An XY-cycle is an XY-chain that closes into a loop. Every cell in it has exactly two candidates, and each cell shares a digit with the next one round the ring. Each cell holds one of its two, so every link between cells has one end that holds its digit, and any other cell that sees both ends loses it.
How to spot it
Look first at the cells with exactly two candidates. This grid has eighteen.
Build an XY-chain through them. From each cell, move to a cell it sees that shares one of its digits, and leave that cell on its other digit. Keep going until you reach a cell that sees the first one and leaves on the first cell's other digit. That last link closes the ring.
Then the ring works as a whole. It doesn't matter where you start or which way you go round. Every dashed link has one end that holds its digit, so every dashed link can strike.
Six cells with two candidates
Row three, column three has a 2 and an 8. It shares the 2 with row three, column five, which has a 2 and a 5. From there the ring runs down column five on the 5, along row seven on the 9, up column six on the 5 and along row six on the 1. That reaches row six, column three, with a 1 and an 8, and column three takes its 8 back to the start.
Suppose row three, column three is the 2
Then row three, column five can't be the 2, since the two cells share row three. Its other candidate is the 5, so it is the 5.
Carry it round the ring
Column five takes the 5 away from row seven, column five, so that cell is the 9. Row seven then makes row seven, column six the 5. Column six makes row six, column six the 1, and row six makes row six, column three the 8. That 8 rules out the 8 at the start, which fits, since the start is the 2.
Now suppose it is the 8
Go round the other way. Column three takes the 8 from row six, column three, so it is the 1. Then row six, column six is the 5, row seven, column six the 9, row seven, column five the 5 and row three, column five the 2. That 2 rules out the 2 at the start, which fits again.
One end of each dashed link holds its digit
Compare the two walks. On the first, row three, column three holds the 2 of the link along row three. On the second, row three, column five does. Row three, column six sees both of those cells, so it can't be the 2.
Remove five candidates
Do the same at every dashed link. Row three, column six also sees both cells of the link in column six, so it loses its 5 as well, and so does row four, column six. Row four, column five loses its 5 to the link in column five, and row four, column three loses its 8 to the link in column three. The links along rows six and seven strike nothing, since no other cell in row six holds a 1 and no other cell in row seven or box eight holds a 9.
Why every dashed link strikes
The ring can be filled in only two ways, the two walks above. A solid line says one of a cell's two candidates is true. A dashed line says its two cells can't both hold the digit they share. The choice at the start fixes every cell, and the ring comes back to the start's other digit, so neither walk runs into a clash.
The two walks disagree on every cell. They agree on one thing: each dashed link has its digit at exactly one end. That is all a strike needs.
In the solution the second walk is the true one. Row three, column three is the 8, and row three, column six is a 3.
How it differs from an XY-chain
An XY-chain is open. Its two ends carry the same digit, and only a cell that sees both ends loses it.
Drop the dashed link in column three. The six cells are then an XY-chain from row three, column three to row six, column three, with an 8 at both ends. It removes the 8 from row four, column three and nothing else. Close the ring and that strike stays, and every other dashed link adds its own. One ring does the work of several chains.
How it differs from an X-cycle
An X-cycle stays on one digit. Its strong links run between cells, where a house has only two places for that digit.
An XY-cycle moves the strong links inside the cells, where a cell has only two candidates. Every link between cells is weak, and its digit changes as you go round: 2, 5, 9, 5, 1 and 8 here. The rule at the end is the one for an X-cycle whose links alternate all the way round. Each weak link has one end that holds its digit, and a cell that sees both ends loses it.
Six cells, six dashed links. Four of the links strike, for five candidates in all.
Without the link in column three the six cells are an XY-chain with an 8 at each end. It strikes only the 8 in row four, column three.
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 3s. Box one has its 3s only in row one, so row one, columns four and six lose their 3s. Two naked pairs follow.
Next the hint shows two XY-chains. The second removes the 2 from row three, column three, one of this ring's six cells. Singles then remove all five candidates the ring strikes.
Each of the five is also struck by an XY-chain on the starting grid: cut the ring at the dashed link that strikes it. The game checks XY-chains before XY-cycles, so the hint seldom shows a cycle. A different route can still be right.
Look for cells with two candidates that join into a ring, each sharing a digit with the next. The ring closes when the last cell sees the first and shares the first cell's other digit. Then strike each shared digit from every cell that sees both cells of its link.
Where this sits on the path
With the five candidates gone, no single follows yet.
XY-cycles come after X-cycles and X-cycles with cell groups in the chapter on chains and coloring. On this grid the game's own order starts with a pointing pair, then two naked pairs, then XY-chains, and the second XY-chain breaks this ring before the game reaches cycles.
XY-cycles with cell groups come next.
On a different grid, this loop joins four cells that each have two candidates. Which of these candidates does it remove?
Find a cell outside the loop that sees both cells of one dashed link and holds that link's digit.
Итог
- Ищите
- Cells with exactly two candidates that join into a ring, each seeing the next and sharing a digit with it.
- Область
- The cells of the ring, and any cell that sees both cells of one of its dashed links.
- Результат
- Each dashed link's digit leaves every other cell that sees both of its cells. One ring can strike in several places at once.
- Признак
- An XY-chain whose last cell sees its first and shares the first cell's other digit.
- Если не сработало
- Check that every cell in the ring has exactly two candidates, and that the last link carries the start's other digit.