## Q: Chain Notation

Everything about Sudoku that doesn't fit in one of the other sections

### Q: Chain Notation

My understanding of chain notation is limited. Plowing through all of the threads on the topic, with their point and counter-point discussions, only leaves me confused ... not enlightened.

This is my (basic) understanding of Implication Chains/Streams and Inference Chains using chain notation. I have questions!

Code: Select all
Sample PM
*-----------------------------------------------------------------------------*
| 6       1       47      | 2       3       8       | 4579    479     4579    |
| 5       9       8       | 4       6       7       | 3       1       2       |
| 3       47      2       | 1       9       5       | 47      8       6       |
|-------------------------+-------------------------+-------------------------|
| 149     5       3       | 7       14      2       | 8       6       49      |
| 24789   2467    4679    | 3       48      469     | 2479    5       1       |
| 124789  2467    4679    | 68      5       1469    | 2479    3       479     |
|-------------------------+-------------------------+-------------------------|
| 249     3       469     |*68      7       146     |*14569   249    *4589    |
| 2479    2467    1       | 5       248     3       | 4679    2479    4789    |
| 247     8       5       | 9       124    *146     |*1467    247     3       |
*-----------------------------------------------------------------------------*

Code: Select all
Implication Chain/Stream: left-to-right

r7c4=6 => r7c9=8 => r7c7=5 => r9c7=1 => r9c6=6 => r7c4=8

Code: Select all
Inference Chain: bi-directional (w/ bivalue-cell information added)

(6=)[r7c4]=8=[r7c9]=5=[r7c7]=1=[r9c7]=6=[r9c6]-6-[r7c4]

Strong Link:  [cell_1]=a=[cell_2]  =>  if (cell_1 is not a) then (cell_2 is     a)
Weak   Link:  [cell_1]-a-[cell_2]  =>  if (cell_1 is     a) then (cell_2 is not a)

Q 1: Is this correct?

Q 2: To present a more complete picture of an Implication Chain/Stream, I'd like to place (l-r) in front of what appears to be an Implication Chain. Is this acceptable? Is there another way that already exists?

Finally, I'd like to present an alternate format for chain notation.

Code: Select all
Inference Chain: bi-directional, alternate format (w/ bivalue-cell information added)

(6=)[r7c4]-8=[r7c9]-5=[r7c7]-1=[r9c7]-6=[r9c6]=6-[r7c4]

Strong Link:  [cell_1]-a=[cell_2]  =>  if (cell_1 is not a) then (cell_2 is     a)
Weak   Link:  [cell_1]=a-[cell_2]  =>  if (cell_1 is     a) then (cell_2 is not a)

Q 3: Is there a case where the alternate format doesn't work? (Why do I have a feeling that I've asked this question before!)
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Re: Q: Chain Notation

daj95376 wrote:Finally, I'd like to present an alternate format for chain notation.

Code: Select all
Inference Chain: bi-directional, alternate format (w/ bivalue-cell information added)

(6=)[r7c4]-8=[r7c9]-5=[r7c7]-1=[r9c7]-6=[r9c6]=6-[r7c4]

Strong Link:  [cell_1]-a=[cell_2]  =>  if (cell_1 is not a) then (cell_2 is     a)
Weak   Link:  [cell_1]=a-[cell_2]  =>  if (cell_1 is     a) then (cell_2 is not a)

Q 3: Is there a case where the alternate format doesn't work? (Why do I have a feeling that I've asked this question before!)

When using the nice loop notation, isn't it sometimes the case that one would write a link in the following way (think of a UR)

[r3c1]=4|3=[r3c8] (which means that if r3c1<>4 then r3c8=3 and if r3c8<>3 then r3c1 = 4)

In this case, I'm not sure how to write it in your notation. If I write

[r3c1]-4|3=[r3c8],

then I get r3c1 <> 4 => r3c8 = 3, but the other direction is lost (and is even incorrect).
re'born

Posts: 551
Joined: 31 May 2007

### Re: Q: Chain Notation

daj95376 wrote:Q 3: Is there a case where the alternate format doesn't work? (Why do I have a feeling that I've asked this question before!)

If you didn't someone else did ... and my answer is the same as before. For your notation, the inferences are incorrect when reading right-to-left.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: Q: Chain Notation

ronk wrote:
daj95376 wrote:Q 3: Is there a case where the alternate format doesn't work? (Why do I have a feeling that I've asked this question before!)

If you didn't someone else did ... and my answer is the same as before. For your notation, the inferences are incorrect when reading right-to-left.

It was probably me. While out running errands, I realized that my notation wasn't bi-directional ... and that someone had told me so in the past. It was probably you in a pm ... and that's why I couldn't find it when I did a search of the forum.

I'm sorry for wasting your time ... AGAIN!!!
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Here is a finned X-Wing puzzle from Mike Barker's contributions to the zoo. It can be solved with Singles and any one of two Sashimi X-Wings.

Code: Select all
6..........7...64.....92.1...4..79...3..1.42.......1.6..32....5.4.5......2.......

#                  Singles
#   c28   -  7     X-Wing (finned/Sashimi)   (note: [r7 c7]<>7)
# r67     -  7     X-Wing (finned/Sashimi)   (note: [r89c8]<>7)
*-----------------------------------------------------------*
| 6     1     2     | 347   37    34    | 5     8     9     |
| 3     9     7     | 1     58    58    | 6     4     2     |
| 4     5     8     | 6     9     2     | 37    1     37    |
|-------------------+-------------------+-------------------|
| 1     6     4     | 38    2     7     | 9     5     38    |
| 78    3     5     | 9     1     6     | 4     2     78    |
| 2     78    9     | 348   358   3458  | 1     37    6     |
|-------------------+-------------------+-------------------|
| 9     78    3     | 2     4     1     | 8-7   6     5     |
| 78    4     6     | 5     378   389   | 2     39-7  1     |
| 5     2     1     | 378   6     389   | 378   39-7  4     |
*-----------------------------------------------------------*
#                  Singles

The two Sashimi X-Wings can be expressed as a chain or with grouped colors.

Code: Select all
Blue    Green  Blue   Green  Blue  -Blue
(7) [r89c8]-[r6c8]=[r6c2]-[r7c2]=[r7c7]-[r89c8]

Did I do this right? I know it's not bi-directional, but I don't see a way around it.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

daj95376 wrote:Here is a finned X-Wing puzzle from Mike Barker's contributions to the zoo. It can be solved with Singles and any one of two Sashimi X-Wings.

Code: Select all
6..........7...64.....92.1...4..79...3..1.42.......1.6..32....5.4.5......2.......

#                  Singles
#   c28   -  7     X-Wing (finned/Sashimi)   (note: [r7 c7]<>7)
# r67     -  7     X-Wing (finned/Sashimi)   (note: [r89c8]<>7)
*-----------------------------------------------------------*
| 6     1     2     | 347   37    34    | 5     8     9     |
| 3     9     7     | 1     58    58    | 6     4     2     |
| 4     5     8     | 6     9     2     | 37    1     37    |
|-------------------+-------------------+-------------------|
| 1     6     4     | 38    2     7     | 9     5     38    |
| 78    3     5     | 9     1     6     | 4     2     78    |
| 2     78    9     | 348   358   3458  | 1     37    6     |
|-------------------+-------------------+-------------------|
| 9     78    3     | 2     4     1     | 8-7   6     5     |
| 78    4     6     | 5     378   389   | 2     39-7  1     |
| 5     2     1     | 378   6     389   | 378   39-7  4     |
*-----------------------------------------------------------*
#                  Singles

The two Sashimi X-Wings can be expressed as a chain or with grouped colors.

Code: Select all
Blue    Green  Blue   Green  Blue  -Blue
(7) [r89c8]-[r6c8]=[r6c2]-[r7c2]=[r7c7]-[r89c8]

Did I do this right? I know it's not bi-directional, but I don't see a way around it.

Code: Select all

*-----------------------------------------------------------*
| 6     1     2     | 347   37    34    | 5     8     9     |
| 3     9     7     | 1     58    58    | 6     4     2     |
| 4     5     8     | 6     9     2     | 37*   1     37*   |
|-------------------+-------------------+-------------------|
| 1     6     4     | 38    2     7     | 9     5     38    |
| 78    3     5     | 9     1     6     | 4     2     78*   |
| 2     78*   9     | 348   358   3458  | 1     37*   6     |
|-------------------+-------------------+-------------------|
| 9     78*   3     | 2     4     1     | 78*   6     5     |
| 78    4     6     | 5     378   389   | 2     379   1     |
| 5     2     1     | 378   6     389   | 378+  379   4     |
*-----------------------------------------------------------*

r9c7 is the only guardian against a length 7 conjugate chain, and therefore r9c7=7.
re'born

Posts: 551
Joined: 31 May 2007

Hmmm! I thought my grouped conjugate chain of length 5 was sufficient. I included the Blue/Green notation because it could be interpreted as grouped Colors, which is akin to how Simple Sudoku handles the 7s.

I haven't implemented Colors (yet) in the new version of my solver that I'm writing. That's why the two Sashimi X-Wings in the solution above. While cross-checking it with chains, I couldn't decide on the best way to represent it as a chain.

Thanks for the alternate solution ... AND ... sorry about your losing all of your posts under your old account. It should have been my account because then there wouldn't have been any great loss!
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

daj95376 wrote:
Code: Select all
Blue    Green  Blue   Green  Blue  -Blue
(7) [r89c8]-[r6c8]=[r6c2]-[r7c2]=[r7c7]-[r89c8]

Did I do this right? I know it's not bi-directional, but I don't see a way around it.

I suppose you could just break it up as two nice loops:
Code: Select all
1. [r89c8]-7-[r6c8]=7=[r6c2]-7-[r7c2]=7=[r7c7]-7-[r89c8], => r89c8<>7

2. [r7c7]-7-[r7c2]=7=[r6c2]-7-[r6c8]=7=[r89c8]-7-[r7c7], => r7c7<>7

Or you could combine everything if you use one of Carcul's famous TILA's
Code: Select all
7-[r9c7]-{TILA: [r7c7]-7-[r7c2]=7=[r6c2]-7-[r6c8]=7=[r89c8]-7-[r7c7]; [r7c7]=7=[r7c2]-7-[r6c2]=7=[r6c8]-7-[r89c8]=7=[r7c7]}, => r9c7=7

I like the idea of thinking of this as a grouped broken wing with r9c7 as the sole guardian.
re'born

Posts: 551
Joined: 31 May 2007