An X-Cycle is a loop built for a single digit X using Strong and Weak Links between that digit's candidates.
It is one of the clearest bridges between familiar single-digit patterns and general chain logic. X-Wings, some Swordfish shapes, Skyscrapers and other patterns can all be viewed through the same link language.
The defining feature is not a rectangle or fish shape. It is the loop of implications.
The building blocks
For one digit X:
- a Strong Link means if candidate A is false, candidate B must be true;
- a Weak Link means if candidate A is true, candidate B must be false.
A standard X-Cycle walks through candidates for X and alternates these link types around a loop.
Continuous X-Cycles
A continuous X-Cycle alternates Strong/Weak perfectly around the entire loop, including the link from the final node back to the first.
That means the loop can be assigned alternating truth states consistently.
The main deduction comes from Weak Links in the loop: if two loop candidates are connected by a Weak Link through a house, any other X candidate in that same house sees both possible alternating states and can be eliminated.
In a simple four-node case, this is effectively an X-Wing written as a loop.
Why an X-Wing can be an X-Cycle
Take the four corners of a valid X-Wing for digit 7.
Each base row supplies a Strong relationship between its two 7 candidates. Each cover column supplies the Weak relationship that prevents both corners from being true.
Walk around the rectangle and the links alternate.
The named X-Wing remains easier to recognize, but the X-Cycle representation reveals the general mechanism.
Discontinuous X-Cycles
A useful loop may contain exactly one place where alternation breaks.
Two important cases are:
Two Strong Links meet
If the same candidate node is reached by two Strong inferences, that candidate must be true.
The discontinuity produces a placement.
Two Weak Links meet
If two Weak inferences meet at one candidate, that candidate cannot be true.
The discontinuity produces an elimination.
These rules are the single-digit form of discontinuous Nice Loop reasoning.
Worked X-Cycle
X-Cycles vs X-Chains
An X-Chain is generally an open chain for one digit. It begins and ends in a way that proves the digit is true at one endpoint or the other, eliminating from common peers.
An X-Cycle closes into a loop. That allows continuous-loop eliminations and discontinuity rules.
The underlying Strong/Weak language is the same.
X-Cycles vs Simple Coloring
Both techniques work on one digit and exploit conjugate relationships.
Coloring builds a network and assigns two colors. X-Cycles trace a specific loop with explicit Strong/Weak links.
A continuous X-Cycle made entirely from conjugate relationships can look almost identical to a coloring proof.
X-Cycles vs AIC
X-Cycles stay on one digit.
AIC allows the chain to change digits through cells and other Strong/Weak relationships.
Conceptually:
X-Cycle = single-digit loop language. AIC = general candidate inference language.
Learning X-Cycles first makes AIC notation much easier to understand.
Short named patterns inside the same language
Skyscraper, Two-String Kite and Turbot Fish can often be represented as short single-digit Chains/Cycles.
Keep the named patterns because they are visually efficient. Use X-Cycle logic when the geometry stops matching a familiar template.
How to search for X-Cycles
- choose a digit with several conjugate pairs;
- mark Strong Links;
- from each Strong endpoint, inspect Weak visibility to another Strong pair;
- continue alternating without reusing a candidate;
- watch for a return to the starting region;
- classify the closure before eliminating anything.
Long loops are harder to find manually, so prioritize short cycles and high-connectivity candidates.
Common mistakes
Mixing digits
That becomes AIC-style reasoning, not an X-Cycle.
Treating every peer relationship as a Strong Link
Peers are Weak by default. Strong requires the relevant exactly-two/forcing condition.
Ignoring the closure link
A path is not a cycle unless the end reconnects logically to the start.
Applying discontinuity rules to multiple breaks
The simple placement/elimination rules assume the loop has the required single discontinuity structure.
Reusing a candidate node
A valid chain representation should not loop through the same candidate repeatedly to manufacture a result.
FAQ
What is the X in X-Cycle?
It represents one fixed Sudoku digit. Every candidate node in the loop is an occurrence of that same digit.
Is X-Cycle the same as X-Wing?
No. An X-Wing is one recognizable pattern; a four-node X-Wing can also be represented as a continuous X-Cycle.
Are X-Cycles a type of Chain?
Yes. They are single-digit chain/loop structures using Strong and Weak Links.
Do X-Cycles require pencil marks?
In practice, yes. Accurate candidate positions are necessary to verify Strong and Weak relationships.
What to learn next
If X-Cycles make sense, continue to AIC and Grouped AIC. They preserve the same inference language while allowing different digits and larger group nodes.