## Avoidable Rectangles - Examples

Advanced methods and approaches for solving Sudoku puzzles

### Avoidable Rectangles - Examples

Has anybody got examples for Avoidable Rectangles Type 2? I can get Type 1 easily enough, but I couldn't produce an example for Type 2 myself.

hobiwan
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### Re: Avoidable Rectangles - Examples

hobiwan wrote:Has anybody got examples for Avoidable Rectangles Type 2?

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`If a UR Type 2 ... ab    abc ab    abcdegenerated to this ... ab    abc ab    acor this ... a     bc b     ac`

What elimination could you make?
ronk
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### Re: Avoidable Rectangles - Examples

ronk wrote:
hobiwan wrote:Has anybody got examples for Avoidable Rectangles Type 2?

Code: Select all
`If a UR Type 2 ... ab    abc ab    abcdegenerated to this ... ab    abc ab    acor this ... a     bc b     ac`

What elimination could you make?

Not sure about your second case (first degenerate), but I was looking for the third: any candidate c seeing both cells containing c in the avoidable rectangle could be eliminated.
hobiwan
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### Re: Avoidable Rectangles - Examples

hobiwan wrote:
ronk wrote:What elimination could you make?

any candidate c seeing both cells containing c in the avoidable rectangle could be eliminated.

Duh, that was one of my dumber questions! I was headed there too, so I'll see what I can find.
ronk
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I finally managed to find one!

Original puzzle:
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`..6........45.27.....8.6.43.3.2.5.6.4.........8.7.............7......12...5497...`

After some steps:
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`.---------------------.---------------------.---------------------.| 38     1257   6     | 19     1347   349   | 25     158    125   || 38     19     4     | 5      13     2     | 7      189    6     || 2579   2579   179   | 8      17     6     | 259    4      3     |:---------------------+---------------------+---------------------:| 179    3      179   | 2      48     5     | 489    6      149   || 4      56     29    | 19     1368   39    | 23589  7      125   || 56     8      129   | 7      1346   349   | 23459  19     1245  |:---------------------+---------------------+---------------------:| 69     469    8     | 3      2      1     | 459    59     7     || 79     479    3     | 6      5      8     | 1      2      49    || 12     12     5     | 4      9      7     | 6      3      8     |'---------------------'---------------------'---------------------'Avoidable Rectangle: 3/1 in r5c46,r7c46 => r5c37,r6c6<>9`
hobiwan
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### Re: Avoidable Rectangles - Examples

hobiwan wrote:I can get Type 1 easily enough, but ...

For which of the below did you search
Code: Select all
`ab    abab    a+Xa     bb     a+X`

I've searched only for the first and the following is the "best of the lot."

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`39.7....51...4..92.5.6..7....3.......2...........9.4..2....5.....5..7.38.14....7.After SSTS only: 3    9   *28   | 7   *28   1    | 6    4    5 1    7    6    | 5    4    38   | 38   9    2 4    5   *28   | 6   *23   9    | 7    18   13----------------+----------------+--------------- 567  4    3    | 128  57   268  | 189  1256 1679 567  2    9    | 138  57   4    | 138  1568 1367 567  8    1    | 23   9    236  | 4    256  367----------------+----------------+--------------- 2    3    7    | 89   68   5    | 19   16   4 9    6    5    | 4    1    7    | 2    3    8 8    1    4    | 239  36   23   | 5    7    69degenerate (28)UR1:r13c35 ==> r3c5<>2`

This example is rather ideal because:
1) it is obtainable with the Simple Sudoku Technique Set (SSTS) only
2) the non-degenerate UR1 is never visible during application of SSTS
3) the puzzle then collapses with cascading singles.

As you've likely perceived, I'm not yet ready to use the "Avoidable Rectangle" term.
ronk
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### Re: Avoidable Rectangles - Examples

ronk wrote:degenerate (28)UR1:r13c35 ==> r3c5<>2

This example is rather ideal because:
1) it is obtainable with the Simple Sudoku Technique Set (SSTS) only
2) the non-degenerate UR1 is never visible during application of SSTS
3) the puzzle then collapses with cascading singles.

Ideal What about the non-degenerate UR where the puzzle is still solved with SSTS?

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` before SSTS performs XY-Wing elimination: non-degenerate (28) UR Type 6 *--------------------------------------------------------------------* | 3      9     *268    | 7     *28     1      | 68     4      5      | | 1      7      68     | 5      4      38     | 368    9      2      | | 4      5     *28     | 6     *238    9      | 7      18     13     | |----------------------+----------------------+----------------------| | 567    4      3      | 128    57     268    | 189    12568  1679   | | 567    2      9      | 138    57     4      | 138    1568   1367   | | 567    8      1      | 23     9      236    | 4      256    367    | |----------------------+----------------------+----------------------| | 2      3      7      | 89     68     5      | 19     16     4      | | 9      6      5      | 4      1      7      | 2      3      8      | | 8      1      4      | 239    36     23     | 5      7      69     | *--------------------------------------------------------------------*`

The XY-Wing can be performed next, followed by a cascade in Singles.
daj95376
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### Re: Avoidable Rectangles - Examples

ronk wrote:For which of the below did you search
Code: Select all
`ab    abab    a+Xa     bb     a+X`

I searched for
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`a     bb     a+X   AR1a     b+Xb     a+X   AR2`

(following Sudopedia's terminology)

I have searched now about 200k random generated puzzles and found 435 puzzles with AR1, but only 13 puzzles containing an AR2.

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`...9.....6.41.7.5.....2518..1.........3......8.26.973...9....4....25...754.....13SinglesNaked Pair: 3,8 in r27c5 => r4c5<>3, r45c5<>8Naked Pair: 3,8 in r27c5 => r4c5<>3, r45c5<>8Uniqueness Test 1: 4/7 in r4c15,r5c15 => r5c5<>4, r5c5<>7Singles.---------------.---------------.---------------.| 1    28   5   | 9    6    38  | 34   7    24  || 6    28   4   | 1    38   7   | 39   5    29  || 9    3    7   | 4    2    5   | 1    8    6   |:---------------+---------------+---------------:| 4    1    6   | 358  7    38  | 59   2    89  || 7    9    3   | 58  *1    2   | 45   6  *-48  || 8    5    2   | 6   *4    9   | 7    3   *1   |:---------------+---------------+---------------:| 2    7    9   | 38   38   1   | 6    4    5   || 3    6    1   | 2    5    4   | 8    9    7   || 5    4    8   | 7    9    6   | 2    1    3   |'---------------'---------------'---------------'Avoidable Rectangle: 1/4 in r5c59,r6c59 => r5c9<>4`

Code: Select all
`......476...86.1.54...5....3........7...8..1...912......6..3.9....71.6....4...8..SinglesHidden Rectangle: 2/5 in r4c37,r5c37 => r4c7<>2Naked Single: r4c7=5XYZ-Wing: 4/9/2 in r5c79,r8c9 => r4c9<>2Singles.---------------.---------------.---------------.| 25   25   8   | 9    3    1   | 4    7    6   || 9    3    7   | 8    6    4   | 1    2    5   || 4    6    1   | 2    5    7   | 39   8    39  |:---------------+---------------+---------------:| 3    1   *2   | 4    7    9   |*5    6    8   || 7    4   *5   | 3    8    6   |*-29  1    29  || 6    8    9   | 1    2    5   | 37   34   347 |:---------------+---------------+---------------:| 8    27   6   | 5    4    3   | 27   9    1   || 25   9    3   | 7    1    8   | 6    45   24  || 1    57   4   | 6    9    2   | 8    35   37  |'---------------'---------------'---------------'Avoidable Rectangle: 2/5 in r4c37,r5c37 => r5c7<>2`

AR2 seems to be rare, most exemplars occur rather late in rather hard puzzles. Only two are early on in the puzzle:
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`.7.8...3...97.16..8.12......2.....9......82......6...46.5....83............4..72.SSTS (including XY-Wing).------------------.------------------.------------------.| 5     7     4    | 8     9     6    | 1     3     2    || 2     3     9    | 7     45    1    | 6     45    8    || 8     6     1    | 2     345   35   | 459   457  -579  |:------------------+------------------+------------------:| 13    2     6    | 35   *7     4    | 8     9    *15   || 4     59    37   | 359  *1     8    | 2     6    *57   || 19    1589  78   | 59    6     2    | 3    1-57   4    |:------------------+------------------+------------------:| 6     49    5    | 1     2     7    | 49    8     3    || 7     489   2    | 6     38    39   | 459   145  1-59  || 139   189   38   | 4     58    59   | 7     2     6    |'------------------'------------------'------------------'Avoidable Rectangle: 7/1 in r4c59,r5c59 => r38c9,r6c8<>5`

With UR1 (again):
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`..45..13.....2....8....76...7.1435.85...7...4.1.8.5....37..1...19.4...........3..SSTSSkyscraper: 9 in r1c9,r9c8 (r19c6) => r2c8,r7c9<>9Uniqueness Test 1: 2/6 in r4c38,r6c38 => r6c8<>2, r6c8<>6SinglesW-Wing: 5/8 in r2c8,r8c3 connected by 8 in r28c7 => r8c8<>5.------------------.------------------.------------------.| 7     26    4    | 5     68    689  | 1     3     29   || 3     56    1    | 69    2     4    | 78    58    579  || 8     25    9    | 3     1     7    | 6     4     25   |:------------------+------------------+------------------:| 9     7     2    | 1     4     3    | 5     6     8    || 5     8     3    | 26    7     26   | 9     1     4    || 4     1     6    | 8     9     5    |*2    *7     3    |:------------------+------------------+------------------:| 26    3     7    | 269   568   1    | 4    25-89  56   || 1     9    5-8   | 4     3    26-8  |*78   *28    567  || 26    4     58   | 7     568   2689 | 3    25-89  1    |'------------------'------------------'------------------'Avoidable Rectangle: 2/7 in r6c78,r8c78 => r79c8,r8c36<>8`

As for your degenerate UR1, I didn't even know it could provide an elimination, and I don't think I fully understand it. Do you have any pointers?
hobiwan
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### Re: Avoidable Rectangles - Examples

ronk wrote:As you've likely perceived, I'm not yet ready to use the "Avoidable Rectangle" term.

IMO, there are 2 good reasons against the 'avoidable rectangle' term. First, the term 'unique rectangle' is already strongly embedded in tutorials, sudopedia and general useage. And second, to my way of thinking, the term 'avoidable rectangle' term doesn't make sense: 'Avoidable' means that something can be avoided in the discretionary sense whereas the need to avoid the unique rectangle is mandatory: 'to be avoided'. If there was some reason to change the name, how about 'forbidden rectangle'!

Devil's advocate point: Just to cover all the bases, if someone were to say that 'avoidable rectangle' refers to the status before the pure rectangle ie. the presence of other digits makes the pure unique rectangle 'avoidable', then it still doesn't really clarify things any better than the term 'unique rectangle, since as long as there are extra digits, it isn't really a rectangle yet anyway. Pushing things that far, a better term would be: a Type 1 Almost Forbidden Rectangle. Hey that actually sounds pretty good!

But no, I think we're better off leaving well enough alone.
DonM
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### Re: Avoidable Rectangles - Examples

DonM wrote:Just to cover all the bases, if someone were to say that 'avoidable rectangle' refers to the status before the pure rectangle ie. the presence of other digits makes the pure unique rectangle 'avoidable', ...

As I understand Adrew Stuart's reasoning when introducing the term it referes to the puzzle maker: The rectangle is avoidable by putting a given into one of the corners. So:
UR: None of the cells solved, therefore the rectangle must be avoided

AR: Two or three of the cells are already solved, but are not givens; since the puzzle maker could have easily avoided it (but didn't), the deduction is valid
I don't really see what's the problem with the name AR (only for cases with solved cells), but I will readily switch to whatever the majority here agrees upon (but then the name should be changed in Sudopedia as well, because AR is already used there as clear defined term).
hobiwan
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Let's face it as descriptive terms both "Unique Rectangle" and "Avoidable Rectangle" suck.

Better terms would be "Un-resolvable" for when there are dual two-digit solutions and "Resolvable" when there aren't, which would add a RR abbreviation to the mix but keep the old UR one, (so it wouldn't really matter how people still thought about them).
Code: Select all
`123 (-2) (-2) | 12123 (-3) (-3) | 13 `
In Eureka these exclusions can be notated as:

(3=12)ALS:r12c1 -[RR]- (12=3)ALS:r12c4 => r2c23 <> 3

(2=13)ALS:r12c1 -[RR]- (13=2)ALS:r12c4 => r1c23 <> 2

The -[]- inference notation indicates that the inference arises as a result of the bracketed pattern. Here the digits and cells involved in the pattern aren't notated as they're obvious from the adjacent nodes, but otherwise they should be included.

I presume that chains involving latent uniqueness patterns over a larger cell set could be notated similarly but I haven't got there yet.
David P Bird
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### Re: Avoidable Rectangles - Examples

hobiwan wrote:As for your degenerate UR1, I didn't even know it could provide an elimination, and I don't think I fully understand it. Do you have any pointers?

Code: Select all
`a     b             ab    abb     a+X           ab    a+X`

'a' may be eliminated from the "a+X" cell for each pattern ... for exactly the same reason. See udosuk's "scenario 2" explanation here.

Assumption of a unique solution is not required for either pattern. The left pattern requires that none of the "filled cells" be a given; the right obviously does not.

when I wrote:I'm not yet ready to use the "Avoidable Rectangle" term ...

... I certainly didn't mean to trigger a naming discussion on this thread.
ronk
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### Re: Avoidable Rectangles - Examples

ronk wrote:
Code: Select all
`a     b             ab    abb     a+X           ab    a+X`

'a' may be eliminated from the "a+X" cell for each pattern ... for exactly the same reason. See udosuk's "scenario 2" explanation here.

Thank you, I skipped that thread because of it's title - apparently a mistake...
hobiwan
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