirmed.
Structurally, this technique differs from the previous three: instead of eliminating candidates directly, it uses a chain of deductions. But it must not be confused with guessing. Guessing tests one possibility and backtracks when it fails. Forcing chains exhaust both branches in full and derive the shared result through pure logic — no backtracking at any point.
Two Types of Forcing Chains
The two most common forms are binary forcing chains and unit forcing chains.
Binary forcing chains: choose a two-candidate cell. Assume it equals A and follow the resulting elimination chain. Then assume B and repeat. If in both cases the same cell takes the same value, that value is confirmed.
Unit forcing chains: a row, column or box has only two possible positions left for a particular digit. Assume each one in turn — if in both cases another cell takes the same value, that value is confirmed.
Visual Example — Binary Forcing Chains
Step-by-Step Application
After X-Wing, Swordfish and XY-Wing have all been exhausted. Forcing chains are powerful but time-consuming — following the chain demands concentration and notes. Short chains (three or four steps) are manageable in your head. For longer ones, paper or the digital notes mode is essential. In Sudokum.Net the N key keeps candidate notes up to date automatically, which makes chain-tracing considerably easier.
Comparison of All Four Techniques
| Technique | Structure | What it does | Difficulty |
|---|---|---|---|
| X-Wing | 2 rows × 2 columns | Eliminates from 2 columns | ★★☆☆☆ |
| Swordfish | 3 rows × 3 columns | Eliminates from 3 columns | ★★★☆☆ |
| XY-Wing | 1 pivot + 2 pincers | Eliminates candidate Z | ★★★☆☆ |
| Forcing Chains | 2 branches, shared result | Confirms the shared deduction | ★★★★☆ |
Which Technique to Use and When
Choosing a technique when you're stuck isn't random. There's a clear order to follow:
- 1Exhaust the basics first: naked single, hidden single, naked pair, pointing pairs. If they work, there's no need to reach for advanced techniques.
- 2Then X-Wing: row-by-row scan for each digit. If two rows have candidates in the same two columns — X-Wing.
- 3Then Swordfish: if no X-Wing turns up, expand to three rows. If three rows' candidates fit into three columns — Swordfish.
- 4XY-Wing: test two-candidate cells as pivots. For each pivot, look for two valid pincers.
- 5Forcing chains as a last resort: if you can see a short chain, try it. Long chains can't be traced without candidate notes.
Frequently Asked Questions
Final Thoughts
All four techniques share the same foundation: systematically narrowing down the spaces where numbers cannot go. X-Wing and Swordfish do that through the symmetry of two or three rows and columns. XY-Wing bridges three cells together. Forcing chains walk both paths and see where they lead.
None of the four involve guessing — but each has its own way of seeing. The first time you spot an X-Wing, that symmetry locks into your brain for good. In XY-Wing, the pivot-pincer relationship becomes something tangible. With forcing chains — holding two branches in mind at once — that process permanently changes the way you read a grid.