Y-Wing in Sudoku: how the three-cell chain works

Y-Wing is an advanced Sudoku technique based on the relationship between three cells with two candidates each.

At first it may seem less intuitive than other techniques, because it is not based on a whole row, column or box, but on a small logical chain. Once you understand the role of the three involved cells, however, the reasoning becomes much clearer.

The technique revolves around a central cell, called the pivot, and two connected cells, called wings. Based on the candidates present in these three cells, it is possible to eliminate a candidate from other cells connected to both wings.

In this guide, we will see what a Y-Wing is, how to recognize it and how to use it without confusing it with pairs or other techniques.

A Y-Wing is a configuration formed by three different cells, each with exactly two candidates.

The main cell is called the pivot. The other two cells are called wings. The pivot must see both wings: this means it must share a row, column or box with the first wing, and a row, column or box with the second wing.

The three cells have this kind of structure:

• pivot: candidates A and B;
• first wing: candidates A and C;
• second wing: candidates B and C.

Candidate C is present in both wings, but not in the pivot: this is exactly the candidate that can be eliminated from some external cells, specifically from cells that see both wings.

The pivot is the central cell of the reasoning. It has two candidates, and each of these candidates is connected to one of the two wings.

If the pivot contains {2,5}, one wing can contain {2,8} and the other {5,8}. The first wing shares 2 with the pivot, the second shares 5 with the pivot. Both wings share candidate 8 with each other, and 8 is the eliminable candidate.

The pivot must see both wings. The two wings, however, do not necessarily need to see each other: they can be in separate units, as long as there are external cells that see both and from which the common candidate can be eliminated.

This distinction is important: the elimination does not happen because the wings are in the same unit, but because an external cell that sees both cannot contain the common candidate.

The strength of Y-Wing comes from the fact that the pivot has only two possibilities.

Imagine a pivot with candidates {2,5}. If the pivot were 2, then the wing containing {2,8} could not be 2 and would have to be 8. If instead the pivot were 5, then the wing containing {5,8} could not be 5 and would have to be 8.

In both cases, at least one of the two wings will necessarily be 8.

As a result, any cell that sees both wings cannot contain 8. Whatever value the pivot takes, at least one of the two wings will be 8; a cell connected to both would therefore certainly conflict with one of the two wings.

This is the logical core of Y-Wing.

To find a Y-Wing, it is useful to look for cells with exactly two candidates. These cells are often called bivalue cells.

Choose a bivalue cell as a possible pivot. Then look for two other bivalue cells that see the pivot and have the right structure: one must share one candidate with the pivot, the other must share the other candidate.

Finally, the two wings must have a candidate in common with each other. That common candidate is the one that might be eliminated from external cells.

For example:

• pivot: {3,7};
• wing 1: {3,9};
• wing 2: {7,9}.

The eliminable candidate is 9, but only from cells that see both wings.

In a Y-Wing, the eliminable candidate is the one common to the two wings, not the one present in the pivot.

In the example pivot {2,5}, wing {2,8} and wing {5,8}, the eliminable candidate is 8.

To eliminate it, however, one precise condition is required: the cell from which you want to eliminate it must see both wings. This means it must share a row, column or box with the first wing and also with the second wing.

It is not enough for it to see the pivot. It is not enough for it to see only one wing. It must see both wing cells, because the elimination is based on the fact that at least one of the two will contain the common candidate.

Imagine these three cells:

• pivot: candidates {1,4};
• wing A: candidates {1,7};
• wing B: candidates {4,7}.

The pivot sees both wings. Wing A shares candidate 1 with the pivot. Wing B shares candidate 4 with the pivot. The two wings share candidate 7 with each other.

Now let us reason it through.

If the pivot were 1, then wing A could not be 1 and would have to be 7. If the pivot were 4, then wing B could not be 4 and would have to be 7.

In both cases, one of the two wings will be 7. Therefore any cell that sees both wing A and wing B cannot contain 7 as a candidate.

From that cell you can therefore eliminate candidate 7.

Y-Wing can be confused with other techniques, but its reasoning is different.

Compared with Naked Pair, you do not have two cells in the same unit with the same two candidates. Instead, you have three cells connected by a logical relationship.

Compared with X-Wing, you are not looking for a rectangle formed by two rows and two columns. You are looking for a chain between one pivot and two wings.

Naked Pair works on a closed pair in the same unit. X-Wing works on the distribution of one candidate across rows and columns. Y-Wing works on three bivalue cells and on a logical consequence between alternatives.

These differences help you understand when to look for each technique.

The first mistake is using a pivot with more than two candidates. For a classic Y-Wing, the pivot must have exactly two candidates.

The second mistake is choosing wings that do not see the pivot. Each wing must share a row, column or box with the pivot.

The third mistake is eliminating the common candidate from cells that see only one wing. The elimination is valid only in cells that see both wings.

The fourth mistake is eliminating one of the pivot candidates instead of the candidate common to the wings. In a Y-Wing, the eliminable candidate is the one shared by the two wings and absent from the pivot.

Y-Wing is an advanced technique based on three cells with two candidates each: one pivot and two wings.

The pivot connects the wings through two different candidates. The wings share a third candidate, which can be eliminated from cells that see both wings.

It is less immediate than X-Wing, but very useful in hard Sudoku. To apply it well, you need updated candidates, attention to the relationships between cells and precision in the eliminations.