# Aligned Pair Exclusion

**Aligned Pair Exclusion** or **APE** is a solving technique in which the solver checks combinations of digits in a pair of cells located in an intersection.

It is a very laborious technique, which may explain why it is not so popular. Many APE moves can also be replicated by the easier XY-Wing and XYZ-Wing techniques. Some APE moves can also be replicated by the ALS-XZ rule.

## Contents

## Examples

Below are two examples that illustrate the APE move.

### Example 1

We are examining the pair 4,9 1,2,5 The following configurations are possible: 4+1 9+1* 4+2 9+2* 4+5 9+5* The combination 1+9 is already present in The same reasoning can be used for the combination 2+9 and Effectively, all combinations which include a |

### Example 2

Here we enumerate all the combinations of the cells 1,3,4,8 3,4,6,8 The enumeration goes as follows: 1+3 1+4 1+6* 1+8 3+4 3+6* 3+8 4+3 4+6* 4+8 8+3 8+4 8+6* Due to the cells |

## Extensions

**Aligned Pair Exclusion** can be extended to **Aligned Triple Exclusion** and so on. In fact, the set of cells to be enumerated does not need to be aligned at all. See **Subset Exclusion**.

Yet another extension for **Aligned Pair Exclusion** is to consider not just bivalue cells, but also pairs of cells having a total of three candidates X, Y and Z. Then we can exclude the X+Y, X+Z and Y+Z combinations. See Andrew Stuart's *Advanced Strategies* (under *Aligned Pair Exclusion - Type 2*) for two examples.

## Relation between APE and ALS-XZ

Consider the APE without any extensions. To apply the APE, we are looking at the intersection of two constraints, one a line and the other a box. Suppose we enumerate the combination of two cells **A** and **B** within the intersection, leading to the elimination of some candidate in **A**. If one of the two constraints has only one cell **X** involved in the application of APE, then we can replicate this APE with an ALS-XZ rule which leads to the same elimination, where the two ALSs are: (a) the cell **X** itself, and (b) the cell **B** plus the cells in the other constraint that participates in the APE.

Hence, Example 1 above can be recast as an ALS-XZ, with the first ALS being **r4c3** and the second ALS being **r126c1**, the restricted common being **1**, eliminating all instances of digit **9** that is seen by all cells in both ALSs that has **9** as a candidate. This eliminates **9** from both **r2c3** and **r5c1**.