The technique ladder on this site stops at the Swordfish, and that stopping point is deliberate: singles through fish will finish the overwhelming majority of published Sudoku, including almost everything labelled “expert”. But a small population of puzzles resists all of it, and solvers who meet one deserve to know what the next country looks like rather than concluding they have hit a wall.
This guide is a map rather than a course. It covers the ideas that structure everything beyond fish, sketches the main techniques, and — more usefully — explains what they all have in common, which is a single concept that makes the whole area comprehensible at once.
The idea underneath everything: links
Every advanced technique past the subsets is built from two relationships between candidates. Learn these two and the rest is variation.
A strong link holds between two candidates when at least one of them must be true. The standard case: a digit has exactly two possible cells in some unit. One of them is the digit's home, so if one is false, the other is true. It is called a conjugate pair, and finding them is mostly a matter of counting occurrences.
A weak link holds between two candidates when at most one of them can be true. Any two candidates for the same digit sharing a unit are weakly linked, since a unit holds one copy of a digit; so are any two different digits in the same cell, since a cell holds one digit. So if one is true, the other is false.
The distinction is worth stating precisely, because it is where beginners in this area go wrong. Strong says “not both false”. Weak says “not both true”. When a link happens to be both — a digit with exactly two homes in a unit is strongly and weakly linked — it is called a bi-location link, and it is the most useful kind because it can be traversed in either direction.
Everything below is a way of stringing these links together and reading off what must follow. Notice that X-Wing already fits: it is two strong links on the same digit, in two rows, connected by their shared columns.
Simple colouring
The gentlest entry point. Pick one digit and find every unit where it has exactly two possible cells. Each such pair is a strong link. Now colour: take a candidate, mark it blue, mark its strongly linked partner green, then continue outward — everything strongly linked to a green cell becomes blue, and vice versa, until the chain runs out.
You now have a network of candidates in two colours, with a guarantee: all the blues are true and all the greens false, or the other way round. Which one you do not know and do not need to. Two conclusions follow:
- Colour appears twice in a unit. If two blue candidates share a row, column or box, blue cannot be the true colour — it would place the same digit twice in one unit. So every blue candidate in the network is false, and green is true. This one deduction can eliminate a dozen candidates at once.
- A cell sees both colours. If some uncoloured candidate shares a unit with a blue candidate and also with a green one, then whichever colour turns out true will eliminate it. So it can be removed immediately, without ever resolving the ambiguity.
Colouring is the best first step past fish because it is visual, it works on one digit at a time like X-Wing, and the second rule especially yields eliminations that no fish pattern would find.
XY-Wing
The most useful of the small chain patterns, and unlike everything above it involves three different digits. You need three cells with exactly two candidates each:
- A pivot holding candidates {A,B}.
- A wing holding {A,C}, sharing a unit with the pivot.
- Another wing holding {B,C}, also sharing a unit with the pivot.
The pivot is either A or B. If A, the first wing cannot be A, so it is C. If B, the second wing cannot be B, so it is C. Either way, one of the two wings is C. Therefore any cell that shares a unit with both wings cannot be C, and the candidate can be struck.
The elimination lands on cells that see both wings — often a single cell, occasionally several. Note that the pivot need not share a unit with anything except the two wings, and the two wings need not see each other. That is why the pattern is easy to walk past: it is three scattered bi-value cells, not a shape.
The XYZ-Wing extends this to a pivot of {A,B,C}, where the eliminated cell must see the pivot as well as both wings.
Remote pairs
A pleasingly simple one. Find a run of cells all holding the same two candidates {A,B}, chained so each shares a unit with the next. The values alternate strictly down the chain: A, B, A, B. So any two cells an even number of steps apart hold the same value, and any two an odd number apart hold opposite values.
Take two cells at opposite ends of an odd-length chain: between them they use up both A and B. Any cell seeing both ends can therefore have neither, and both candidates are eliminated from it. This is really just colouring applied to bi-value cells, which is a good illustration of how much of the deep end is one idea in different clothing.
Forcing chains
The general form, and the point at which technique names stop mattering. Pick a candidate and follow the consequences: if this cell is 4, then that cell cannot be 4, so it must be 7, so this third cell cannot be 7, and so on down the chain of strong and weak links. Two outcomes are useful:
- The chain contradicts itself. If assuming a candidate leads to a cell with no legal digits, or a unit needing two copies of one digit, the assumption is false and the candidate can be eliminated.
- Two chains converge. If a cell can only be 4 or 7, and both assumptions force some third cell to be 2, then that cell is 2 regardless — a deduction reached without ever resolving the original ambiguity.
Nice Loops, AICs (alternating inference chains) and the various Wings are all disciplined, named versions of this, with rules about how links must alternate so the reasoning stays valid. They are genuinely powerful, and there is a real question about whether they are genuinely enjoyable — which brings us to the honest part.
Where deduction shades into search
A forcing chain is legitimate logic. Every step is sound, and the conclusion is certain. But following a fifteen-step chain by hand is, experientially, close to trial and error with bookkeeping. You assume something, push it a long way, and see whether it explodes. The reasoning is valid; the feeling is search.
This is where solvers legitimately disagree about what Sudoku is. One camp holds that any chain of valid inference is fair, and depth is just skill. The other holds that a technique should be seen rather than executed, and that once you are maintaining a stack of assumptions on scrap paper, the puzzle has stopped being a puzzle and become an exercise in manual computation.
Neither position is wrong, and the disagreement explains why puzzle sources differ so much in where they draw the line. It also explains this site's choice: the technique picker stops at Swordfish because everything up to and including it can be genuinely spotted — trained into recognition, so it appears on the board rather than being computed. Techniques that require a stack are excluded on that basis rather than because they are too hard.
Worth adding: a puzzle requiring long chains is often just an indifferently constructed puzzle. Dig clues out of a random grid until it is minimal and you will sometimes produce a grid with one brutal chain-only step and nothing else of interest. That is a construction artefact, not a design achievement.
Before you go looking for any of this
A caution that applies more often than anything above. When a puzzle stalls, the probability that you need a forcing chain is far lower than the probability that you have missed something ordinary. In rough order of likelihood, a stall means:
- An un-followed cascade of singles somewhere on the board.
- An Omission, very likely running down a column you scanned less carefully than the rows.
- A Hidden Pair or triplet in a busy unit.
- An X-Wing or Swordfish on a digit you have not tested.
- Genuinely, a chain.
Solvers who have just learned about colouring reliably start finding reasons to use it on puzzles that a locked candidate would have opened in five seconds. The common mistakes guide covers this failure mode and its relatives.
If you want to pressure-test whether you actually need any of this, the hardest puzzles in the library are the ones whose peak required technique is highest, and the technique combination pages drill puzzles needing two advanced patterns at once — which is where most of the difficulty in practical solving genuinely lives, rather than in exotic chains.