X-Cycles are the passport to grids of level 9 and beyond. When the X-Wing, the W-Wing and the SwordFish stop unlocking anything, these take over.
The principle fits in one sentence: you follow a single digit around a closed chain that alternates strong links and weak links. The technique has three variants, called rules 1, 2 and 3. This page covers rule 1, the one where the chain alternates perfectly all the way round.
This is a hard technique, as hard to grasp as it is to spot in a grid. Do not expect to master it in one reading. It does, however, rest on a single idea, and that idea takes five minutes to explain.
🎬 The Domino33 video on rule 1
Theory and practice on video, with worked examples. The text below covers the same ground, at your own pace.
▶ X-Cycles, rule 1: theory and practice through examples.
Over 120 videos on Sudoku techniques, from beginner to expert. English audio track and subtitles.
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Strong links and weak links
Everything rests on this distinction. Take it seriously: if it stays vague, nothing that follows will work.
We reason about one digit only, say the 5, and about where it can go. Within a given unit, row, column or box, two cells can be related in two ways.
| Type of link | Condition | What it guarantees |
|---|---|---|
| Strong link | The digit has only these two spots in the unit | One of the two cells is true. If one is false, the other is true. |
| Weak link | The digit has other spots in the unit | At most one of the two is true. If one is true, the other is false. |
Hold on to the asymmetry, it drives the whole mechanism. A strong link guarantees that one of the two cells is true. A weak link only guarantees that they cannot both be true, and it may be that neither is.
A strong link is also a weak link, but not the other way round. A strong link can therefore always be used where a chain calls for a weak one. That is what makes loops easier to find than they look.
Part 2
Rule 1, the continuous loop
Build a chain on that single digit: strong link, weak link, strong link, weak link, and so on. If you manage to close the chain back onto its starting cell while keeping the alternation right through the last link, you have a continuous loop. That is rule 1.
Such a loop necessarily has an even number of cells, as many strong links as weak ones.
A continuous six-cell loop: three strong links, three weak links, and the return to the starting point keeps the alternation.
What the loop gives you
When the alternation is perfect, every weak link in the loop behaves like a strong one. In other words, for each of them, one of its two cells is necessarily true.
The consequence follows at once: in the unit of every weak link, all the other cells lose that digit. They cannot hold it, since one of the two loop cells will.
Continuous loop: for each weak link, eliminate the digit from every other cell of the unit carrying that link.
Note carefully where the eliminations land. They never touch the loop cells, which stay untouched, but the cells around them in the units of the weak links.
Part 3
You already know an X-Cycle
Here is the good news. The X-Wing you already use is an X-Cycle, the shortest one there is.
An X-Wing is four cells at the corners of a rectangle. The two rows are strong links, since the digit has only two spots in each. The two columns are weak links. Strong, weak, strong, weak: the alternation is perfect, the loop is continuous.
The 5 has only two spots in R2 and in R6, on columns 2 and 8. Strong links in the rows, weak links in the columns.
Apply rule 1: the two weak links are columns 2 and 8, so the 5 is eliminated from the rest of those two columns. That is exactly the X-Wing elimination, reached by another route.
Remember this equivalence, it beats an abstract definition: an X-Cycle is an X-Wing allowed to be longer and not to form a rectangle.
Part 4
A six-cell loop
Now for what the X-Wing cannot do. Still on the 5, with six cells this time.
Three strong links in rows 2, 5 and 8. Three weak links in columns 8, 4 and 2.
Follow the loop round:
- R2C2 → R2C8, strong link: in row 2, the 5 has only these two spots.
- R2C8 → R5C8, weak link, in column 8.
- R5C8 → R5C4, strong link: in row 5, the 5 has only these two spots.
- R5C4 → R8C4, weak link, in column 4.
- R8C4 → R8C2, strong link: in row 8, the 5 has only these two spots.
- R8C2 → R2C2, weak link, in column 2. The loop is closed.
Six links, three strong and three weak, perfectly alternating right back to the start. This is a continuous loop, rule 1 applies.
The three weak links sit in columns 8, 4 and 2. The 5 is therefore eliminated from every other cell of those three columns, here R7C8, R1C4 and R6C2, and from any other cell of those columns where it is a candidate.
Suppose the 5 sits in R6C2. It then drives the 5 out of R2C2 and R8C2. The strong link in row 2 therefore forces the 5 into R2C8, and the one in row 8 forces it into R8C4. Row 5 is now stuck: both of its spots, R5C8 and R5C4, are ruled out by their columns. Row 5 can no longer place its 5, which is impossible. So the 5 is not in R6C2.
Part 5
How to find one
This is the real obstacle. Understanding rule 1 takes a quarter of an hour, spotting it in a grid takes a method.
1. Pick the right digit
Do not search at random. Take the digit with the most strong links, that is, the most rows, columns and boxes where it has only two spots. A digit with five or six strong links is an excellent candidate, one with two will lead nowhere.
2. Map the strong links
On a blank grid, mark only that digit's spots and join those that form strong links. You get a network far easier to read than the full grid. That is where loops appear.
3. Start from a strong link and try to close
Pick a strong link, leave one end by a weak link, look for a strong link at the far end, and carry on. At each step, one question only: can I get back to my starting cell keeping the alternation?
4. Check before eliminating
Before any elimination, walk the whole loop again naming each link out loud. One slip in the alternation and you do not have rule 1 but a discontinuous chain, which falls under rules 2 or 3, with different conclusions.
Always look for an X-Wing first. It is the shortest loop, and if one exists it spares you exploring six or eight-cell chains for nothing.
🧩 Interactive exercise
Your turn. The digit followed is the 3. In rows 1 and 4 it has only two spots: columns 3 and 7. So you have a continuous four-cell loop.
Part 6
Quiz
Four questions to check what you have taken in. Good luck!
❓ Frequently asked questions
An X-Cycle is a closed chain alternating strong and weak links, built on one single digit. Rule 1 covers the case where the alternation is perfect all the way round the loop, which is called a continuous loop. It then lets you eliminate that digit in the unit of each weak link, outside the two loop cells.
A strong link joins two cells of one unit where the digit has only those two possible spots: one of the two is necessarily true. A weak link joins two cells of one unit where the digit has other spots: at most one of the two is true, and it may be that neither is.
Yes, the simplest of them. An X-Wing is a four-cell loop whose two rows are strong links and whose two columns are weak links. Rule 1 applied to that loop gives exactly the X-Wing eliminations.
Just one. That is what separates X-Cycles from chains such as the XY-Chain, which involve several digits. The whole loop is read off the map of a single candidate's spots.
From level 9 onwards. Below that, the classic advanced techniques, X-Wing, Y-Wing, W-Wing, XYZ-Wing, SwordFish and Sashimi X-Wing, almost always suffice. X-Cycles are for the blockages that resist all the others.
📋 In short
The essentials of rule 1
- One digit only, followed from end to end of the chain
- Strong link: only two spots, one of the two is true
- Weak link: other spots exist, at most one of the two is true
- Rule 1: the loop closes keeping the alternation
- Action: eliminate the digit in the unit of each weak link, outside the loop
- Landmark: the X-Wing is the shortest X-Cycle
X-Cycles extend the advanced techniques. They assume the X-Wing and the SwordFish are in place, along with rigorous candidate notation. The vocabulary used here is defined in the glossary.
📘 Go further with the Domino33 books
X-Cycles are among the sophisticated techniques detailed in the memento, alongside Unique Rectangles and 3D-Medusa.
