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

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


2 Chain Logic Rules
I am a big fan of 2chain 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 2chain logic. I am going to begin by presenting the 2chain logic rules that I use. In doing this I am going to use the following definitions. First of all I call the two chains Achain and Bchain.
I call a locked set of cells LSa for the Achain and LSb for the Bchain. I call a single cell Ca for the Achain and Cb for the Bchain. 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 2chain logic techniques before I have. It is definitely not a mainstream technique but I don’t know why. In my 2chain posts in the old forum, I used only the Locked Sets Or Logic Rule, but at that time I called it “inclusiveor 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 Achain and Bchain 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 Swings and hybrid wings use 2 chains that start out from the bivalue pivot cell. In fact example 1 is an extended hybridwing.
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 Achain and y for the Bchain, 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 SWing which is also a 245 continuous loop. This continuous loops contains 5 cells which are part of the 2chain 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 hybridwing with pivot cell r7c4 and both the Achain and Bchain 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 Bchain. The Locked Sets Logic Rule applies to row 9. The AChain 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 Bchain: (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 
 5689 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 
 2689B 189A 126789B 5 4789B 49A 12489B 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

Posted: Mon Aug 08, 2011 1:28 pm Post subject: 


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 Achain or the Bchain 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 Achain and Bchain cells in box 47.
Achain: (4)r9c2=(2)r2c2=(9)r2c9 or Bchain: (4)r9c4r3c4=(4)r3c5=(1)r2c5=(78)r2c18 and (5)r2c4.
 
r5c2=(4)r6c1 (5)r9c3r7c1=(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)r9c3r7c1=(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 SWing 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 12789A 4  589B 1359B 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 
 457AB 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

Posted: Mon Aug 08, 2011 7:21 pm Post subject: 


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 = (26)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: 3285 Location: near Detroit, Michigan, USA

Posted: Mon Aug 08, 2011 10:38 pm Post subject: 


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

Posted: Mon Aug 08, 2011 11:11 pm Post subject: 


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: 3285 Location: near Detroit, Michigan, USA

Posted: Tue Aug 09, 2011 12:42 am Post subject: 


Just adding.
Keith 

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ronk
Joined: 07 May 2006 Posts: 398

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


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 hybridwing with pivot cell r7c4 and both the Achain and Bchain 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 Bchain. The Locked Sets Logic Rule applies to row 9. The AChain 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 Bchain: (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 doublylinked sets) that cause the locked set(s) in row 9? 

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