22 April 2007, Grouped AIC's and Rings
Posted: Wed Apr 25, 2007 8:40 pm
Start position for the 22 April 2007 Nightmare:
I found this one tougher than most of the Sunday Nightmares. I had occasion to use several "grouped" Alternating Inference Chains (AIC's), in which the nodes of the chains are not necessarily single cells, but possibly groups of cells. Grouped (multi-cell) nodes may be of two types -- nodes which contain Almost Locked Sets (ALS's), and grouped nodes for a single digit.
In an AIC, a digit X may be used to weakly link an ALS node to any other type of node if both nodes contain X as a candidate and all occurrences of candidate X in the two nodes lie in some common house. Thus the digit X cannot exist in both nodes. When digit X weakly links two ALS nodes, it's usually called a "restricted common" digit for those ALS's.
A multi-cell node for a single digit, say digit X, consists of two or three aligned cells, all containing X as a candidate, in some box. Again, a node of this type is weakly linked to another node of any type, containing X as a candidate, if all X candidates in the two nodes lie in some common house. Thus X cannot exist in both nodes. The link is a strong conjugate link (which can be used as a link of weak inference or strong inference in an AIC) if the two nodes together contain all X candidates in that house. Thus, one or the other of the two nodes must contain X.
Sometimes an AIC will loop back to its starting point to form a ring structure in which the alternation of links of strong and weak inference is continuously maintained around the loop. Here I'll use the term "AIC ring" for such a structure. It's a good idea to learn to recognize these as they can sometimes produce many eliminations. Generally, for any AIC ring, we can conclude that the links of weak inference in the ring are also links of strong inference as well. One way to see this is to imagine breaking the ring at any weak link. The remaining chain is an AIC, beginning and ending with a strong link, which connects the two nodes of the broken link and therefore establishes a strong link between them. There are some AIC rings in this solution.
Since this solution is somewhat lengthy, when possible I've tried to use one diagram to illustrate several successive steps. To avoid excessive clutter and confusion, I usually haven't tried to label all cells involved in the various chains or patterns, or show all the eliminations from all steps. When I have shown specific eliminations, I've followed the convention of listing all eliminated digits to the right of the "-" symbol.
Getting down to specifics, basic techniques bring the grid to this position:
Uniqueness-based techniques often produce good yield so the solution begins with the "unique corner" eliminations shown in r2c1, from the "34" based UR pattern in r23c19 (marked with "*"). This leaves (4)r2c9 as a hidden single in row 2. With digit 3 thus eliminated from r2c19, there is an X-wing for digit 3 in r24c36 (marked with "#"), giving the eliminations shown in r15c3, r56c6.
Note that an X-wing is a simple example of an AIC ring. For this one we have:
After basic follow-up:
Here we have the grouped AIC
It's acceptable to group the first two nodes together, as shown below, and in fact this would be the way to view the pattern as an ALS XZ rule position.
With the candidates in r9c3 reduced to "58," there is a naked "578" triple in r9c359, which eliminates (5)r9c2.
This reduces the r9c2 candidates to "26," so we have a short XY chain:
Here we have another AIC ring, this one containing two grouped nodes. The nodes in the ring are marked with letters A (cell r2c1) through F (cells r2c46). Nodes B and F are the only multi-cell nodes -- node B is an ALS node, and node F is a grouped node for digit 5.
In this position there is a grouped AIC for digit 6. In the diagram, the nodes in the chain are labelled A (r5c1) through F (r5c9), with node B being the only grouped node.
Note that this could be "closed up" to form an AIC ring for digit 6, but this doesn't add anything in this case since all the links of weak inference in the chain are already strong conjugate links.
Independent of this AIC, there is also this non-grouped AIC:
Here we have:
Also, after the eliminations of (2)r5c3 and (2)r6c2 (which are shown in the diagram) from the preceding two chains, there is a short XY chain:
Here we have:
Code: Select all
. . .|4 . 8|. . 1
. . .|. 6 .|2 7 .
. . 6|. . 2|8 5 .
-----+-----+-----
7 . .|. . .|1 . 9
. 8 .|. . .|. 4 .
9 . 4|. . .|. . 5
-----+-----+-----
. 4 9|2 . .|5 . .
. 3 7|. 8 .|. . .
1 . .|3 . 9|. . .
In an AIC, a digit X may be used to weakly link an ALS node to any other type of node if both nodes contain X as a candidate and all occurrences of candidate X in the two nodes lie in some common house. Thus the digit X cannot exist in both nodes. When digit X weakly links two ALS nodes, it's usually called a "restricted common" digit for those ALS's.
A multi-cell node for a single digit, say digit X, consists of two or three aligned cells, all containing X as a candidate, in some box. Again, a node of this type is weakly linked to another node of any type, containing X as a candidate, if all X candidates in the two nodes lie in some common house. Thus X cannot exist in both nodes. The link is a strong conjugate link (which can be used as a link of weak inference or strong inference in an AIC) if the two nodes together contain all X candidates in that house. Thus, one or the other of the two nodes must contain X.
Sometimes an AIC will loop back to its starting point to form a ring structure in which the alternation of links of strong and weak inference is continuously maintained around the loop. Here I'll use the term "AIC ring" for such a structure. It's a good idea to learn to recognize these as they can sometimes produce many eliminations. Generally, for any AIC ring, we can conclude that the links of weak inference in the ring are also links of strong inference as well. One way to see this is to imagine breaking the ring at any weak link. The remaining chain is an AIC, beginning and ending with a strong link, which connects the two nodes of the broken link and therefore establishes a strong link between them. There are some AIC rings in this solution.
Since this solution is somewhat lengthy, when possible I've tried to use one diagram to illustrate several successive steps. To avoid excessive clutter and confusion, I usually haven't tried to label all cells involved in the various chains or patterns, or show all the eliminations from all steps. When I have shown specific eliminations, I've followed the convention of listing all eliminated digits to the right of the "-" symbol.
Getting down to specifics, basic techniques bring the grid to this position:
Code: Select all
.---------------------.-----------------------.---------------------.
| 235 257 25-3 | 4 357 8 | 6 9 1 |
|*58-34 159 #1358 | 159 6 #135 | 2 7 *4-3 |
|*34 179 6 | 179 179 2 | 8 5 *34 |
:---------------------+-----------------------+---------------------:
| 7 256 #235 | 568 4 #356 | 1 268 9 |
| 2356 8 125-3 | 15679 123579 1567-3 | 37 4 26 |
| 9 126 4 | 1678 1237 167-3 | 37 268 5 |
:---------------------+-----------------------+---------------------:
| 68 4 9 | 2 17 167 | 5 3 78 |
| 256 3 7 | 56 8 4 | 9 1 26 |
| 1 256 258 | 3 57 9 | 4 26 78 |
'---------------------'-----------------------'---------------------'
Note that an X-wing is a simple example of an AIC ring. For this one we have:
- (3): r2c3 = r2c6 - r4c6 = r4c3 - r2c3 => AIC ring => r15c3, r56c6 <> 3
After basic follow-up:
Code: Select all
.---------------------.---------------------.---------------------.
| 235 257 25 | 4 357 8 | 6 9 1 |
| 58 159 1358 | 159 6 135 | 2 7 4 |
| 4 179 6 | 179 179 2 | 8 5 3 |
:---------------------+---------------------+---------------------:
| 7 256 235 | 568 4 356 | 1 268 9 |
| 2356 8 125 | 15679 123579 1567 | 37 4 26 |
| 9 126 4 | 1678 1237 167 | 37 268 5 |
:---------------------+---------------------+---------------------:
| 68 4 9 | 2 17 167 | 5 3 78 |
| 256 3 7 | 56 8 4 | 9 1 26 |
| 1 256 58-2 | 3 57 9 | 4 26 78 |
'---------------------'---------------------'---------------------'
- (2=5)r1c3 - (5=8)r2c1 - (8=562)r78c1|r9c2 => r9c3 <> 2
It's acceptable to group the first two nodes together, as shown below, and in fact this would be the way to view the pattern as an ALS XZ rule position.
- (2=58)r1c3|r2c1 - (8=562)r78c1|r9c2 => r9c3 <> 2
With the candidates in r9c3 reduced to "58," there is a naked "578" triple in r9c359, which eliminates (5)r9c2.
This reduces the r9c2 candidates to "26," so we have a short XY chain:
- (2=5)r1c3 - (5=8)r2c1 - (8=6)r7c1 - (6=2)r9c2 => r1c2 <> 2
Code: Select all
.----------------------.-----------------------.---------------------.
| 23-5 57 25 | 4 E357 8 | 6 9 1 |
| A58 19-5 138-5 |F159 6 F135 | 2 7 4 |
| 4 179 6 | 179 179 2 | 8 5 3 |
:----------------------+-----------------------+---------------------:
| 7 256 235 | 568 4 356 | 1 268 9 |
| 236-5 8 125 | 15679 12379-5 1567 | 37 4 26 |
| 9 126 4 | 1678 1237 167 | 37 268 5 |
:----------------------+-----------------------+---------------------:
| B68 4 9 | 2 17 167 | 5 3 78 |
| B256 3 7 |C56 8 4 | 9 1 26 |
| 1 B26 58 | 3 D57 9 | 4 26 78 |
'----------------------'-----------------------'---------------------'
- (5=8)r2c1 - (8=265)r78c1|r9c2 - (5)r8c4 = (5)r9c5 - (5)r1c5 = (5)r2c46 - (5)r2c1 => AIC ring
- (5)r9c5 = (5)r1c5, eliminating (5)r5c5
- (5)r2c46 = (5)r2c1, eliminating (5)r2c23
Code: Select all
.------------------.----------------------.------------------.
| 23 57 25 | 4 357 8 | 6 9 1 |
| 58 19 138 | 159 6 135 | 2 7 4 |
| 4 179 6 | 179 179 2 | 8 5 3 |
:------------------+----------------------+------------------:
| 7 B256 235 | 568 4 356 | 1 268 9 |
|A236 8 125 | 1579-6 12379 157-6 | 37 4 F26 |
| 9 B126 4 | 1678 1237 167 | 37 268 5 |
:------------------+----------------------+------------------:
| 68 4 9 | 2 17 167 | 5 3 78 |
| 256 3 7 | 56 8 4 | 9 1 E26 |
| 1 C26 58 | 3 57 9 | 4 D26 78 |
'------------------'----------------------'------------------'
- (6): r5c1 = r46c2 - r9c2 = r9c8 - r8c9 = r5c9 => r5c46 <> 6
Note that this could be "closed up" to form an AIC ring for digit 6, but this doesn't add anything in this case since all the links of weak inference in the chain are already strong conjugate links.
Independent of this AIC, there is also this non-grouped AIC:
- (5=2)r1c3 - (2=3)r1c1 - (3)r2c3 = (3)r4c3 => r4c3 <> 5
- (3=7)r6c7 - (7=3)r5c7 - (3=62)r5c19 - (2)r5c5 = (2)r6c5 => r6c5 <> 3
Code: Select all
.-----------------.---------------.---------------.
| 23 57 25 | 4 357 8 | 6 9 1 |
| 58 19 138 | 159 6 135 | 2 7 4 |
| 4 179 6 | 179 179 2 | 8 5 3 |
:-----------------+---------------+---------------:
| 7 256 23 | 568 4 356 | 1 268 9 |
| 236 8 15-2 | 159 1239 15 | 7 4 26 |
| 9 16-2 4 | 1678 127 167 | 3 268 5 |
:-----------------+---------------+---------------:
| 68 4 9 | 2 17 167 | 5 3 78 |
| 256 3 7 | 56 8 4 | 9 1 26 |
| 1 26 58 | 3 57 9 | 4 26 78 |
'-----------------'---------------'---------------'
- (2=3)r4c3 - (3)r4c6 = (3-2)r5c5 = (2)r6c5 => r6c2 <> 2
- (2=3)r4c3 - (3=62)r5c19 => r5c3 <> 2
Also, after the eliminations of (2)r5c3 and (2)r6c2 (which are shown in the diagram) from the preceding two chains, there is a short XY chain:
- (6=1)r6c2 - (1=5)r5c3 - (5=8)r9c3 - (8=6)r7c1 => r5c1, r9c2 <> 6
Code: Select all
.---------------.-----------------.---------------.
| 23 57 25 | 4 357 8 | 6 9 1 |
| 58 9 138 | 15 6 13-5 | 2 7 4 |
| 4 17 6 | 17 9 2 | 8 5 3 |
:---------------+-----------------+---------------:
| 7 56 23 | 68-5 4 356 | 1 28 9 |
| 23 8 15 | 9 23 15 | 7 4 6 |
| 9 16 4 | 1678 127 167 | 3 28 5 |
:---------------+-----------------+---------------:
| 68 4 9 | 2 17 167 | 5 3 78 |
| 56 3 7 | 56 8 4 | 9 1 2 |
| 1 2 58 | 3 57 9 | 4 6 78 |
'---------------'-----------------'---------------'
- (5=1)r2c4 - (1)r2c3 = (1-5)r5c3 = (5)r5c6 => r2c6, r4c4 <> 5
- (3=1)r2c6 - (1=5)r2c4 - (5=6)r8c4 - (6=8)r4c4 - (2=3)r4c3 => r2c3, r4c6 <> 3