dailysudoku.com

[Bud]

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.

[Bud]

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.

[daj95376]

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.

[keith]

[daj95376]

[keith]

Just adding.

Keith

[ronk]