Advanced Sudoku Techniques: Solve Expert Puzzles Without Guessing
Advanced Sudoku techniques are the candidate-elimination patterns that expert grids are built to require: X-Wing, Swordfish, Jellyfish, XY-Wing, XYZ-Wing, W-Wing, unique rectangles, simple colouring and forcing chains. They do not place digits directly. They remove candidates, and those removals unlock the ordinary singles that finish the puzzle. Learn them in this order, and no expert Sudoku will ever force you to guess.
This is the reference for every technique above the basics. Each one is explained by what it looks like on the grid, what it lets you erase, and when it is worth hunting for. If a grid still beats you, our free Sudoku solver handles any expert puzzle instantly.
Before you start: the prerequisites
Every technique below operates on candidates, not on empty cells. That means two conditions must already be met:
- All the basics are exhausted. Naked singles, hidden singles, cross-hatching, naked and hidden pairs, pointing pairs and box-line reduction. If any of those is still available, use it — it is always cheaper. The full ladder is in our guide to how to solve Sudoku step by step.
- Your pencil marks are complete and correct. Not partial. A single stale candidate makes every pattern below unreliable, and the contradiction it causes will appear ten placements later where you cannot trace it.
Expert solvers spend more time maintaining candidates than hunting patterns, and that is the right ratio.
X-Wing
The first genuinely advanced technique, and the one to learn first because it is the most common.
The pattern. Take one digit. Find two rows in which it has exactly two possible cells. If those four cells fall in the same two columns, you have an X-Wing.
Why it works. The four cells form a rectangle. In row one, the digit is in the left column or the right column. Whichever it is, row two must take the other, because the two rows cannot both use the same column. So the digit occupies two opposite corners either way.
What you erase. The digit, from every other cell in those two columns.
The pattern works in both orientations: start from two columns with exactly two positions each, aligned in the same two rows, and eliminate along the rows instead.
Swordfish and Jellyfish
Swordfish is X-Wing extended to three. Take a digit that appears in at most three cells in each of three rows, with all those cells confined to the same three columns. The digit must occupy one cell in each row and one in each column, so it can be erased from those three columns everywhere else.
The "at most" matters: a row with only two positions still counts, which is why Swordfish is easier to miss than X-Wing.
Jellyfish takes it to four rows and four columns. It is genuinely rare, and in practice a grid needing a Jellyfish usually yields to a chain first.
XY-Wing
The most useful of the wing patterns, and the first that works on cells rather than on a single digit.
The pattern. Three cells with exactly two candidates each. A pivot holding {X,Y}. Two pincers, one holding {X,Z} and one holding {Y,Z}, each sharing a unit with the pivot — though not necessarily with each other.
Why it works. The pivot is either X or Y. If it is X, the first pincer cannot be X, so it is Z. If it is Y, the second pincer cannot be Y, so it is Z. Either way, one of the two pincers is Z.
What you erase. Z, from every cell that sees both pincers.
XYZ-Wing
A variant where the pivot holds three candidates {X,Y,Z} and the pincers hold {X,Z} and {Y,Z}. Now one of the three cells must be Z, so Z is eliminated from every cell that sees all three. Rarer, and the eliminations are narrower.
W-Wing
Two cells with identical candidates {X,Y} that do not see each other, linked by a strong link on X — a unit where X has only two possible cells, one seeing each of them. Then Y can be erased from every cell seeing both.
Unique rectangle
This technique uses a fact about the puzzle rather than about the grid: a valid Sudoku has exactly one solution.
The pattern. Four cells forming a rectangle, sitting in exactly two rows, two columns and two boxes, all four holding the same two candidates {X,Y}.
Why it works. If that configuration were completed, X and Y could be swapped diagonally and the grid would still be valid — giving two solutions. Since the puzzle has only one, the configuration is impossible.
What you erase. Depending on the variant, one of the corners must take a third candidate, letting you remove X and Y from it.
Simple colouring
When a digit has exactly two possible cells in a unit, those two form a conjugate pair: exactly one is true. Colouring follows the consequences.
- Pick a digit and find all its conjugate pairs.
- Start from any cell, colour it blue, and colour its conjugate partner green.
- Follow the links outward, alternating colours across the chain.
- Rule one: if two cells of the same colour share a unit, that colour is entirely false — erase the digit from every cell of that colour.
- Rule two: any cell that sees both a blue and a green cell cannot hold the digit, since one of the two is true.
Colouring is where most solvers stop, and it is also where most expert grids give way.
Forcing chains
The general form of every pattern above. Pick a cell with two candidates, assume the first, and follow the forced consequences as far as they go. Then do the same with the second.
- If both assumptions force the same digit into the same cell, that placement is certain regardless.
- If one assumption leads to a contradiction, the other candidate is the answer.
Forcing chains will solve anything, but they are slow and error-prone on paper. Treat them as the last resort, after colouring has failed.
Remote pairs and the BUG technique
Remote pairs
A chain of cells that all hold the same two candidates, each linked to the next by a shared unit. Because the values alternate along the chain, any cell that sees both ends of an even-length chain cannot hold either digit. Remote pairs are easy to miss because the cells look unremarkable individually — you have to notice the repetition of the same pair.
BUG+1 (Bivalue Universal Grave)
If every unsolved cell in the grid holds exactly two candidates except one cell holding three, that third candidate in the odd cell must be the answer. A grid where all cells hold two candidates would have multiple solutions, which a valid puzzle cannot have. BUG+1 appears near the end of expert grids and resolves them instantly.
Skyscraper and Two-String Kite
Both are simplified single-digit chains. A skyscraper is two rows where a digit has exactly two positions, sharing one column; the two off-column cells then eliminate the digit from any cell seeing both. A two-string kite is the same idea using one row and one column linked through a box. They catch many eliminations that a full X-Wing search misses.
How to know which technique a grid needs
Rather than trying patterns at random, read the shape of the candidate grid. It tells you what is available.
| What you see | What to try |
|---|---|
| Many cells with exactly two candidates | Naked pairs, XY-Wing, remote pairs, BUG+1 |
| One digit with two positions in several rows | X-Wing, skyscraper, simple colouring |
| A digit confined to one line inside a box | Pointing pair |
| A digit confined to one box inside a line | Box-line reduction |
| Four cells sharing two rows and two columns | Unique rectangle |
| Nothing obvious anywhere | Rebuild candidates — you have an error |
Practising advanced techniques deliberately
- One technique per week. Spend seven days hunting only X-Wings, even on grids that do not need one. Scattered exposure to eight techniques teaches none of them.
- Mark the pattern when you find it. Circle the four corners of an X-Wing, or the three cells of an XY-Wing, before you erase anything. Seeing the shape is what makes it automatic.
- Re-solve the same expert grid a week later. You will spot in seconds what took twenty minutes the first time. That is the pattern becoming perceptual rather than analytical.
- Use the solver as a diagnostic. When you are stuck, solve the grid, find the first cell you could not place, then work out which pattern would have revealed it. That single question teaches more than ten completed puzzles.
- Solve on paper. Apps that auto-fill and auto-eliminate candidates remove the exact skill expert grids test.
What these techniques cost you
Every technique on this page trades time for certainty, and the trade is not always worth making.
A full X-Wing search means checking nine digits across nine rows and nine columns — a minute or two of careful reading. On a grid that actually needed a hidden single, that minute is wasted. This is why the order matters more than the techniques themselves: always exhaust the cheaper step before paying for the expensive one.
Experienced solvers are not faster because they know more patterns. They are faster because they know which pattern the grid is likely to contain, and they look for that one first. Reading the candidate grid, as in the table above, is what produces that judgement.
Notation you will meet elsewhere
- r4c7 — row 4, column 7. The standard way to name a cell.
- b3 — box 3, numbering boxes left to right, top to bottom.
- Strong link — a unit where a digit has exactly two possible cells: one of them is true.
- Weak link — two cells that see each other: at most one of them is true.
- Conjugate pair — another name for a strong link on a single digit.
- Fish — the family containing X-Wing (2), Swordfish (3) and Jellyfish (4).
- Elimination — removing a candidate, as opposed to placing a digit. Advanced techniques produce eliminations, not placements.
Knowing this vocabulary makes every Sudoku forum, solver log and technique guide readable, which is the fastest way to keep improving once the named patterns above are automatic.
Which technique to learn in which order
| Order | Technique | How often it appears | Difficulty to spot |
|---|---|---|---|
| 1 | X-Wing | Common on expert grids | Moderate |
| 2 | Simple colouring | Common | Moderate |
| 3 | XY-Wing | Fairly common | Harder |
| 4 | Swordfish | Occasional | Hard |
| 5 | Unique rectangle | Occasional | Easy once known |
| 6 | XYZ-Wing / W-Wing | Uncommon | Hard |
| 7 | Forcing chains | Always available | Slow, error-prone |
| 8 | Jellyfish | Rare | Very hard |
How to actually hunt for these patterns
- Scan by digit, not by cell. X-Wing, Swordfish and colouring all concern a single digit across the whole grid. Take 1 to 9 in turn.
- Count positions per unit. For each digit, note the rows and columns where it has exactly two possible cells. Those are the seeds of every fish and every colouring chain.
- List the bi-value cells. Cells with exactly two candidates are where XY-Wings, W-Wings and unique rectangles live. Circle them.
- Look for rectangles. Any four cells sharing two rows and two columns are worth a second look, especially if they carry the same pair.
- Colour before you chain. Colouring is faster than forcing chains and catches most of the same eliminations.
Never guess
Every technique on this page exists so that you never have to. A guess on an expert grid does not fail immediately — it fails ten placements later, in a different region, with no way to trace which move was wrong. The usual result is starting over.
If you cannot prove a placement, you have not found the elimination yet. That is not a reason to guess; it is a reason to scan by digit again.
Frequently asked questions
What are advanced Sudoku techniques?
Candidate-elimination patterns used when basic scanning stops working: X-Wing, Swordfish, Jellyfish, XY-Wing, XYZ-Wing, W-Wing, unique rectangles, simple colouring and forcing chains. They remove candidates rather than placing digits directly.
What is an X-Wing in Sudoku?
A digit with exactly two possible cells in each of two rows, with all four cells in the same two columns. The digit occupies opposite corners of that rectangle, so it can be eliminated from those two columns everywhere else.
What is a Swordfish?
X-Wing extended to three rows and three columns. A digit confined to at most three cells in each of three rows, all inside the same three columns, can be erased from those columns elsewhere.
What is an XY-Wing?
Three two-candidate cells: a pivot holding X and Y, and two pincers holding X,Z and Y,Z, each sharing a unit with the pivot. One pincer must be Z, so Z is eliminated from every cell that sees both pincers.
What is the difference between XY-Wing and XYZ-Wing?
In an XY-Wing the pivot has two candidates and the eliminations apply to cells seeing both pincers. In an XYZ-Wing the pivot has three candidates including Z, so Z can only be erased from cells that see all three of the pattern's cells.
What is simple colouring?
Following chains of conjugate pairs for one digit, alternating two colours. If two cells of the same colour share a unit, that colour is false. Any cell seeing both colours cannot hold the digit.
What is a unique rectangle?
Four cells in two rows, two columns and two boxes all holding the same two candidates. That configuration would allow two solutions, which a valid Sudoku cannot have, so one corner must take a different candidate.
Which advanced technique should I learn first?
X-Wing, then simple colouring, then XY-Wing. Those three cover the large majority of expert grids. Swordfish, Jellyfish and the rarer wings can wait until the first three are automatic.
Do I need pencil marks to use these techniques?
Yes, complete and accurate ones. Every pattern here is defined by candidate positions, so a technique you have not written down is a technique you cannot see. Partial marks hide exactly the patterns you are hunting.
Is guessing ever required on an expert Sudoku?
No. Every valid Sudoku is solvable by logic alone. Forcing chains will resolve anything the named patterns miss, though they are slower and easier to get wrong on paper.
Why do these techniques not place a digit directly?
Because they remove candidates rather than confirm them. The value comes from what the removal exposes: once candidates are erased, ordinary naked and hidden singles reappear and carry the grid forward.
What is a forcing chain?
You assume one candidate in a two-candidate cell and follow the forced consequences, then repeat with the other. If both paths force the same digit into the same cell, it is certain. If one path contradicts itself, the other candidate is correct.
Are advanced techniques needed on newspaper Sudoku?
Rarely. Most published grids, including those labelled hard, are solvable with pointing pairs and box-line reduction. Advanced patterns become necessary at the expert and diabolical ratings.
How do I check whether I applied a technique correctly?
Solve the grid with a Sudoku solver and compare it to your candidates. If a digit you eliminated appears in the solution, the pattern was misapplied or a pencil mark was stale.
Put them to work
Take an expert grid, rebuild your candidates carefully, then scan digit by digit for two-position rows. That single habit finds X-Wings, Swordfish and colouring chains all at once. And when a grid still will not yield, our free Sudoku solver shows you the answer in milliseconds.
Next, read: How to Solve Hard Sudoku · How to Solve Sudoku — The Complete Guide · Free Sudoku Solver
