This is an interesting strategy since it overlaps with Y-Wings and XYZ-Wings but uses very different logic. APE logic will solve an XY-Wing (3 bi-values) and an XYZ-Wing (bi-value <-> tri-value <-> bi-value).There are two types of APE - the normal APE and Extended APE.
Aligned Pair Exclusion - Type 1
The Aligned Pair Exclusion can be succinctly stated: Any two cells aligned on a row or column within the same box CANNOT duplicate the contents of any two-candidate cell they both see. The Y-Wing strategy has some diagrams (see Figure 2) to show how cells can see other cells along the row, column or box and how they intersect or overlap. In Figure 1 X and Y are two cells and the yellow shading shows the common cells they can both 'see'.
Lets consider all the possible pairs of numbers in X and Y. These are:
3 and 2 (in X and Y)3 and 55 and 25 and 57 and 27 and 5
Now is obvious that 5 and 5 can't be a solution to X and Y. If any of the other pair solutions were true we'd be able to remove those solutions from the candidates in all the other yellow squares. The strategy asks us to look at all the bi-value cells X and Y can 'see'. Cells marked A, B and C containing 2/7 and 3/5 and 5/7 match some of the options we have for X and Y. Any of these pairs would remove ALL candidates from one of A, B or C which is illogical, captain. This means we can exclude them from possible solutions for X and Y. This leaves us with a shorter list: 3 and 2 (in X and Y)5 and 2
What are we left with? According to our new list Y can only take the value 2 so we can remove 5.We can also remove the 7 from X. This helps us solve the Sudoku.
Credits - Rod Hagglund first popularised this method. A good thread with a double example and walk-through is here
Aligned Pair Exclusion - Type 2
The Extended Aligned Pair Exclusion includes tri-values spread over two cells as part of the attack. APE 2 Says that any two cells with only abc excludes combinations ab, ac and bc from the pair under consideration. This example is very clear since the two-cell tri-value is convieniently 4/5/6 in both cells. (see next example for alternative tri-value formations).
Lets consider all the possible pairs of numbers in X and Y first. These are:
1 and 4 (in X and Y)1 and 61 and 84 and 4 (impossible)4 and 64 and 86 and 46 and 6 (impossible)6 and 8
Cell R5C6 marked C removes a 1/4 pair. Now the tri-value: These are 4/5, 4/6 and 5/6. Removing these from the possibles for X but Y leaves us:
1/4/6 remains in X but Y is reduced to 6/8. Why should this work? Well, 5 is not really part of the tri-value that effects our APE. The key combination is 4/6 and that does the damage. Pretend that X is 4 and Y is 6 (or the other way round). This would leave A and B both equalling 5. Thats illegal which is why 4/6 is a combination we can remove from possible pairs in X and Y.
1 and 6 (in X and Y)1 and 84 and 86 and 8
In this second example the tri-value contained in A and B is 2/7/8. The only common value is 2. Nevertheless, the abc combinations are 2/7, 2/8 or 7/8. All the possible pairs of numbers in X and Y are. 7 and 5 (in X and Y)7 and 98 and 58 and 78 and 9
Our one tri-value which matches these is 7/8. If we remove 7/8 from our list Y is reduced to 5 and 9. We get a naked pair and the rest of the Sudoku solves.
No comments:
Post a Comment