W-Wing — the same pair, twice, joined by one link

The short version

A W-Wing is two cells that hold the exact same pair — say {2, 3} — and cannot see each other. On its own that is nothing: two unconnected pairs tell you nothing about each other.

The connection comes from outside. Find a strong link on one of the two digits: a line where that digit has exactly two homes, one of which each of your pairs can see.

Now the two pairs are tied together. Whichever way the link falls, at least one of the two cells has to take the other digit. So any cell that sees both of them cannot hold that other digit.

The shape

Four cells, two roles:

• Pair A at R1C7, candidates {2, 3}.
• Pair B at R5C3, candidates {2, 3} — the identical pair, and deliberately not visible from A.
• Link ends at R7C7 and R7C3, the only two homes for 3 in row 7.
• A sees one end (same column as R7C7); B sees the other (same column as R7C3).

Note what is not required: the two pairs do not see each other, the link does not have to be a row, and the link digit is not the one you eliminate. You eliminate the other digit of the pair.

Why it works

The proof is one implication, run in one direction, and it is short enough to hold in your head.

3 in row 7 has exactly two homes: R7C7 and R7C3. So exactly one of them is 3.

• Suppose R1C7 is not 2. Its pair leaves only 3, so R1C7 = 3.
• R1C7 shares column 7 with R7C7, so R7C7 cannot be 3.
• Row 7 still needs a 3, and R7C3 is the only home left. So R7C3 = 3.
• R5C3 shares column 3 with R7C3, so R5C3 cannot be 3. Its pair leaves only 2. So R5C3 = 2.

That is the whole argument. Either R1C7 is 2, or it is not 2 — and in that case R5C3 is 2. One of the two pairs must be the shared digit. Never neither.

So: eliminate 2 from every cell that sees both R1C7 and R5C3.

A real board

The pair is {2, 3}, and the two cells holding it are R1C7 and R5C3. They do not see each other — different row, different column, different box — so on their own they say nothing.

The link is in row 7: the digit 3 has exactly two homes there, R7C7 and R7C3. That is a strong link: one of those two cells is 3, and there is no third option.

The connections line up:

• R1C7 shares column 7 with R7C7.
• R5C3 shares column 3 with R7C3.

So each pair sees one end of the link, and the argument above applies: one of R1C7 and R5C3 is 2.

Now look at R5C7, which holds {1, 2, 7, 9}. It shares row 5 with R5C3 and column 7 with R1C7, so it sees both pairs. Since one of them is always 2, R5C7 cannot be 2.

One candidate removed, nothing placed — and the cell is closer to collapsing. That is what Expert-level boards are made of.

How to scan for one

W-Wings are found by pair, not by digit, which is why they are easy to miss when you are scanning the way fish patterns taught you to scan.

1. List the bivalue cells. Every empty cell with exactly two candidates. With full pencil marks this takes a minute and it is the only inventory this technique needs.
2. Group them by pair. Two cells carrying the same {2, 3} is the starting shape — not an XY-Wing, which needs three different pairs.
3. Check they cannot see each other. If they can, that is a naked pair and much simpler. W-Wing only exists for pairs that are blind to each other.
4. Hunt the link. Take one digit of the pair and look for a line — row, column or box — where that digit has exactly two homes, one visible from each pair. This is the expensive step, so do it digit by digit.
5. Collect the eliminations. Every cell that sees both pairs loses the other digit. Often that is exactly one cell, which is why the technique feels underwhelming until it unlocks the board.

The productive habit: when a digit has only two homes left in a line, do not stop at "that line is constrained". Ask what those two cells are adjacent to. That is where the link's value comes from.

W-Wing vs XY-Wing

The two names sound related and the logic is genuinely similar — both prove that one of two cells must hold a particular digit. The pictures are not.

The practical difference: an XY-Wing is found by walking candidate chains out from a pivot. A W-Wing is found by grouping identical pairs and then looking outward for a link. Same conclusion, opposite search direction.

A W-Wing is also weaker than it looks. It usually removes one candidate from one cell, whereas an XY-Wing removes a digit from every cell that sees both wings. What the W-Wing buys you is reach: it can connect two cells that are nowhere near each other, which the XY-Wing cannot do.

The traps

Using a digit that is not in the pair. The link must be on one of the two digits in the pair. A strong link on some unrelated digit proves nothing about these cells.

Forgetting that exactly one pair sees each end. If both pairs see the same end of the link, you have a weaker pattern — the link forces nothing about the other cell. The two ends must be split between the two pairs.

Treating a shared line as a pair. If the two cells with {2, 3} can see each other, use the naked pair. Reaching for a W-Wing there is wasted effort, and reporting it as one is wrong.

Eliminating the link digit. The conclusion is about the other digit of the pair. In the board above, the link is on 3 and the elimination is 2. Getting this backwards produces a wrong elimination, which breaks the puzzle rather than merely wasting a move.

Reading the pair off a half-filled mark-up. If a cell shows {2, 3} only because you have not finished marking candidates, the pattern is not there. A W-Wing needs it to be genuinely true that the cell has exactly two candidates.

Where to use it

W-Wings sit one step beyond the fish family, and they are usually found late — after singles are exhausted, after the obvious pairs and triples are gone, and often after a skyscraper or an XY-Wing has already fired.

They are worth the effort for one specific reason: they connect cells that nothing else connects. When a board has been reduced to isolated bivalue cells scattered across the grid, W-Wing is the technique that starts talking between them again.

If you are learning this band, learn it after the XY-Wing and before the forcing chain. It is the cheapest example of "two things that look unrelated are actually linked", which is the idea every chain technique after it is built on.

How often does this technique actually show up? See its measured numbers →