Two cells with the same two candidates that do not see each other, plus a digit with exactly two places in some house, one place seeing each cell. One of those places holds the digit, so at least one of the two cells is left with its other candidate. Any cell that sees both cells loses that candidate.
How to spot it
The Y-wing needed a pivot in the middle. A W-wing has none. Its two cells never see each other, and something else has to tie them together.
Look first for two cells with the same two candidates, say A and B, that share no row, column or box. Then pick one of the two digits, A, and look for a house where A has exactly two places. One place must see the first cell and the other place the second. Two places for one digit in a house are called a strong link, because one of the two must hold it. It is the same pattern the X-wing is built from.
When you have both, any cell that sees both of your cells loses B.
Find two cells with the same pair
Four empty cells hold exactly 1 and 5. Take the cell in row five, column nine and the cell in row six, column five. They share no row, no column and no box, so they do not see each other. Row five, column six has 1 and 5 as well, but it shares a row with one of them and a box with the other.
Find a link on one of the two digits
Row seven has exactly two places for a 5, in column five and column nine. Column five also holds the cell in row six, and column nine holds the cell in row five. So each place sees one cell. Both places hold three candidates, which does not matter. Only the count of 5s in the row does.
Follow the link
One of the two places in row seven is the 5. If it is column five, the cell in row six, column five shares that column and cannot be 5, so it is 1. If it is column nine, the cell in row five, column nine is 1 for the same reason. Either way, one of the two cells holds the 1.
Strike the 1 from cells that see both
A cell that sees both of them would share a house with that 1, so it cannot be 1. Three empty cells with a 1 see both. Row five, column six shares a row with one cell and a box with the other. Row six, columns seven and nine share a row with one and a box with the other. All three lose their 1.
Read what the strike left behind
Row five, column six had 1 and 5. Now it is a 5, a naked single. Before the strike this grid had no single of either kind. Nothing earlier on the path removed a mark either, from pointing pairs through to the Y-wing and the XYZ-wing. The W-wing is the first technique that does anything here.
Why one of them is enough
The argument splits on the link, not on the two cells. The 5 in row seven has two places, so it is in one or the other. Each place sees one of the cells and takes the 5 away from it. A cell with two candidates that loses one is left with the other, so each case puts a 1 into one of the cells. A cell that sees both sees that 1, whichever cell it is in.
It is tempting to read this as one cell being 1 and the other 5. The argument does not say that. On this grid both are 1. The 5 in row seven is in column five, which forces the cell in row six. Nothing forces the cell in row five, yet in the solution it is 1 as well. The strike only needs one of them.
The two cells must not see each other. Two cells with the same two candidates in one house are a naked pair, which comes much earlier and is read differently. Row five, column six and row five, column nine are one here. They are the only empty cells in row five, so that pair strikes nothing.
A row with three places
Go back to the board before the strike. Row seven has a 1 in column five and a 1 in column nine, and each of those places sees one of the two cells, just as the 5s did. It looks like a second link, this time on the 1.
It is not one, because row seven has a third place for the 1, in column three. The 1 could sit there, and then neither cell would be forced. Treat it as a link anyway and you would strike the 5 from row five, column six, from row six, column eight and from row six, column nine. Two of those three cells are 5. On this grid the 1 in row seven is in column three, the place that breaks the link.
Before you use a link, count the places for its digit. It needs exactly two.
The 5 has two places in row seven. One must hold it, so one of the two cells is 1.
The 1 has three places in row seven. The one in column three sees neither cell, so nothing is forced.
One pair, three links
The link can be in a row, a column or a box. On the board before the strike, take the cell in row five, column nine with the cell in row nine, column six. Both hold 1 and 5, and they do not see each other.
Three different links on the 5 tie them together. Row seven works again, since its place in column five shares a box with row nine, column six. Column eight has two places for the 5, in rows six and nine. The one in row six shares a box with the first cell, and the one in row nine shares a row with the second. Box nine also has two places for the 5, in row seven, column nine and row nine, column eight.
All three give the same strike. Only one cell sees both, row five, column six, and it loses its 1, which agrees with the first W-wing.
Find two cells with the same two candidates that do not see each other. Pick one of the digits and look for a house where it has exactly two places, one seeing each cell. Then strike the other digit from every cell that sees both. Three places in the house is not a link.
Where this sits on the path
A W-wing places nothing. It removes candidates and leaves the placing to the singles. On the grid above it made one naked single, in row five.
It closes the wings. The Y-wing and the XYZ-wing join their cells through a pivot that sees them all. The W-wing joins two cells that cannot see each other, through a digit with two places somewhere else.
Simple coloring comes next. It follows one digit from link to link across the grid, using the same two-place links.
On the board before the strike, the cell in row six, column five and the cell in row nine, column six both hold 1 and 5. Column eight has exactly two places for a 5, in rows six and nine. Which 1 can you strike?
Work out which cell each place in column eight sees. Then look for a cell that sees both of the cells with 1 and 5.
الخلاصة
- ابحث عن
- Two cells with the same two candidates that do not see each other, and a house where one of those digits has exactly two places, one seeing each cell.
- النطاق
- Every cell that sees both cells. The two cells and the two places of the link keep their candidates.
- النتيجة
- The other digit comes out of every cell that sees both cells.
- العلامة
- Late in a puzzle, with the same pair of candidates turning up in cells far apart.
- إذا لم تنجح
- If the digit has three or more places in the house, there is no link. Try the other digit of the pair, or another house, row, column or box.