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