## Remote pairs, with an offset?

Advanced methods and approaches for solving Sudoku puzzles

### Remote pairs, with an offset?

Code: Select all
`. . . | . 2 4 | . 9 .. . 6 | 8 . . | 3 . .. 7 . | . . . | . . .---------------------. . . | . 3 9 | 8 . .. 9 4 | 5 . 8 | 1 2 .. . 8 | 4 1 . | . . .---------------------. . . | . . . | . 6 .. . 5 | . . 3 | 4 . .. 6 . | 7 5 . | . . .`

This puzzle that I posted can with simple methods get to here.

I would use skyscraper in c2r67 versus c8r69 to eliminate r7c9 3 and r9c3 3.

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567 *235   8     | 4     1     67   | 5679 *357   35679  |:--------------------+------------------+--------------------:| 12378 *1238  1237  | 9     4     12   | 257   6     1257-3 || 1279   12    5     | 6     8     3    | 4     17    1279   || 4      6     129-3 | 7     5     12   | 29   *138   12389  |'--------------------'------------------'--------------------'`

This gives us this position. Here is the question.
What do you call this method. To me it is a pattern that involves remote pairs. I can see that singles can work in this same pattern, so pairs may not be in the answer.
r9c6 and r8c2 are twinned, they are the same.
r9c6 and r7c6 are opposites.
That means that what r2c8 and r7c6 both see are not possible.

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I do not claim to be the early scout on this. I saw it posted a year ago, the general idea, here: http://www.sudoku.frihost.net/forum/viewtopic.php?t=1412

There the pattern goes further.

Code: Select all
`.------------------.------------------.------------------.| 4    #37    9    | 1     8     5    | 2    *37    6    || 8     3567  367  | 346   346   2    | 9     1    *357  || 2     356   1    | 9     7     36   | 8     4    *35   |:------------------+------------------+------------------:| 5     6-37  2    | 8     346   3467 | 1     9    #37   || 1     4     367  | 3567  356   9    | 36    2     8    || 9     8     367  | 367   2     1    | 5     367   4    |:------------------+------------------+------------------:| 67    1     5    | 3467  9     8    | 346   36    2    || 67    2     48   | 34567 1     3467 | 346   58    9    || 3     9     48   | 2     456   46   | 7     58    1    |'------------------'------------------'------------------'`

So, is this two short chains?

Is there a name for the second "step" that is not "chain"?
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

wapati wrote:This gives us this position. Here is the question.
What do you call this method. To me it is a pattern that involves remote pairs. I can see that singles can work in this same pattern, so pairs may not be in the answer.
r9c6 and r8c2 are twinned, they are the same.
r9c6 and r7c6 are opposites.
That means that what r2c8 and r7c6 both see are not possible.

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I don't see how r8c2 and r9c6 are the same. However, I can see that there is a Y-wing style that at least eliminates the 2's in r7c123.

wapati wrote:
Code: Select all
`.------------------.------------------.------------------.| 4    #37    9    | 1     8     5    | 2    *37    6    || 8     3567  367  | 346   346   2    | 9     1    *357  || 2     356   1    | 9     7     36   | 8     4    *35   |:------------------+------------------+------------------:| 5     6-37  2    | 8     346   3467 | 1     9    #37   || 1     4     367  | 3567  356   9    | 36    2     8    || 9     8     367  | 367   2     1    | 5     367   4    |:------------------+------------------+------------------:| 67    1     5    | 3467  9     8    | 346   36    2    || 67    2     48   | 34567 1     3467 | 346   58    9    || 3     9     48   | 2     456   46   | 7     58    1    |'------------------'------------------'------------------'`

So, is this two short chains?

Is there a name for the second "step" that is not "chain"?

It seems to be that these are just examples of 2-string kites.
[Edit: The 7 elimination is an example of a 2-string kite, the 3 elimination is definitely not. ]
[Edit2: Changed the y-wing elimination thanks to ravel's comment.]
Last edited by re'born on Mon Jun 11, 2007 5:35 am, edited 2 times in total.
re'born

Posts: 551
Joined: 31 May 2007

### Re: Remote pairs, with an offset?

re'born wrote:
wapati wrote:

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I don't see how r8c2 and r9c6 are the same. .

Set r8c2 as 1, r9c6 must be one.
Set r9c6 as 2, r8c2 must be two.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

re'born wrote:
It seems to be that these are just examples of 2-string kites.

A 2 string kite eliminates 1 candidate.

I am thinking that you need six to show me what I did in one logical step.

I don't see any, BTW, but I have missed them before. <sigh>
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

wapati wrote:
re'born wrote:
wapati wrote:

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I don't see how r8c2 and r9c6 are the same. .

Set r8c2 as 1, r9c6 must be one.
Set r9c6 as 2, r8c2 must be two.

Yes, but if r8c2=2, why should r9c6=2?
re'born

Posts: 551
Joined: 31 May 2007

### Re: Remote pairs, with an offset?

re'born wrote:
I don't see how r8c2 and r9c6 are the same.

From the other puzzle.....

r4c9 is 3 r1c8 is 3.
r1c8 is 7 r9c4 is 7.

It isn't a hard pattern. Using it can be tough.

Code: Select all
`.------------------.------------------.------------------.| 4    #37    9    | 1     8     5    | 2    *37    6    || 8     3567  367  | 346   346   2    | 9     1    *357  || 2     356   1    | 9     7     36   | 8     4    *35   |:------------------+------------------+------------------:| 5     6-37  2    | 8     346   3467 | 1     9    #37   || 1     4     367  | 3567  356   9    | 36    2     8    || 9     8     367  | 367   2     1    | 5     367   4    |:------------------+------------------+------------------:| 67    1     5    | 3467  9     8    | 346   36    2    || 67    2     48   | 34567 1     3467 | 346   58    9    || 3     9     48   | 2     456   46   | 7     58    1    |'------------------'------------------'------------------' `
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

re'born wrote:
wapati wrote:
re'born wrote:
wapati wrote:

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I don't see how r8c2 and r9c6 are the same. .

Set r8c2 as 1, r9c6 must be one.
Set r9c6 as 2, r8c2 must be two.

Yes, but if r8c2=2, why should r9c6=2?

You have hit upon why it is not well known.

If you look I have accounted for all cases of the digits in these two cells.
It is not obvious, yep.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

wapati wrote:
re'born wrote:
I don't see how r8c2 and r9c6 are the same.

From the other puzzle.....

r4c9 is 3 r1c8 is 3.
r1c8 is 7 r9c4 is 7.

It isn't a hard pattern. Using it can be tough.

Code: Select all
`.------------------.------------------.------------------.| 4    #37    9    | 1     8     5    | 2    *37    6    || 8     3567  367  | 346   346   2    | 9     1    *357  || 2     356   1    | 9     7     36   | 8     4    *35   |:------------------+------------------+------------------:| 5     6-37  2    | 8     346   3467 | 1     9    #37   || 1     4     367  | 3567  356   9    | 36    2     8    || 9     8     367  | 367   2     1    | 5     367   4    |:------------------+------------------+------------------:| 67    1     5    | 3467  9     8    | 346   36    2    || 67    2     48   | 34567 1     3467 | 346   58    9    || 3     9     48   | 2     456   46   | 7     58    1    |'------------------'------------------'------------------' `

Looking at your setup again with the idea that you want it done in one step, I realize this is a pattern I mentioned in a post as my alter ego. Here for posterity is a reprint of said post

re'born nee rep'nA (5/21/2007) wrote:
wapati wrote:Here is an odd one. It is trivial if you use uniqueness once.
I was going to toss it because I know that if I got stuck I would jump on the UR. Others may have more grit and find this a great puzzle?

It is quite hard and full of big fish if you use "won't power".
(Will power is a dumb phrase considering that you need to "won't", not "will" ! )

Anyways, easy UR, hard without!
Code: Select all
`9 . 8 | 4 3 . | . 2 .. . 2 | . . 8 | . 3 6. . 3 | 9 . . | . . .---------------------8 . . | . 6 . | . . .4 . . | 8 . 5 | . . 2. . . | . 4 . | . . 3---------------------. . . | . . 9 | 3 . .2 5 . | 3 . . | 4 . .. 9 . | . 5 4 | 2 . 7`

Yes, it is easy with UR, but it doesn't have to be too hard without. Doing a little 3D coloring, I found
Code: Select all
` *-----------------------------------------------------------* | 9     167   8     | 4     3     167   | 157   2     15    | | 17    4     2     | 5     17    8     | 9     3     6     | | 5     167   3     | 9     127   1267  | 17    4     8     | |-------------------+-------------------+-------------------| | 8     127   59    | 127   6     3     | 15    1579  4     | | 4     3     1A7a  | 8     9     5     | 6     1a7A  2     | | 6     127   59    | 127   4     127   | 8     59    3     | |-------------------+-------------------+-------------------| | 17    8     4     | 1267  127   9     | 3     156   15    | | 2     5     167A  | 3     8     1A7a  | 4     1-6   9     | | 3     9     16    | 16    5     4     | 2     8     7     | *-----------------------------------------------------------*`

[r8c8]-1-[r8c6]-7-[r8c3]=7=[r5c3]-7-[r5c8]-1-[r8c8], which gives r8c8<>1, solving the puzzle.

Alternatively, one could notice the 'naked pair covering an xyz-wing':
Code: Select all
` *-----------------------------------------------------------* | 9     167   8     | 4     3     167   | 157   2     15    | | 17    4     2     | 5     17    8     | 9     3     6     | | 5     167   3     | 9     127   1267  | 17    4     8     | |-------------------+-------------------+-------------------| | 8     127   59    | 127   6     3     | 15    1579  4     | | 4     3     17    | 8     9     5     | 6     17    2     | | 6     127   59    | 127   4     127   | 8     59    3     | |-------------------+-------------------+-------------------| | 17#   8     4     | 1267- 127-  9     | 3     156*  15*   | | 2     5     167-  | 3     8     17#   | 4     16*   9     | | 3     9     16    | 16    5     4     | 2     8     7     | *-----------------------------------------------------------*`

I suppose it is the ALS xz-rule with A={1567} on r7c189, B={167} on r8c68, x = 6, z = 7. But this is a pattern I've seen before and so recognize as an xyz-wing (or in this case even a naked triple in a box) where the arms both see a bivalue cell (with the same candidates) sharing the normal elimination candidate for the xyz-wing. More generally, it might look like:

Code: Select all
` *  .  .  | .  .  .  |  .  .  . *  wx .  | .  xz .  |  .  xyz . *  .  .  | .  .  .  |  .  .  .----------+----------+---------- .  .  .  | .  .  .  |  .  .  . .  .  .  | .  .  .  |  .  .  . .  .  .  | .  .  .  |  .  .  .----------+----------+---------- .  *  .  | .  .  .  |  .  .  . wx *  .  | .  .  .  |  .  xy  . .  *  .  | .  .  .  |  .  .  .`

where we can eliminate w from the *'d cells.
re'born

Posts: 551
Joined: 31 May 2007

### Re: Remote pairs, with an offset?

wapati wrote:
re'born wrote:
wapati wrote:
re'born wrote:
wapati wrote:

Code: Select all
`.--------------------.------------------.--------------------.| 58     58    13    | 13    2     4    | 67    9     67     || 29     4     6     | 8     79    57   | 3     15    125    || 13     7     1239  | 13    69    56   | 25    48    48     |:--------------------+------------------+--------------------:| 1567   15    17    | 2     3     9    | 8     457   4567   || 367    9     4     | 5     67    8    | 1     2     367    || 23567  235   8     | 4     1     67   | 5679  357   35679  |:--------------------+------------------+--------------------:| 378-12 38-12 37-12 | 9     4    #12   | 257   6     1257   || 1279  #12    5     | 6     8     3    | 4     17    1279   || 4      6    *129   | 7     5    *12   |*29    138   12389  |'--------------------'------------------'--------------------'`

I don't see how r8c2 and r9c6 are the same. .

Set r8c2 as 1, r9c6 must be one.
Set r9c6 as 2, r8c2 must be two.

Yes, but if r8c2=2, why should r9c6=2?

You have hit upon why it is not well known.

If you look I have accounted for all cases of the digits in these two cells.
It is not obvious, yep.

Indulge me with an explanation as I'm at a loss. If r8c2=2 and r9c6=1, it doesn't contradict any your two statements since those only take as hypotheses r8c2=1 or r9c6=2.
re'born

Posts: 551
Joined: 31 May 2007

Indulge me with an explanation as I'm at a loss. If r8c2=2 and r9c6=1, it doesn't contradict any your two statements since those only take as hypotheses r8c2=1 or r9c6=2.

We have two cells that can be only 1 or 2.

If the first is one the second must be one, I showed that.

If the second is two, can the first be one? Well, no, if the first was one the second would be one. Thus the first is two, if the second is two.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

wapati wrote:
Indulge me with an explanation as I'm at a loss. If r8c2=2 and r9c6=1, it doesn't contradict any your two statements since those only take as hypotheses r8c2=1 or r9c6=2.

We have two cells that can be only 1 or 2.

If the first is one the second must be one, I showed that.

If the second is two, can the first be one? Well, no, if the first was one the second would be one. Thus the first is two, if the second is two.

What you are saying is only true if you are writing

first=r8c2
second=r9c6.

You did not show that if r9c6=1, then r8c2=1.
re'born

Posts: 551
Joined: 31 May 2007

re'born wrote:
wapati wrote:
Indulge me with an explanation as I'm at a loss. If r8c2=2 and r9c6=1, it doesn't contradict any your two statements since those only take as hypotheses r8c2=1 or r9c6=2.

We have two cells that can be only 1 or 2.

If the first is one the second must be one, I showed that.

If the second is two, can the first be one? Well, no, if the first was one the second would be one. Thus the first is two, if the second is two.

What you are saying is only true if you are writing

first=r8c2
second=r9c6.

You did not show that if r9c6=1, then r8c2=1.

This is why it is not well used.

It does show that if r8c2=2 then r9c6 must be 2, thus the question cannot arise.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

### Re: Remote pairs, with an offset?

[Deleted: Post is currently half-baked. I'll repost it when it's done cooking.]
Last edited by re'born on Sun Jun 10, 2007 10:38 pm, edited 3 times in total.
re'born

Posts: 551
Joined: 31 May 2007

re'born wrote:
What you are saying is only true if you are writing

first=r8c2
second=r9c6.

Wrong.

It is binary. Swap them as you will.
wapati
2010 Supporter

Posts: 527
Joined: 13 September 2006

wapati wrote:
re'born wrote:
What you are saying is only true if you are writing

first=r8c2
second=r9c6.

Wrong.

It is binary. Swap them as you will.

But that requires a proof!

Assume you have two cells A and B who can take the values x and y. If I have the statements

A=x => B=x
B=y => A=y

it does not imply that if A=y, then B=x or that if B=x then A=y. You would need some extra information.

Here is a proof of what I'm saying based on your example.

Assume r8c2=2. Then r6c1=2 => r2c2=9 => r3c3<>9 => r9c3=9 => r9c6=1. Therefore, r8c2=2 does not imply r9c6=1 and these two cells do not have to have the same values (the fact that they will in the solution is irrelevant).
re'born

Posts: 551
Joined: 31 May 2007

Next