An X-cycle is an X-chain that closes into a loop on one digit. If its strong and weak links alternate all the way round, each weak link has one end that holds the digit, so a cell that sees both cells of a weak link loses it. If two weak links meet at one cell, that cell loses the digit.
How to spot it
Start as for an X-chain. Pick a digit. A house with only two places for it gives a strong link, and two cells that hold it and see each other give a weak link.
Then look for a path of those links that comes back to its first cell. Two kinds of loop give a result.
In a continuous loop the links alternate strong and weak all the way round. Each weak link then has exactly one end that holds the digit, so any other cell that sees both cells of a weak link loses it.
In a discontinuous loop the alternation breaks at one cell, where two weak links meet. That cell can't hold the digit. The loop on this grid is of this kind.
Suppose row eight, column one is the 7
Row eight, column one has a 3, a 6 and a 7. Suppose it is the 7. Row eight has four places for the 7, so the link along the row is weak. Row eight, column four shares that row, so it is not the 7.
Column four puts the 7 in row six
Column four has only two places for the 7, rows six and eight. Row eight is ruled out, so row six, column four is the 7.
Row six rules out column two
Row six has three places for the 7, in columns two, four and six. Column four now holds it, so row six, column two does not.
Column two puts the 7 in row nine
Column two has two places for the 7, rows six and nine. Row six is out, so row nine, column two is the 7.
Back at the start
Row nine, column two shares box seven with row eight, column one, so they can't both be the 7. Yet the walk made both of them 7s. The loop has closed through a second weak link and contradicts itself, so the assumption was wrong.
Remove the 7 from row eight, column one
Row eight, column one is not the 7. It is the cell where two weak links meet, one along row eight and one in box seven, and the loop breaks its alternation there. The cell keeps its 3 and its 6.
Why the meeting cell loses the digit
Walk the loop the other way and it ends the same, with a 7 at row eight, column four, in the start's own row.
A weak link only tells you something when one of its cells holds the digit. At the start both links are weak, so the assumption sends a "not the 7" out in both directions, and each one comes back as a 7 that sees the start. Assume the start is not the 7 and nothing follows. That is why this kind of loop only ever removes the digit from one cell.
In the solution row eight, column one is a 3. The 7s of the loop are at row six, column four and row nine, column two.
The continuous case
When the links alternate strong and weak all the way round, the loop has an even number of cells and nowhere for two links of one kind to meet. Then exactly every second cell of the loop holds the digit, one way round or the other. Either way, each weak link has exactly one end that holds it.
So a cell outside the loop that sees both cells of a weak link loses the digit, and the loop does that at every weak link at once. An X-wing is a continuous loop of four cells: two strong links in two rows and two weak links in two columns.
No loop on this grid alternates all the way round.
How it differs from an X-chain
Leave row eight, column one out of the loop. The other four cells, from row eight, column four through row six to row nine, column two, form a chain that starts and ends on a strong link. That is a short AIC, the shortest X-chain. One of its ends holds the 7, and row eight, column one sees both ends, so its 7 goes. The result is the same.
Every strike an X-cycle makes can also be read off a shorter chain inside the loop. In a continuous loop, cut it at the weak link the cell sees. In a discontinuous loop, leave out the cell where the weak links meet. What the loop changes is where you look. An X-chain asks which cells see both of its ends. A loop asks where its weak links are.
Five cells, five links. Strong and weak alternate everywhere except at row eight, column one, where row eight and box seven both give weak links.
Leave out the start and the rest is a short AIC. Row eight, column one sees both its ends, and so does row nine, column five.
What the in-game hint shows
The in-game hint may show a different or simpler step first. On this grid it opens with a pointing pair on the 2s. Box eight has its 2s only in row seven, so row seven, column one loses its 2.
Next the hint shows an X-chain on the 7s that removes the 7 from row nine, column five. Row nine then has its 7s only in box seven, and a claiming group removes the 7 from row eight, column one, the same 7 this loop removes.
The game checks chains before cycles, and a chain or a simpler step reaches whatever an X-cycle strikes, so the hint seldom shows a cycle. A different route can still be right.
Follow one digit round a closed path of strong and weak links. If they alternate all the way round, strike the digit from any cell that sees both cells of a weak link. If two weak links meet at one cell, strike it from that cell.
Where this sits on the path
With the 7 gone, row eight, column one is down to a 3 and a 6, and no single follows yet.
X-cycles follow 3D Medusa in the chapter on chains and coloring. On this grid the game's own order starts with a pointing pair, then an X-chain, then a claiming group, and that claiming group removes the same 7 as this loop.
X-cycles with cell groups come next, then XY-cycles.
On a different grid, this loop joins seven cells that hold a 3. Which 3 does it remove?
Find the cell where two dashed links meet.
Recap
- Look for
- A path on one digit, of strong and weak links in turn, that comes back to its own first cell.
- Scope
- One digit. The cells of the loop, and any cell that sees both cells of one of its weak links.
- Result
- If the links alternate all the way round, the digit leaves every cell that sees both cells of a weak link. If two weak links meet at one cell, the digit leaves that cell.
- Tell-tale
- Strong links on one digit that join up into a ring, with weak links between them.
- If it fails
- Check that every solid link sits in a house with exactly two places for the digit, and that the loop closes on its first cell.