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2 Chain Logic Rules with Examples

 
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Bud



Joined: 06 May 2010
Posts: 47
Location: Tampa, Florida

PostPosted: Mon Aug 08, 2011 1:18 pm    Post subject: 2 Chain Logic Rules with Examples Reply with quote

2 Chain Logic Rules

I am a big fan of 2-chain logic, primarily because it can result in a lot of cell eliminations. It can also result in forcing but that topic is well covered in all of the solving technique sites and will not be covered here. From my experience cell eliminations are much more likely to occur with 2-chain logic. I am going to begin by presenting the 2-chain logic rules that I use. In doing this I am going to use the following definitions. First of all I call the two chains A-chain and B-chain.
I call a locked set of cells LSa for the A-chain and LSb for the B-chain. I call a single cell Ca for the A-chain and Cb for the B-chain. Note that in order to keep track of which cells belong with each chain, I use coloring or A and B labeling. It should be noted that Aran and probably others have used 2-chain logic techniques before I have. It is definitely not a mainstream technique but I don’t know why. In my 2-chain posts in the old forum, I used only the Locked Sets Or Logic Rule, but at that time I called it “inclusive-or logic”. Those examples were lost when the site crashed, but the examples in this post are more powerful because I have increased the number of logic rules I use.

Linkage Rule: In order to be useful the A-chain and B-chain must be linked in such a way that either one chain or the other is true. I use three different types of linkage to accomplish this.
1) Bivalue cell with digits xy. Chain A starts out with x and Chain B starts out with y. All extended S-wings and hybrid wings use 2 chains that start out from the bivalue pivot cell. In fact example 1 is an extended hybrid-wing.
2) Conjugate x. Each chain starts out with the x on opposite conjugate cells. Example 2 uses this type of linkage.
3) Weak link x. Each chain starts out with not x on opposite weak link cells. Extended turbot fish patterns and extended purple cows use 2 chains with this type of linkage. Both of these are covered in one of my previous posts.

And Logic Rule: If any cell Ca=x and Cb=x, then x can be eliminated from any cell that sees both of these cells cannot be x. This follows from the fact that either Ca=x or Cb=x must be true. There are 2 instances of this in example 1.

Locked Sets Or Logic Rule: If LSa and LSb are in the same house, then none of the digits in LSa can be in the cells of LSb and vice versa. This follows from the fact that either LSa or LSb must be true. Both Example 1 and Example 2 get most of their call eliminations from this rule. In Example 2 all of the cells in row 2 are either in LSa and LSb which means that these are also hidden locked sets for row 2.

Hidden Bivalue Cell Rules
1) If a single cell is x for the A-chain and y for the B-chain, then all digits except xy can be eliminated from the cell. This occurs in Example 2
2) If Ca=x and Cb=y are in the same house and if there is a strong link z between these cells, then Ca=xz and Cb=yz and all other degits can be eliminated from these cells.at least part of the 2 chain pattern is a continuous loop. Simple examples of this rule are the 3 digit continuous loops that can occur inside of the S=Wing and the Purple Cow.
In example 2 rule 1 creates a bivalue cell which is the pivot for a 45 S-Wing which is also a 245 continuous loop. This continuous loops contains 5 cells which are part of the 2-chain pattern.
After using either rule, It is a good idea to check for continuous loops in which part of the of the loop includes these hidden bivalue cells.

Example 1. Chains start at bivalue cell.

100007000060900701700800000003008040070000320020300500000002006400106070000500003

After making basic moves the puzzle is as shown in the diagram. Part of this example is an extended hybrid-wing with pivot cell r7c4 and both the A-chain and B-chain start at this cell. Here the 4 digit in the pivot cell is the first cell in the A=chain and the 7 digit is the first cell in the B-chain. The Locked Sets Logic Rule applies to row 9. The A-Chain branch in column 4 is used to create And Logic Rules in which cells (6)r5c4 and (2)r1c4 are combined with row 9 cells (2)r9c7 and (6)r9c1 which => r5c1<>6 and r1c7<2>(9)r9c6=>(18)r9c28 or B-chain: (7)r7c4 -r9c5=>(7)r9c3=>(6)r9c1=>(2)r9c7
||
(6)r5c4=>(2)r1c4

For row 9 LSa=189 in c286 and LSb =267 in c137. Thus from the Locked Set Or Logic Rule r9c3,r9c7 <>189 and r9c1<>89. Note that the total number of eliminations is 10 with 5 digits, a powerful move.

Code:

+-------------------------+-------------------+---------------------+
| 1        34589  2589    | 26A  256     7    |-2689     35689  245 |
| 2358     6      258     | 9    245     345  | 7        358    1   |
| 7        3459   259     | 8    1256    135  | 269      3569   245 |
+-------------------------+-------------------+---------------------+
| 569     159     3       | 267  125679  8    | 16       4      79  |
| 5-689    7      145689  | 46A  14569   1459 | 3        2      89  |
| 689      2      14689   | 3    14679   149  | 5        16     789 |
+-------------------------+-------------------+---------------------+
| 3589    13589  15789    | 47AB 34789   2    | 1489     1589   6   |
| 4       3589   2589     | 1    389     6    | 289      7      25  |
| 26-8-9B 189A  -1267-8-9B| 5    4789B   49A  |-124-8-9B 189A   3   |
+-------------------------+-------------------+---------------------+


Last edited by Bud on Mon Aug 08, 2011 1:46 pm; edited 2 times in total
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Bud



Joined: 06 May 2010
Posts: 47
Location: Tampa, Florida

PostPosted: Mon Aug 08, 2011 1:28 pm    Post subject: Reply with quote

Example 2. Chains start at conjugate.

300270005004006000050000000200004093900000006030100008003000020000300500600081937

After making basic moves the puzzle is as shown in the diagram. In this example I start one chain at each cell of the 4 conjugate in row 9. This ensures that either the A-chain or the B-chain must be true. The chains give me two locked sets LSb=1345678 or LSa=29 in row 2, plus a hidden bivalue cell rule 1 (245 continuous loop branch consisting of the 5 A-chain and B-chain cells in box 47.

A-chain: (4)r9c2=(2)r2c2=(9)r2c9 or B-chain: (4)r9c4-r3c4=(4)r3c5=(1)r2c5=(78)r2c18 and (5)r2c4.
| ||
r5c2=(4)r6c1 (5)r9c3-r7c1=(5)r6c1

This => r2c2<>178 and r245<>9 (Lsa or LSb) and r6c1<>7 (hidden bivalue cell rule 1) and r78c2<>4 (Continuous Loop). Score=9 eliminations in 5 digits.

To verify the continuous loop (5)r9c3-r7c1=(5)r6c1=(4)r5c2=(2)r9c2. I prefer to find it the following way. The hidden cell rule 1 makes r6c1 a bivalue cell 45 which is a pivot for a 45 S-Wing that uses all the A and B chain cells in box 47 and is also a 245 continuous loop because of the strong 2 link in row 9.

Code:

+------------------------+---------------------+-----------------+
| 3      1689     1689   | 2     7       89    | 1468   148   5  |
| 178B  -12-7-89A 4      | 58-9B 135-9B  6     | 1378   178B  29A|
| 178    5        126789 | 489B  1349B   389   | 13678  1678  29 |
+------------------------+---------------------+-----------------+
| 2      1678     15678  | 5678  56      4     | 17     9     3  |
| 9      1478A    1578   | 78    235     23578 | 1247   1457  6  |
| 45-7AB 3        567    | 1     269     2579  | 247    457   8  |
+------------------------+---------------------+-----------------+
| 14578B 14789    3      | 5679  4569    579   | 68     2     14 |
| 1478   14789    1789   | 3     2469    279   | 5      68    14 |
| 6      24A      25B    | 45B   8       1     | 9      3     7  |
+------------------------+---------------------+-----------------+
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Mon Aug 08, 2011 7:21 pm    Post subject: Reply with quote

Your 2 Chain Logic Rules strongly resemble the very old definition of Double Implication Chain, which is mentioned as the second scenario in Sudopedia's section on Solving Techniques.

Sudopedia wrote:
Double Implication Chain | DIC

There are 2 interpretations circulating. The first originates from a reliable source [1].
    * A Forcing Chain that has implications in both directions.

    * 2 Forcing Chains starting from a bivalue cell or a bilocal unit showing a verity.


BTW: There's an interesting (and useful) chain in your first puzzle.

Code:
 grid for Example 1 (after basics)
 +--------------------------------------------------------------------------------+
 |  1       34589   2589    |  26      256     7       |  2689    35689   245     |
 |  235     6       258     |  9       245     345     |  7       358     1       |
 |  7       3459    259     |  8       1256    135     |  269     3569    245     |
 |--------------------------+--------------------------+--------------------------|
 |  569     159     3       |  267     125679  8       |  16      4       79      |
 |  5689    7       146     |  46      14569   1459    |  3       2       89      |
 |  689     2       146     |  3       14679   149     |  5       16      789     |
 |--------------------------+--------------------------+--------------------------|
 |  359     13589   17      |  47      34789   2       |  1489    1589    6       |
 |  4       3589    2589    |  1       389     6       |  289     7       25      |
 |  269     189     167     |  5       4789    49      |  12489   189     3       |
 +--------------------------------------------------------------------------------+
 # 131 eliminations remain

(5689=1)r456c1+r4c2 - (1=6)r4c7 - (6=289)r13+8c7 - (2)r9c7 = (2-6)r9c1 = (6)r9c3 => r56c3<>6

Note: I use (+) to separate the cells for the single value from other cells in the naked subset.



Note: It's considered poor form to not reduce a puzzle through basics before discussing an advanced step/technique.
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Mon Aug 08, 2011 10:38 pm    Post subject: Reply with quote

daj95376 wrote:
Note: It's considered poor form to not reduce a puzzle through basics before discussing an advanced step/technique.


And, furthermore, to ignore simpler "advanced" techniques when describing a complex method.

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Mon Aug 08, 2011 11:11 pm    Post subject: Reply with quote

keith wrote:
daj95376 wrote:
Note: It's considered poor form to not reduce a puzzle through basics before discussing an advanced step/technique.


And, furthermore, to ignore simpler "advanced" techniques when describing a complex method.

Were you adding to my comment, or did I miss something?
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Tue Aug 09, 2011 12:42 am    Post subject: Reply with quote

Just adding.

Keith
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ronk



Joined: 07 May 2006
Posts: 397

PostPosted: Wed Aug 10, 2011 7:35 pm    Post subject: Re: 2 Chain Logic Rules with Examples Reply with quote

Bud wrote:
Example 1. Chains start at bivalue cell.

100007000060900701700800000003008040070000320020300500000002006400106070000500003

After making basic moves the puzzle is as shown in the diagram. Part of this example is an extended hybrid-wing with pivot cell r7c4 and both the A-chain and B-chain start at this cell. Here the 4 digit in the pivot cell is the first cell in the A=chain and the 7 digit is the first cell in the B-chain. The Locked Sets Logic Rule applies to row 9. The A-Chain branch in column 4 is used to create And Logic Rules in which cells (6)r5c4 and (2)r1c4 are combined with row 9 cells (2)r9c7 and (6)r9c1 which => r5c1<>6 and r1c7<2>(9)r9c6=>(18)r9c28 or B-chain: (7)r7c4 -r9c5=>(7)r9c3=>(6)r9c1=>(2)r9c7
||
(6)r5c4=>(2)r1c4

For row 9 LSa=189 in c286 and LSb =267 in c137. Thus from the Locked Set Or Logic Rule r9c3,r9c7 <>189 and r9c1<>89. Note that the total number of eliminations is 10 with 5 digits, a powerful move.

Bud, where in that writeup do you identify the continuous loop (or doubly-linked sets) that cause the locked set(s) in row 9?
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