An X-chain with cell groups is an X-chain in which a node can be two or three cells that share a box and a row or column. The group stands for "the digit is in one of these cells" and links like a single cell. That carries the chain through a box with three places for the digit.
How to spot it
Start as for an X-chain, with one digit and links that alternate strong and weak, strong at each end. Often the chain reaches a box with three places for the digit, and a plain strong link needs exactly two.
Look at how the three places fall. If two of them share a row or a column and the third sits outside it, treat those two as one node, a cell group. The digit is in one of its cells or in neither. The box then has two sides, the group and the third cell, and they form a strong link.
A weak link into a group needs a cell that sees every cell of it. Any cell holding the digit that sees both ends of the chain loses it.
Open with column three
Take the 7. Column three has only two places for it, row one and row eight. Those two cells form a strong link, and one of them is the 7.
Step into a group across a weak link
Row eight has four places for the 7, in columns two, three, seven and nine. Row eight, column three sees both row eight, column seven and row eight, column nine, so if it holds the 7, neither of them can. A weak link joins it to the two of them taken together, drawn as one outlined group.
Leave the group on a strong link
Box nine has three places for the 7, so no two single cells there form a strong link. But all three sit in two rows. If the 7 of box nine is not in the group in row eight, it is in row seven, column nine. A weak link then goes up column nine to row six, column nine, and row six, with two places, ends the chain on a strong link at row six, column one.
Follow the 7 from the top
Suppose row one, column three is not the 7. Then column three puts the 7 in row eight, column three. The weak link in row eight rules out both cells of the group. Box nine puts the 7 in row seven, column nine. Column nine rules out row six, column nine. Row six puts the 7 in row six, column one. So if one end is not the 7, the other end is.
Strike the 7 that sees both ends
Row one, column one shares row one with one end and column one with the other. Whichever end holds the 7, it is in a house with row one, column one, so the 7 comes out of that cell. Row one now has one place left for the 7, row one, column three. That is a hidden single, and the grid had no single anywhere before the strike.
Why a group links like one cell
Box nine's three 7s sit in rows seven and eight. If neither 7 in row eight is the real one, the 7 of box nine must be in row seven. A strong link needs nothing more, and it holds even though one side is two cells.
The weak link into the group works because row eight, column three shares row eight with both of its cells. A 7 there leaves neither of them a 7.
So a group has to sit inside one box and along one row or column. A cell in that row then sees all of it, and the box splits cleanly around it.
The same chain without the group
Try the chain with single cells. The weak link along row eight can reach row eight, column nine on its own. The next link, to row seven, column nine, is not strong. Box nine has a third place for the 7 in row eight, column seven, and column nine has five. If row eight, column nine is not the 7, nothing puts it in row seven, column nine.
The solution shows the gap. The 7 of box nine is in row eight, column seven, so neither of those two cells holds it. Row eight, column seven alone fails the same way, and no plain X-chain exists on this grid for any digit.
The group and row seven, column nine hold all three places. If the group lacks the 7, the single cell has it.
Row eight, column nine and row seven, column nine cannot both be the 7. Neither has to be, because row eight, column seven is left.
What counts as a group
A group is every place for the digit where one box meets one row or one column, so two or three cells. Row eight meets box nine in three cells. Row eight, column eight is already a 4, so the group here is the other two. Had that cell still held a 7, it would belong to the group too.
A group can also be an end of the chain. Then a cell must see every cell of the group to lose the digit. The quiz below ends on this same group.
A group that crosses a box
Row eight has four places for the 7. If row eight, column two is not the 7, one of the other three is, so that much is a valid strong link. The trouble starts when those three, in columns three, seven and nine, are treated as a group.
From them a weak link seems to reach row seven, columns one and two, a proper group in box seven. A strong link along row seven goes to row seven, column nine, and the chain runs on through column nine and row six to row six, column one. Row four, column two sees both ends. Strike its 7 and you remove the real one, because in the solution row four, column two is the 7.
The weak link is the break. Row seven, columns one and two see row eight, column three, which shares their box, but not columns seven or nine of row eight. Keep every group inside one box, and check that a weak link sees each of its cells.
When an X-chain stalls at a box with three places for the digit, see whether two of them share a row or column that the third avoids. If they do, take them as one group, with the third cell as its strong partner. A weak link into the group must see every cell of it. Strike the digit from any cell that sees both ends.
Where this sits on the path
An X-chain with cell groups places nothing. It removes candidates and leaves the placing to the singles.
It is the third step of the chapter on chains and coloring, right after the plain X-chain. On this grid nothing earlier on the path removes a candidate, the plain X-chain included.
Short AICs come next.
This chain runs from row one, column three to the group in box nine. Which 7 does it remove?
Locate the ends first. The cell you want holds a 7 and sees the single cell and both cells of the group.
Итог
- Ищите
- Strong links on one digit that stall at a box with three places, where two of the three share a row or column and the third does not.
- Область
- One digit. The chain's cells and groups, and any other cell holding that digit that sees both ends.
- Результат
- The digit comes out of every cell that sees every cell of both ends.
- Признак
- An X-chain that needs one more strong link and finds a box with three places for the digit.
- Если не сработало
- Check each group. It sits in one box and one row or column and holds every place for the digit where they meet. A weak link into it must see all of its cells.