The New Sudoku Players' Forum

[JeffInCA]

I'm trying to get my mind around the concept of coloring and conjugates, and I have a few questions given the following diagram which lists all of the candidates for 8 in a puzzle

Code: Select all

 *-----------------------------------------------------------*
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | .     .     .     |
 |-------------------+-------------------+-------------------|
 | .     8A    .     | .     8     .     | .     .     8     |
 | .     8a    .     | 8B    .     .     | .     .     8     |
 | .     .     .     | .     8     .     | .     .     8     |
 |-------------------+-------------------+-------------------|
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | 8b    8B    .     | .     .     .     |
 | .     .     .     | .     .     .     | .     .     .     |
 *-----------------------------------------------------------*


I have identified two separate conjugate chains, one identified with Aa the other with Bb

My questions are the following:

1) What if anything can we deduce about the 8's in c5 based on these two chains using coloring principles?

2) There is another conjugate pair at r6c5 and r6c9 that, if I'm understanding the concept correctly, is a separate chain from the other two. Would we then label these as Cc? Is there any value in doing so, as I'm not sure how the principles apply to 3 separate chains, if at all

3) Back to the 5 cells labeled with the AaBb? Do these form a Turbot Fish pattern?

4) From a concepts perspective is it correct to say that the Turbot Fish pattern is a specialized case of the coloring / conjugate concept?

Thanks in advance for your help

Jeff

[jphamel]

I'm not to much experienced either but I would try for r4c5 and r5c9

I might be wrong but removing r4c5 would link all the chain allowing you to remove r5c9

If anyone knows better, let me know I'm still learning too.

JPHamel

[tso]

1) What if anything can we deduce about the 8's in c5 based on these two chains using coloring principles?

Nothing.

2) There is another conjugate pair at r6c5 and r6c9 that, if I'm understanding the concept correctly, is a separate chain from the other two. Would we then label these as Cc? Is there any value in doing so, as I'm not sure how the principles apply to 3 separate chains, if at all

Yes there is a third chain. Any number of chains may be identified. In this case, no deduction can be made.

3) Back to the 5 cells labeled with the AaBb? Do these form a Turbot Fish pattern?

No. Turbot fish are five cells that form a closed loop -- each cell is connected to two others. Among these five cells, r4c2 and r8c5 are only connected to one other cell.

4) From a concepts perspective is it correct to say that the Turbot Fish pattern is a specialized case of the coloring / conjugate concept?

I would say yes.

[jphamel]

Yes there is a third chain. Any number of chains may be identified. In this case, no deduction can be made.

You mean that when there is 3 or more chains no deduction can be made?

If so, that is new to me, I didn't see this mentioned anywhere. Or I did not read all the docs I could find.

JPHamel

[tso]

Yes there is a third chain. Any number of chains may be identified. In this case, no deduction can be made.

You mean that when there is 3 or more chains no deduction can be made?

If so, that is new to me, I didn't see this mentioned anywhere. Or I did not read all the docs I could find.

JPHamel

No -- just that in this particular case, no deduction can be made.

[Jeff]

Hi JeffInCA, the post on x-cycle provides answers to all your questions. x-cycle generalises simple colouring and turbot fish.

The 3 chains you have identified consist of conjugate links. Simple colouring is the only technique that makes use of pure conjugate links. Since there is no close loop of pure conjugate links, there is no deduction due to simple colouring.

Turbot fish uses conjugate links as well as unconditional links where:

a conjugate link has exactly 2 candidate in the unit
an unconditional link has 2 or more candidates in the unit

These definitions tell us that a conjugate link can be treated as an unconditional link.

Let a conjugate link be U or C
Let an unconditional link be U

A deduction is possible if one of the following chain pattern (with alternative U-link and C-link) can be identified:

[cell 1]-U-[cell 2]-C-[cell 3]-U-[cell 4]-C-[cell 5]-U-[cell 1]
implies candidate exclusion at cell 1

[cell 1]-C-[cell 2]-U-[cell 3]-C-[cell 4]-U-[cell 5]-C-[cell 1]
implies candidate inclusion at cell 1

In your candidate grid, although there are 3 conjugate link segments, no such chain patterns can be found. Therefore, no deduction due to turbot fish either.

[JeffInCA]

Thanks for all of your replies.

I've been looking at this a bit further. I actually took this example from a puzzle from an earlier post that indicated the puzzle was solvable using Turbot Fish

After some further inspection I think the Turbot Fish is in the 7's as shown below

Code: Select all

 *-----------------------------------------------------------*
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | .     7     .     | 7     7a    .     |
 | .     .     .     | 7A    .     .     | 7a    .     .     |
 |-------------------+-------------------+-------------------|
 | .     .     7     | .     7     .     | 7     .     .     |
 | .     .     .     | 7a    .     .     | .     7A    .     |
 | .     .     7     | .     7     .     | 7     .     .     |
 |-------------------+-------------------+-------------------|
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | .     .     .     |
 *-----------------------------------------------------------*


I've performed a "coloring" on the cells using Aa. And if I'm not mistaken, these 5 cells also form a Turbot Fish.

Using coloring, I believe we are able to conclude that A=true and a=false since r2c8 and r3c7 can't both be true.

This also let's us conclude that 7's can be placed in the "A" cells at r3c4 and r5c8 (as well as at r2c7 since r2c8 & r3c7 must be false).

If I'm understanding correctly, using Turbot Fish essentially leads us essentially the same conclusion.

At the end of all of this, it seems to me that the general simple coloring principles are easier to follow than the more specific Turbot Fish pattern, which requires a little more thought and inspection to spot.

However, I'm very interested to here others' opinions on this topic.

Jeff

[Jeff]

Using coloring, I believe we are able to conclude that A=true and a=false since r2c8 and r3c7 can't both be true. ..............If I'm understanding correctly, using Turbot Fish essentially leads us essentially the same conclusion.

Hi Jeff, This is because simple colouring of length 5 is a subset of turbot fish. Don't forget turbot fish considers unconditional links as well as conjugate links.

[JeffInCA]

So are you saying that Turbot Fish is more powerful than Simple Coloring?

I guess I'm having trouble "spotting" the Turbot Fish pattern vs. simple coloring which is so far very intuitive for me.

I'm trying to decide if I should learn to memorize how to spot the Turbot Fish pattern or just rely on my understanding of the colroing principle.

If you have any additional insight, I'd appreciate it.

Jeff

[Jeff]

So are you saying that Turbot Fish is more powerful than Simple Coloring?

Absolutely.

I guess I'm having trouble "spotting" the Turbot Fish pattern vs. simple coloring which is so far very intuitive for me.

I'm trying to decide if I should learn to memorize how to spot the Turbot Fish pattern or just rely on my understanding of the colroing principle.

My advice is to learn and understand the generalised principle and then reconcile it with simple colouring and turbot fish. These techniques are just a special case of forcing chains called x-cycle.

[JeffInCA]

Thanks. I'll read up on that.

OK. One more example. Sorry if I'm getting redundant here but I just want to make sure I am understanding all the facets of this.

Given the following set of candidate cells for "3"

Code: Select all

 *-----------------------------------------------------------*
 | .     .     .     | .     .     .     | .     .     .     |
 | 3a    .     .     | .     .     .     | .     3A    .     |
 | .     .     3A    | .     .     .     | .     3a    .     |
 |-------------------+-------------------+-------------------|
 | .     .     3a    | .     3A    .     | .     .     .     |
 | 3     .     .     | .     3a    .     | .     3b    .     |
 | .     3b    .     | .     .     .     | 3B    .     .     |
 |-------------------+-------------------+-------------------|
 | .     .     .     | .     .     .     | .     .     .     |
 | 3b    3B    .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | 3b    .     3B    |
 *-----------------------------------------------------------*


There are 2 conjugate chains denoted by coloring with Aa and Bb. Also, there is one candidate cell at r5c1 that is not a part of either chain.

From what I can tell, there is no deduction that can be made from this layout on the basis of simple coloring, as tempting as it is to try to conclude something in box 4, this seems to be non-deterministic, and the solution must lie elsewhere.

Am I right?

[Jeff]

From what I can tell, there is no deduction that can be made from this layout on the basis of simple coloring, as tempting as it is to try to conclude something in box 4, this seems to be non-deterministic, and the solution must lie elsewhere.

Simple colouring implies r5c8<>3.

Nice loop notation of this x-cycle is:

[r5c8]-3-[r5c5]=3=[r4c5]-3-[r4c3]=3=[r3c3]-3-[r2c1]=3=[r2c8]-3-[r5c8] => r5c8<>3

[JeffInCA]

Oops... I had a typo in my grid. 3 is not a candidate in r5c8, but rather r5c9.

Try again with this one. Sorry.

Code: Select all

 *-----------------------------------------------------------*
 | .     .     .     | .     .     .     | .     .     .     |
 | 3a    .     .     | .     .     .     | .     3A    .     |
 | .     .     3A    | .     .     .     | .     3a    .     |
 |-------------------+-------------------+-------------------|
 | .     .     3a    | .     3A    .     | .     .     .     |
 | 3     .     .     | .     3a    .     | .     .     3b    |
 | .     3b    .     | .     .     .     | 3B    .     .     |
 |-------------------+-------------------+-------------------|
 | .     .     .     | .     .     .     | .     .     .     |
 | 3b    3B    .     | .     .     .     | .     .     .     |
 | .     .     .     | .     .     .     | 3b    .     3B    |
 *-----------------------------------------------------------*



[Jeff]

Try again with this one.

You are right. There is no deduction possible on the basis of simple colouring nor x-cycle.

[JeffInCA]

Cool. Thanks for the confirmation.