# ALS-XY-Wing

The ALS-XY-Wing rule is a solving technique that uses three Almost Locked Sets.

Make sure you understand the concept of Almost Locked Sets (ALSs) before you proceed.

## How it works

Say we have three Almost Locked Sets A, B and C. Suppose

Then for any digit X that is distinct from Y and Z and is a common candidate for A and B, we can eliminate X from any cell that sees all cells belonging to either A or B and having X as a candidate.

## Relations to other techniques

Observe that this technique generalizes XY-Wing, and hence the name ALS-XY-Wing. Note that each of the three bivalue cells in the XY-Wing is itself an ALS, hence we can set C as the pivot cell, and the other two pincer cells as A and B respectively.

Recall that the XY-Wing is a chain of length three, and can be generalized to XY-Chain, which is a chain of arbitrarily length. Similarly, we can generalize ALS-XY-Wing to ALS-XY-Chain, which is a chain whose nodes are ALSs instead of single cells.

## Example

This example is taken from Ron Moore's walkthrough for Ruud's 2007 May 5 Nightmare.

The ALS-XYZ-Wing is constructed as follows:

• The ALS A is the cells colored green in box 8,
• The ALS B is the cells colored light blue in box 3,
• The ALS C is the cell colored yellow in box 9,
• The restricted common Y between A and C is 2, and
• The restricted common Z between B and C is 1.

The cells r7c46 in ALS A and the cell r2c9 in ALS B contain the candidate 5. Since all of r7c46 and r2c9 have a common peer r7c9, we can eliminate 5 from r7c9.

A possible Eureka notation for this elimination is:

```(5=24791)ALS:r12c89,r3c8 - (1=2)r7c8 - (216=5)ALS:r7c46,r9c5 => r7c9 <> 5
```

Note, the reader must check the grid to ensure that r7c9 can see all the 5's in both ALS's and similarly for the links involving digits 1 and 2.

Using the ALS (245)r12c9 and a group node for (2)r12c8 we could also use ALS nodes in an AIC:

```(5=42)ALS:r12c9 - (2)r12c8 = (2)r7c8 - (216=5)ALS:r7c46,r9c5 => r7c9 <> 5
```

which provides greater freedom for the locations of candidates in box 3.

In this example, since the pivot ALS C is a single cell, we can replicate the same elimination using Death Blossom instead.

## Notes on notation

Unlike ALS-XZ and XY-Wing, which eliminates the digit Z, the ALS-XY-Wing eliminates the digit X instead. Unfortunately, this inconsistent notation is widely used by the Sudoku community.