This definitive and easy-to-explain guide is all about Aligned pair Exclusion and Subset exclusion Strategies along with examples, Types, and Techniques.
So let’s get started with the definition of Subset Exclusion.
Contents
Subset Exclusion
Let’s define an “Almost Locked Set” (“ALS”) as a collection of n Cells in the same region (Row, Column, or Square) that include precisely (n+1) Candidates: A “Bi-Value” Cell or two “Bi-Value” Cells having the same candidate, for example.
In the Sudoku community, the term Subset Exclusion is not often used. This page serves as a link between Death Blossom and Aligned Pair Exclusion.
Extending Aligned Pair Exclusion to Aligned Triple Exclusion and so on is possible. In reality, the cells to be counted do not need to be aligned in any way. Let’s call the resulting approach Subset Exclusion for want of a better term.
Example of Subset Exclusion
As in Aligned Pair Exclusion, we may count the number of potential pairings of blue cells. From top to bottom, the following is a list of potential combinations:
5+2+4
5+2+6
5+6+4
5+7+4*
5+7+6*
5+9+4*
5+9+6*
7+2+4
7+2+6
7+6+4
7+9+4
7+9+6
The highlighted combinations result in no candidates for any of the yellow cells. Thus they may be deleted from the list. As a consequence, we can delete seven from the r2c4 code.
Aligned Pair Exclusion
This is an intriguing method known as APE and is often referred to as Subset Exclusion. It has some similarities to Y-Wings, XYZ-Wings, and WXYZ-Wings, but its reasoning is entirely different. Because the overlap isn’t exact, they’re worth keeping an eye out for in a pinch.
A base pair of cells is always present (which now show up as grey cells on the solver). In one of the two cells, at least one elimination will occur. The solver will also display a range of colored cells that represent the items employed in the elimination process. I used to differentiate between APE type 1 and type 2, which both employed bi-value cells and 2-cell Almost Locked Sets (ALS).
However, the solver will now identify a wider range of ALSs, including 3-cell ALSs, and because these are just extensions of the same logic, the solver will return the first of any it finds. Thus, the rationale is slightly different, but I’ll explain why in the examples that follow.
Types of Aligned Pair Exclusion
Type 1
Aligned Pair Exclusion Example 1:– Type 1
(Y-Wing must be unchecked): Example of a Load or: From the Beginning
The Aligned Pair Exclusion is as follows: Any two cells that can view each other CANNOT replicate any Almost Locked Set contents with which they both see and share candidates.
Consider the most basic scenario: two bi-value cells assaulting the couple. In the diagram, I’ve also included the Y-Wing to indicate a simpler approach to doing the same task – but only in some circumstances.
All potential pairings of numbers that fit in [G2/G3] are considered. These are the G2 and G3 versions:
2 and 2 (impossible)
2 and 5
2 and 8
4 and 2
4 and 5
4 and 8
What if solutions 2 and 8 were tried? That would replicate G9 and leave it empty. H1 would also be emptied by 4 and 8. We’re left with a collection of options that looks somewhat like this:
2 and 2 (impossible)
2 and 5
2 and 8 (impossible)
4 and 2
4 and 5
4 and 8 (impossible)
Have you noticed that there are no 8s remaining in any of the pairings? As a result, we may eliminate eight from our base pair.
Example 2 of Aligned Pair Exclusion:– Type 1
Load or: From the Beginning As part of the assault in the next example, tri-values are dispersed across two cells. Any two cells with just ABC eliminate combinations ab, ac, and bc from the base pair under consideration, which we can employ double cells. This clever approach substantially expands APE’s use, which would otherwise be limited to a poor man’s Y-Wing.
The 2-cell ALS in [A1, B3] comprises 1/3/7. Hence the pairings 1,3, 1,7, and 3,7 would cripple the solution for that ALS.
Let’s look at all of the potential number pairings in our base pair [C2/C3]. These are the following:
1 & 3 – excluded by A1 + B3
1 and 4 – excluded by C5
1 and 9
3 and 3
3 and 4
3 and 9
8 and 3 – excluded by C9
8 and 4
8 and 9
Now we have to be a little more cautious. Although three has been ruled out as a possible answer in C3, 3 + 4 and 3 + 9 are still acceptable options. So far, we haven’t been able to remove three from C2. Myth is to be credited. The idea for ABC = ab/ac/bc originates from Jellies.
In an APE assault, there might be more than two, three, or four ALSs of various sizes. For the sake of simplicity, I’ve considered two examples.
Exclusion of Aligned Pairs Example 3:– Type 1
Load or: From the Beginning, We come to discover a 3-cell ALS plus a bi-value in H3 hitting A3/B3 farther into the same problem. In [A1, C1, C3], the ALS contains the four integers 1,4,7,9, which the solver interprets as a quadruple combination. Abcd comprises the letters ab, ac, ad, bc, bd, and cd. Returning to the first pair: The A3/B3 combinations are as follows:
4 & 3
4 & 7 – excluded by [A1,C1,C3]
5 & 3
5 and 7 – excluded by H3
7 and 3
7 and 7 (impossible)
The tough part about the 3-cell ALS isn’t that the base pair can’t empty it (since it’s two cells and the ALS is three). Instead, it’s the reality that a 4 in A3 and a 7 in B3 solution would leave just two options to fill three cells. That’s enough to rule out the possibility of a match.
Aligned Pair Exclusion: Type 2
Even if the pair is not aligned, Aligned Pair Exclusion can operate. Although it may appear to be a joke, it is too late to rename this strategy:) Maybe ‘Subset Exclusion’ would be a better term. There is a small logical difference, but I have discovered several examples that support the strategy’s utility.
Aligned Pair Exclusion Example 1:– Type 2
(Y-Wings are turned off): Example of a Load or: From the Beginning, The simplest version of APE2 repeats the Y-Wing with just two bi-value cells. However, I offer an example to show how APE2 works.
The Y-Wing is shown first, with A1 – A4 (the pivot) – B6 as the foundation. Therefore, eight must appear in either A1 or B6, excluding it from B1 and B2.
But, using the non-aligned pair A4 and B1, let’s follow the APE reasoning. (Note: We may also reduce the eight by selecting A1 and B2.) Using the following combinations, A4 and B1 form a pair:
1 and 1 – POSSIBLE!
1 and 6
1 and 8 – excluded by A1
9 and 1
9 and 6
9 and 8 – excluded by B6
The main distinction between APE 1 and APE 2 is that non-aligned pairings allow the same candidate to be a solution in both cells. So 1 and 1 is a foregone conclusion. It’s not like it matters in this scenario. Because of the other exclusions, we won’t have an 8 in B1, as we had hoped.
Aligned Pair Exclusion Example 2:– Type 2
Example of a Load or: From the Beginning
Here’s a more advanced APE that doesn’t have a Wing option. B1 and C7 are being attacked by two bi-value cells and one two-cell ALS. Let’s make a list of the possible combinations between those cells:
1 & 4 – excluded by B9
1 & 5 – excluded by B8
1 & 7 – excluded by [C1 + C3]
2 and 4
2 and 5
2 and 7
7 and 4
7 and 5
7 and 7 – Permitted!
One is removed from B1. The same formation also removes one from B2.
Aligned Pair Exclusion Example 3:– Type 2
Example of a Load or: From the Beginning
Finally, a non-aligned pair was created using a bi-value cell and a three-cell ALS.
Later in the solution process, a second good APE uses a 2-cell ALS and a 3-cell ALS. Finally, the puzzle may be loaded using the links below the diagram.
An Eight-Cell Aligned Pair
Load Example or: From the Beginning: An Eight-Cell Aligned Pair Klaus Brenner’s Sudoku is one of my favorite ways to conclude these essays. He’s turned discovering unique and attractive instances into an art form, and in the case of Aligned Pairs, he’s accomplished the almost impossible. An Aligned Pair elimination with eight cells! We had discovered some five-cell instances and wondered if a six-cell or maybe a seven-cell version could exist.
This is the first and only eight-cell configuration that has been discovered. Fortunately, the solver is up to the task.
Advanced-Aligned Pair Exclusion
This is an intriguing approach since it employs a logic similar to Y-Wings and XYZ-Wings yet is extremely different. APE logic will be used to solve an XY-Wing (3 bi-values) and an XYZ-Wing (bi-value -> tri-value -> bi-value). Normal APE and Extended APE are the two forms of APE.
Advanced Aligned Pair Exclusion – Type 1
The Aligned Pair Exclusion may be summarized: Any two cells aligned on a row or column within the same box CANNOT view the same two-candidate cell. The Y-Wing approach includes various illustrations (see Figure 2) that explain how cells may see each other along the row, column, or box and how they meet or overlap. Figure 1 illustrates X and Y cells with yellow shading indicating the shared cells they can both see.’ Let’s look at all of the potential number pairings in X & Y.
It should now be clear that 5 and 5 cannot be a solution to X and Y. We’d be able to eliminate those answers from the possibilities in all the other yellow squares if any of the other pair solutions were true. The technique instructs us to examine all bi-value cells that X and Y can see.’ Cells designated A, B, and C with 2/7, 3/5, and 5/7 correspond to some X and Y alternatives. This is absurd because any of these combos would exclude ALL candidates from one of A, B, or C. This implies we can exclude them out of X and Y’s probable answers.
So, where do we go from here? We can eliminate five since Y can only take the value two according to our revised list. We can also take the 7 out of X. This assists us in completing the Sudoku puzzle.
Advanced Aligned Pair Exclusion – Type 2
As part of the assault, the Extended Aligned Pair Exclusion contains tri-values distributed across two cells. For example, APE 2 states that any two cells with only ABC are excluded from the pair in question, as are ab, ac, and bc combinations. Because the two-cell tri-value is conveniently 4/5/6 in both cells, this example is quite evident. (Alternative tri-value constructions are shown in the next example.) Let’s start with all of the possible pairs of numbers in X and Y. These are the following:
1, 2, 3, and 4 (in X and Y)
1 and 61, as well as 84 and 4 (impossible)
4, 64, 86, 46, 6 (impossible)
6 and 8
A quarter pair is removed from cell R5C6 marked C. Then there’s the tri-value: 4/5, 4/6, and 5/6 are the numbers. Taking them out of the equation for X but not for Y leaves us with:
X is kept at 1/4/6, while Y is decreased to 6/8. Why should this be successful? 5 isn’t part of the tri-value that influences our APE. The harm is done by the key combination 4/6. Assume that X is four and Y is six (or the other way round). A & B are both equal to 5. That’s against the law, which is why we can rule out 4/6 as a potential pair in X & Y.
The tri-value included in A and B in this second case is 2/7/8. The only value in common is 2. The ABC combinations, however, are 2/7, 2/8, or 7/8. There are all possible number pairings in X and Y. 7 & 5 (in X & Y)7 and 98 and 58 and 78 and 9 (in X and Y)7 and 98 and 58 and 78 and 9 (in X and Y)7 and
7/8 is the only tri-value that fits these. Y is reduced to 5 and 9 if we delete 7/8 from our list. The rest of the Sudoku is solved after we acquire a nude pair.
Aligned Pair Exclusion Technique
A pair of cells at the intersection of a box and a column, or a box and a row, are excluded using the Aligned Pair Exclusion method. In this method, all candidate combinations for a pair are verified. This combination for this pair is not permitted if a cell views both cells of the pair and has precisely two candidates that the pair may have. If no possibilities are permitted for a certain digit in one of the pair’s cells, this candidate is excluded from that cell.
All potential candidate combinations for the pair R8C3 and R9C3 are shown in the sudoku diagram below:
1-2
1-6
1-7
2-2 (invalid)
2-6 (conflicted with R1C3)
2-7 (conflicted with R9C1)
For obvious reasons, the 2-2 pair is not permitted. All eligible candidates would be excluded from the R1C3 cell if the pair was set to 2-6. Pair 2-7 and the cell R9C1 are in a similar position. As can be seen, in the R8C3 cell, all possibilities containing the digit two have been excluded. As a result, digit two can be removed from this cell. Only digit one may be inserted in the R8C3 cell after this removal.
Aligned Pair Exclusion (ALS) is a candidate elimination approach because an ALS can only give up one number before being stuck with n numbers for n cells and unable to give up another. Therefore, when you see a pair of cells in a chute with remainders containing ALS some of the same numbers, it’s a good idea to double-check that the pair can’t include true candidates of two numbers from any ALS remainder.
Pick one of the cells and test all combinations of two with candidates from the other cell for each of its candidates. There are several recommended tabulations in Sudoku literature for doing this, but here is one that is quite compact, a logical statement for the potential combinations. From the explanation of the contents of Sue de Coq’s chute, you’re familiar with the usage of algebraic addition (+) for logical “or” and multiplication for logical “and.”
Aligned Pair Exclusion Technique
The aligned cells of SWr9 share units and numbers with three ALS, bv 18, 78, and 15, from KrazyDad’s Insane collection, volume 4, book 4, # 5.
We start with 1r9c3, permitting 71, since bv 15r9c8 and 18r8c1 restrict 51 and 81, respectively. Then, for the complete description of 71+17+57, we include the combinations that conclude in 7.
For the two cells, the phrase now specifies all potential solution values combinations. For example, eight must be removed from r1c1 based on the values on the left.
We use rounded squares to denote the elimination pair and then go on to the new 8-clue.
There is a way around this. Any candidate not among the ALS candidates will be removed if you notice that one cell, on the left or right, has all of its numbers on the list, in this example 1 and 7 on the right.
If we hadn’t gotten to it first, you could ask if r8 would have caused an APE. All applicants would be able to stay if the combinations of r8c1 and r8c3 were 23+31+32+82+83. This is a common occurrence with APE efforts.
In my previous example, you may have noted that my APE is also a short XY ANL or XY wing and that r1c1 is the hinge cell for a victimless WXYZ wing. That is not an issue.
Andrew Stuart’s sudokuwiki.org treatment on APE includes a variant of APE he refers to as Type 2. It is predicated on the notion that ALS limitations do not require the target cells to be aligned to induce eliminations. When both cells share a unit with an ALS comprising both numbers, a combination of numbers in the target cells is prohibited. It doesn’t need to be the same unit or even the same ALS.
This potential is illustrated by one of Andrew’s examples, which has been translated into Sysudoku marking. Each target cell shares a unit with the same two ALS in this setup. Make certain you understand why combinations 14, 15, and 17 are not allowed.
For each target cell, there are two options. The 1-candidate is likewise omitted from r2c2, as Andrew points out.
What about the r3c9 code? Six permits all NWr2 combinations, just as 2 and 3 allow all NEr3 combinations. Eliminations in the other target are only achievable when one of the target cells is “covered” by ALS.
When basic removals are less comprehensive, additional APE removals are conceivable, although they are uncommon in Sysudoku evaluations. APE is worth noting, but a thorough search would include putting ALS along banks and towers on a duplicate of the basic grid, then selecting and testing target cells. Before XY and X-panel chains, fish, and color, that’s too much labor for too little gain.
Before turning to trials, I will take a more systematic look at APE, particularly Type 2, when previous reviews are updated. From the KrazyDad Insane assessment of the latest puzzle, v.4, b.10, #5, here’s a Type 2 example. The initial review post attempted Single Alternate SdC NWr1 = 437, which failed, allowing Sue de Coq NWr1 = (7+3)(5+8) to delete 8r2c2.
In this case, the two cells r8c46 operate as a single target cell, similar to ALS SWr8 and ALS Sr7. The SASdC experiment is avoided because the combinations (4+8+9)1 with target r7c1 are rejected by the two ALS. The second SASdC experiment, also known as a coloring trial, is still required.
In Type 2 arrangements, there are three ways that ALS might limit target cell placement.
Two ALS are shared by two target cells on the North bank, similar to Stuart’s example. The identical layout is depicted on the East tower. However, it is vertically oriented. Targets share only one (blue) ALS in the West tower. A distinct green ALS attacks each target. Again, there is a possibility that ALS restrictions will overlap.
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