24 June 2007: Almost XYZ Wing
Posted: Tue Jun 26, 2007 1:40 am
Start position for the 24 June 2007 Nightmare:
In the "Almost Swordfish" thread for the 1 June, 2007 Nightmare, I mentioned that the approach there could be used with patterns besides fish. In this puzzle there is an "Almost XYZ Wing" pattern which can be used. I occasionally come across these, but in most cases they aren't really useful, either because they don't really advance the solution, or there is a more natural way to arrive at the same elimination. In this case, it turns out that there is an ALS XZ rule reduction which gives the same elimination, but the sets are fairly large and I didn't see it until I looked at the Sudocue solver's solution. The elimination is quite useful, so I'll consider this a reasonable example to illustrate the idea.
After initial basics and several naked or hidden subsets, the following position is reached:
There are two independent AIC's in this position. The cells marked with "%" form a relatively simple grouped chain for digit 5:
In the "Almost Swordfish" thread I used the term "spoiler candidate" to denote a candidate, which, if it were not present, would cause some familiar pattern to be created. That term reflects my perhaps unsophisticated intuition, while the term "surplus candidate for the pattern" sounds more conventional and polished, so I'll use that here. Consider the cells marked with "#" above -- r6c8, r59c9. With the annotation in r9c9, I've distinguished (2)r9c9 as we would a surplus candidate for a UR pattern, for example. Observe that if this surplus candidate is false, an XYZ wing pattern exists in the "#" cells. This XYZ wing would eliminate (8)r4c9. On the other hand, if (2)r9c9 is true, there is what could be called an "adjunct chain" to the XYZ wing pattern which also eliminates (8)r4c9.
We could also use a different adjunct chain:
Now we have:
The previous elimination allows us to write this AIC:
Remark: In the Almost Swordfish thread, I did find an Almost XYZ wing in the first position I gave, but I didn't mention it because it didn't seem to advance the solution. For any readers who want to try their hand at spotting these patterns, you might see if you can spot an Almost XYZ wing in that position. Hint: Be sure to do this before the elimination of (7)r3c5 which I show there, as that destroys the potential XYZ wing. As in this puzzle, there may be alternative "adjunct chains" which can be used.
Code: Select all
080000000002000836360000700000002300000900010709006005150000000000708900004003000
. 8 .|. . .|. . .
. . 2|. . .|8 3 6
3 6 .|. . .|7 . .
-----+-----+-----
. . .|. . 2|3 . .
. . .|9 . .|. 1 .
7 . 9|. . 6|. . 5
-----+-----+-----
1 5 .|. . .|. . .
. . .|7 . 8|9 . .
. . 4|. . 3|. . .
After initial basics and several naked or hidden subsets, the following position is reached:
Code: Select all
·--------------------·----------------------·-------------------·
| 459 8 157 | 36 36 %4579 | 15 2459 1249 |
| %459 79 2 | %45 %4579 1 | 8 3 6 |
| 3 6 15 | 28 28 %459 | 7 459 149 |
·--------------------+----------------------+-------------------·
| 568 14 568 | 1458 14578 2 | 3 6789 *79-8 |
| *268-5 234 3568 | 9 34578 %457 | 246 1 #278 |
| 7 1234 9 | 1348 1348 6 | 24 #28 5 |
·--------------------+----------------------+-------------------·
| 1 5 78 | 246 2469 49 | 26 78 3 |
| 26 23 36 | 7 15 8 | 9 45 14 |
| 89 79 4 | 1256 1256 3 | 15 2678 #78+2 |
·--------------------·----------------------·-------------------·
- (5): r2c1 = r2c45 - r13c6 = r5c6 => r5c1 <> 5
In the "Almost Swordfish" thread I used the term "spoiler candidate" to denote a candidate, which, if it were not present, would cause some familiar pattern to be created. That term reflects my perhaps unsophisticated intuition, while the term "surplus candidate for the pattern" sounds more conventional and polished, so I'll use that here. Consider the cells marked with "#" above -- r6c8, r59c9. With the annotation in r9c9, I've distinguished (2)r9c9 as we would a surplus candidate for a UR pattern, for example. Observe that if this surplus candidate is false, an XYZ wing pattern exists in the "#" cells. This XYZ wing would eliminate (8)r4c9. On the other hand, if (2)r9c9 is true, there is what could be called an "adjunct chain" to the XYZ wing pattern which also eliminates (8)r4c9.
- (2)r9c9 => not (2)r1c9 => (2)r1c8 => (8)r6c8
- (8)[XYZ wing r6c8|r59c9] = (2)r9c9 - (2)r1c9 = (2)r1c8 - (2=8)r6c8 => r4c9 <> 8
We could also use a different adjunct chain:
- (8)[XYZ wing r6c8|r59c9] = (2)r9c9 - (2=6)r7c7 - (6=248)r56c7|r6c8 => r4c9 <> 8.
Now we have:
Code: Select all
·------------------·--------------------·-------------------·
| 459 8 #157 | 36 36 #4579 | 15 2459 1249 |
| 459 79 2 | 45 4579 1 | 8 3 6 |
| 3 6 15 | 28 28 459 | 7 459 149 |
·------------------+--------------------+-------------------·
| 568 14 568 | 1458 14578 2 | 3 6789 79 |
| 268 234 3568 | 9 34578 #457 | 246 1 *28-7 |
| 7 1234 9 | 1348 1348 6 | 24 28 5 |
·------------------+--------------------+-------------------·
| 1 5 #78 | 246 2469 49 | 26 #78 3 |
| 26 23 36 | 7 15 8 | 9 45 14 |
| 89 79 4 | 1256 1256 3 | 15 2678 #278 |
·------------------·--------------------·-------------------·
- (7)r5c6 = (7)r1c6 - (7)r1c3 = (7)r7c3 - (7=8)r7c8 - (8)r9c9 = (8)r5c9 => r5c9 <> 7
Remark: In the Almost Swordfish thread, I did find an Almost XYZ wing in the first position I gave, but I didn't mention it because it didn't seem to advance the solution. For any readers who want to try their hand at spotting these patterns, you might see if you can spot an Almost XYZ wing in that position. Hint: Be sure to do this before the elimination of (7)r3c5 which I show there, as that destroys the potential XYZ wing. As in this puzzle, there may be alternative "adjunct chains" which can be used.