## Almost Unique Rectangles: description and use in nice loops

Advanced methods and approaches for solving Sudoku puzzles
Sorry, idiotic post deleted, Ron
Last edited by ronk on Mon Jan 09, 2006 9:22 am, edited 1 time in total.
ronk
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Hi Jeff.

Obviously your loop is more straightforward, but the purpose of the example was to illustrate the difference between the links in ALS and AUR. Even then, I have already edited the example.

Regards, Carcul
Carcul

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Hi Ronk.

Ronk wrote:If r4c7=1, then r1c8<>9

... are both valid. (I haven't actually seen a proof for the later though.)

I don't think this last one is valid. The only thing that we can say with certain is, if there is only one solution, we cannot have r1c8<>9 and r4c7<>1. So, if r4c7<>1 => r1c8=9, if r1c8<>9 => r4c7=1. But we could also have r1c8=9 and r4c7=1. This does not happen in this particular grid, but consider the following one:

Code: Select all
` 3   1  9  | 6  7   48 | 248  25 458  6   7  2  | 38 5   348| 48   1  9  5   8  4  | 2  9   1  | 7    6  3    -----------+-----------+------------  278 4  137| 9  238 5  | 6    27 18  28  6  35 | 7  1   38 | 2348 9  458  9   23 157| 4  238 6  | 238  57 15 -----------+-----------+------------  4   9  8  | 5  6   2  | 1    3  7  27  23 37 | 1  4   9  | 5    8  6  1   5  6  | 38 38  7  | 9    4  2 `

We have a type 2 AUR in cells r1c6, r1c7, r2c6, and r2c7, and in the solution of this grid it turns out that r2c6=3 and r1c7=2.

Regards, Carcul
Carcul

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Carcul wrote:
Ronk wrote:If r4c7=1, then r1c8<>9

... are both valid. (I haven't actually seen a proof for the later though.)

I don't think this last one is valid. The only thing that we can say with certain is, if there is only one solution, we cannot have r1c8<>9 and r4c7<>1. So, if r4c7<>1 => r1c8=9, if r1c8<>9 => r4c7=1.

You're quick! I realized that and deleted my idiotic post ... but you'd already quoted me.
ronk
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Carcul wrote:We have a type 2 AUR in cells r1c6, r1c7, r2c6, and r2c7, and in the solution of this grid it turns out that r2c6=3 and r1c7=2.

Using BUG-like notation, we have an "AUR+2" grid ...
Code: Select all
` 3    1   9   | 6   7    48  | 48+2  25  458   6    7   2   | 38  5    48+3| 48    1   9   5    8   4   | 2   9    1   | 7     6   3     --------------+--------------+---------------  278  4   137 | 9   238  5   | 6     27  18   28   6   5   | 7   1    38  | 2348  9   458   9    23  157 | 4   28   6   | 238   57  15  --------------+--------------+---------------  4    9   8   | 5   6    2   | 1     3   7   27   23  37  | 1   4    9   | 5     8   6   1    5   6   | 38  38   7   | 9     4   2  `

... and note that r2c6=3 => r2c4=8 => r1c6=4 => r1c7<>4. Combined with the trivial r1c7=3 => r1c7<>4, we have r1c7<>4.

[edit: Even better is ...
r2c6=3 => r5c6=8 => r6c5=2 => r4c5<>2
r1c7=2 => r1c8=5 => r6c8=7 => r4c8=2 => r4c5<>2
which solve the puzzle.]

For now, I'll refrain from attempting nice loops for these.

Ron
ronk
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ronk wrote:My starting implication chains are based on the fact that for this type 2 AUR example ... either r1c8=9 or r4c7=1 and therefore ...

r4c7=1 => r4c9<>1 => r4c9=4 => r4c8<>4 => r4c8=8
r4c7<>1 => r1c8=9 => r1c8<>8 => r4c8=8 => r4c8<>4

Where is the error in logic? Is is possible such a simple deduction is not expressible with a nice loop simpler than the one you posted? Tell me it isn't so!

Hi Ronk and Carcul, Perhaps this simple deduction could be expressed with the following nice loop:

[r4c8]-4-[r4c9]-1-(AUR:[r4c7][r1c7][r4c8]-4|8-[r1c8])=8=[r4c8] => r4c8<>4 => r4c8=8

where (AUR:[r4c7][r1c7][r4c8]-4|8-[r1c8]) is the AUR+2 implying that if r4c7, r1c7 & r4c8=48, then r1c8<>48
Note: the candidate '9' in r1c8 is insignificant for this nice-loop’s propagation.

Carcul, I don't disagree with your notation, but just to suggest an alternative for brainstorming.
Jeff

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Hi Jeff.

Jeff wrote:[r4c8]-4-[r4c9]-1-(AUR:[r4c7][r1c7][r4c8]-4|8-[r1c8])=8=[r4c8] => r4c8<>4 => r4c8=8

I don't want to be boring or arrogant, but I still don't agree with this loop. If r4c8=4 then there is no contradiction in the cells that you have used in the loop.

Jeff wrote:Carcul, I don't disagree with your notation, but just to suggest an alternative for brainstorming.

Yes, I now my notation might be confusing sometimes (or manytimes), but I prefer to write a "condensed" notation and then explain separately the origin of a link if necessary (this is just a personal preference).
BTW, what is the exact meaning of the word "brainstorming"?
And how about the exercise of AURs that I posted above, refering to a grid in the thread "Nice loops for elementary level players - the x-cycle", what do you think?

Regards, Carcul
Carcul

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Carcul wrote:
Jeff wrote:[r4c8]-4-[r4c9]-1-(AUR:[r4c7][r1c7][r4c8]-4|8-[r1c8])=8=[r4c8] => r4c8<>4 => r4c8=8

I don't want to be boring or arrogant, but I still don't agree with this loop. If r4c8=4 then there is no contradiction in the cells that you have used in the loop.

Hi Carcul, Both implication steams in the loop implies that r4c8<>4. What could imply r4c8=4?

Carcul wrote:Yes, I now my notation might be confusing sometimes (or manytimes), but I prefer to write a "condensed" notation and then explain separately the origin of a link if necessary (this is just a personal preference).

I do not disagree with your notation. Most importantly, it obeys the nice loop propagation rules.

Carcul wrote:what is the exact meaning of the word "brainstorming"?

Brainstorming means random thinking to induce ideation and inspiration.

Carcul wrote:And how about the exercise of AURs that I posted above, refering to a grid in the thread "Nice loops for elementary level players - the x-cycle", what do you think?

Pep's example is a continuous grouped x-cycle. It will be a good example for my other thread. I might work on this first.
Jeff

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I have edited the original post with a description and example of Type 3 Almost Unique Rectangles.

Carcul
Carcul

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here is another example
Code: Select all
`+----------------+----------------+----------------+| 4    1258 12   | 15   9    7    | 6    158  3    | | 38   158  9    | 6    134  34   | 7    158  2    | | 367  156  1367 | 135  8    2    | 9    15   4    | +----------------+----------------+----------------+| 9    3    1267 |^17   1246 46   | 8   ^67   5    | | 2678 268  4    | 9   *236  5    |*23   67   1    | | 267  126  5    | 137 *1236 8    |*23   4    9    | +----------------+----------------+----------------+| 2356 7    236  | 4    356  36   | 1    9    8    | | 1    4    8    | 2    7    9    | 5    3    6    | | 356  9    36   | 8    356  1    | 4    2    7    | +----------------+----------------+----------------+ we get  r4c6<>6`
bennys

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Hi Bennys.

Thanks, this is a very good example. In chain notation we have:

[r4c6]-6-[r5c5|r6c5]-1-[r4c4]-7-[r4c8]-6-[r4c6] => r4c6<>6.

Regards, Carcul
Carcul

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Carcul wrote:In chain notation we have:

[r4c6]-6-[r5c5|r6c5]-1-[r4c4]-7-[r4c8]-6-[r4c6] => r4c6<>6.

Very nice. Because of AUR requirements, r5c5 and r6c5 "behave" like a single bivalued cell.

But did you notice that when viewed as a bivalue, we also have the almost locked set xz-rule ... A = {167}, B = {16}, x = 1, z = 6, so z can be eliminated in *all* other cells that 'see' all z in the sets?

So r4c5<>6 too. [edit: In this particular puzzle, r4c6<>6 alone solves the puzzle.]

Ron
ronk
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Hi Ronk.

No, I didn't notice that.

Ronk wrote:r5c5 and r6c5 "behave" like a single bivalued cell.

This is an interesting observation. Very good interpretation Ronk. We have now a "different" kind of the ALS xz-rule.
By curiosity, I have now take a look at the examples of Bennys regarding the xz-rule and I have noted that all of them can be expressed by (weak) nice loops, as already pointed out by Jeff. This is the case too for the present grid. Hence, we have two equivalent forms to see the same logical argument.

Regards, Carcul
Carcul

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Hi Carcul,

In your AUR type numbering, which type number is for the simplest of AURs ... the AUR+1?

This example has an AUR (45) + 1 at cells r1c26 and r3c26:
Code: Select all
` 1     45+2  9     | 6     3     45    | 7     25    8    235   8     35    | 7     1     9     | 25    4     6    6     45    7     | 2     8     45    | 3     1     9   -------------------+-------------------+----------------- 345   25    45+23 | 45    9     7     | 6     8     1    45    7     1     | 3     6     8     | 245   9     25   9     6     8     | 1     45    2     | 45    3     7   -------------------+-------------------+----------------- 245   3     6     | 9     45    1     | 8     7     245  8     1     45+2  | 45    7     6     | 9     25    3    7     9     45    | 8     2     3     | 1     6     45  `

Did I miss the type numbering, or do you consider it a subset of type 2?

TIA, Ron

P.S. For ease of type number recognition, I wish there was a one-to-one correspondence between the N of 'type N' and the N of 'AUR+N'. For example, AUR+1 <=> type 1, AUR+2 <=> type 2, etc.
ronk
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Hi Ronk.

Ronk wrote:In your AUR type numbering, which type number is for the simplest of AURs ... the AUR+1?

This example has an AUR (45) + 1 at cells r1c26 and r3c26:

This is not an almost unique rectangle. The example in your grid is already an unique rectangle (of type 1), which imply immediatly r1c2=2.
Please check the definition of AUR that I have writen in the original post.

Regards, Carcul
Carcul

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