127.0.0.1 at 09:49, 27 October 2021

2021-10-27T09:49:35Z

← Older revision		Revision as of 09:49, 27 October 2021
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	== Relation between APE and ALS-XZ ==		== Relation between APE and ALS-XZ ==
	Consider the APE without any extensions. To apply the APE, we are looking at the [[intersection]] of two [[constraint]]s, one a [[line]] and the other a [[box]]. Suppose we enumerate the combination of two [[cell]]s '''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 [[ALS]]s are: (a) the cell '''X''' itself, and (b) the cell '''B''' plus the cells in the other constraint that participates in the APE.		Consider the APE without any extensions. To apply the APE, we are looking at the [[intersection]] of two [[constraint]]s, one a [[line]] and the other a [[box]]. Suppose we enumerate the combination of two [[cell]]s '''A''' and '''B''' within the intersection, leading to the [[eliminate\|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 [[ALS]]s 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 [[ALS]]s that has '''9''' as a candidate. This eliminates '''9''' from both '''r2c3''' and '''r5c1'''. Also, Example 2 is an example of a doubly-linked ALS-XZ with r12c4 = 468 ALS and r78c4r9c56 = 13468 ALS. The restricted common digits are 4 and 8. This provides the following cell eliminations: r7c456<>6, r3c6<>6, r4c5<>4, r7c8<>3, and r7c6<>1. What is needed here is an axample that cannot be solved with another technique which provides more cell eliminations.		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 [[ALS]]s that has '''9''' as a candidate. This eliminates '''9''' from both '''r2c3''' and '''r5c1'''. Also, Example 2 is an example of a doubly-linked ALS-XZ with r12c4 = 468 ALS and r78c4r9c56 = 13468 ALS. The restricted common digits are 4 and 8. This provides the following cell eliminations: r7c456<>6, r3c6<>6, r4c5<>4, r7c8<>3, and r7c6<>1. What is needed here is an axample that cannot be solved with another technique which provides more cell eliminations.

127.0.0.1: Created page with "'''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]..."

2021-10-25T17:07:23Z

Created page with "'''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]..."

New page

'''Aligned Pair Exclusion''' or '''APE''' is a [[solving technique]] in which the solver checks combinations of [[digit]]s in a pair of [[cell]]s 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.

== Examples ==
Below are two examples that illustrate the APE move.

=== Example 1 ===
{| cellspacing="8" cellpadding="0"
|
[[Image:APE.png|APE example]]
|
We are examining the pair '''r5c1''' and '''r6c1'''. Their respective candidates are:
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 '''r4c3''', which can [[see]] both cells. This combination would eliminate all candidates from this cell, so it must be invalid.

The same reasoning can be used for the combination 2+9 and '''r2c1''', and also for 5+9 and '''r1c1'''.

Effectively, all combinations which include a '''9''' in '''r5c1''' are invalid. As a result, we can eliminate this candidate. The remaining candidate '''4''' can be placed in '''r5c1'''.
|}

=== Example 2 ===
{| cellspacing="8" cellpadding="0"
|
[[Image:APE2.png|APE example]]
|
Here we enumerate all the combinations of the cells '''r7c4''' and '''r8c4'''. Their respective candidates are:
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 '''r1c4''', '''r2c4''', '''r9c5''', '''r9c6''', four of the combinations are not permissible. When we strike off these illegal combinations, we find that all combinations with '''6''' are removed. Hence we can remove '''6''' from '''r8c4'''.
|}

== 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 [http://scanraid.com/AdvanStrategies.htm#APE 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 [[constraint]]s, one a [[line]] and the other a [[box]]. Suppose we enumerate the combination of two [[cell]]s '''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 [[ALS]]s 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 [[ALS]]s that has '''9''' as a candidate. This eliminates '''9''' from both '''r2c3''' and '''r5c1'''. Also, Example 2 is an example of a doubly-linked ALS-XZ with r12c4 = 468 ALS and r78c4r9c56 = 13468 ALS. The restricted common digits are 4 and 8. This provides the following cell eliminations: r7c456<>6, r3c6<>6, r4c5<>4, r7c8<>3, and r7c6<>1. What is needed here is an axample that cannot be solved with another technique which provides more cell eliminations.

== Alternative Techniques ==
* [[XY-Wing]]
* [[XYZ-Wing]]
* [[ALS-XZ]]
* [[Subset Counting]]

== See Also ==
* [[Aligned Triple Exclusion]]
* [[Subset Exclusion]]

[[Category:Solving Techniques]]

Aligned Pair Exclusion - Revision history

127.0.0.1 at 09:49, 27 October 2021

127.0.0.1: Created page with "'''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]..."