The New Sudoku Players' Forum

[DoubleB72]

1,6 3 5 2 8 7 1,4,6 6,9 1,4,9
1,8 2 9 4 3 6 1,5,7,8 7,8 1,5,8
7 4 6,8 1 9 5 6,8 3 2
------------------------------------------------------------------
2,4,9 1 2,4,7 5 6 4,8 2,7,8 7,8,9 3
4,9 6,7,8 3,4,6,7 3,9 2 4,8 1,5 6,7,8,9 1,5
5 6,7,8 2,3,6,7 3,9 7 1 2,6,8 4 8,9
----------------------------------------------------------------
3 9 4,8 6 1 2 4,8 5 7
2,4 6,7 2,4,6,7 8 5 3 9 1 4,6
6,8 5 1 7 4 9 3 2 6,8

[Havard]

Hi. When you post a grid, try to make sure it looks something like this:

Code: Select all

16   3    5    | 2    8    7    | 146  69   149
18   2    9    | 4    3    6    | 1578 78   158
7    4    68   | 1    9    5    | 68   3    2
---------------+----------------+---------------
249  1    247  | 5    6    48   | 278  789  3
49   678  3467 | 39   2    48   | 15   6789 15
5    678  2367 | 39   7    1    | 268  4    89
---------------+----------------+---------------
3    9    48   | 6    1    2    | 48   5    7
24   67   2467 | 8    5    3    | 9    1    46
68   5    1    | 7    4    9    | 3    2    68


And this puzzle solves easily with the use of a Unique Rectangle:

Code: Select all

Unique Rectangle type 3:
16     3      5      | 2      8      7      | 146    69     149
18-    2      9      | 4      3      6      | 1578U# 78#    158U#
7      4      68     | 1      9      5      | 68-    3      2
---------------------+----------------------+---------------------
249    1      247    | 5      6      48     | 278    789    3
49     678    3467   | 39     2      48     | 15U    6789   15U
5      68     236    | 39     7      1      | 268    4      89
---------------------+----------------------+---------------------
3      9      48     | 6      1      2      | 48     5      7
24     67     2467   | 8      5      3      | 9      1      46
68     5      1      | 7      4      9      | 3      2      68


(actually a very good example of a Type 3 UR!)

If you search this forum, you will find a lot of good stuff about these!

Havard

[Sped]

Harvard -

Nice work on the unique rectangle solution.

It's elegant, and simple enough that even I can understand it.

I'm glad I didn't post my brutal solution, which involved an X Wing, multiple coloring, a lengthy XY chain, etc.

I've got to start looking for URs.

[Havard]

Thanks! UR is higly recommended!

Havard

[DoubleB72]

i dont get it

[DoubleB72]

i have been reading about rectangles and they are sort of making sense. Would someone be so kind to explain the next step here, as far as what can be eliminated and why? I think I may understand it from there.
Thanks

[Sped]

i have been reading about rectangles and they are sort of making sense. Would someone be so kind to explain the next step here, as far as what can be eliminated and why? I think I may understand it from there.
Thanks

Code: Select all

 
 *-----------------------------------------------------------*
 | 16    3     5     | 2     8     7     | 146   69    149   |
 | 18    2     9     | 4     3     6     | 1578* 78    158*  |
 | 7     4     68    | 1     9     5     | 68    3     2     |
 |-------------------+-------------------+-------------------|
 | 249   1     247   | 5     6     48    | 278   789   3     |
 | 49    678   3467  | 39    2     48    | 15*   6789  15*   |
 | 5     68    236   | 39    7     1     | 268   4     89    |
 |-------------------+-------------------+-------------------|
 | 3     9     48    | 6     1     2     | 48    5     7     |
 | 24    67    2467  | 8     5     3     | 9     1     46    |
 | 68    5     1     | 7     4     9     | 3     2     68    |
 *-----------------------------------------------------------*


The thing about Unique Rectangles is this.. There can't be a "deadly pattern"

What is a deadly pattern? It's an arrangement of candidates that would allow a puzzle to have more than one solution.

Look at the 15 pair in row 5 of box 6. Then notice that the 15 pair appears in the same columns in box 3, along with some other candidates.

The deadly pattern would be this:

Code: Select all

 

 IMPOSSIBLE!!!  PUZZLE WOULD HAVE MULTIPLE SOLUTIONS!!!
 *-----------------------------------------------------------*
 | 16    3     5     | 2     8     7     | 46    69    49    |
 | 18    2     9     | 4     3     6     | 15*   78    15*   |
 | 7     4     68    | 1     9     5     | 68    3     2     |
 |-------------------+-------------------+-------------------|
 | 249   1     247   | 5     6     48    | 278   789   3     |
 | 49    678   3467  | 39    2     48    | 15*   6789  15*   |
 | 5     68    236   | 39    7     1     | 268   4     89    |
 |-------------------+-------------------+-------------------|
 | 3     9     48    | 6     1     2     | 48    5     7     |
 | 24    67    2467  | 8     5     3     | 9     1     46    |
 | 68    5     1     | 7     4     9     | 3     2     68    |
 *-----------------------------------------------------------*


The 15 pairs are in a nice rectangle and occupy 2 rows, 2 columns, and 2 boxes.
It's pretty, but there is a problem with it. The puzzle can be solved 2 ways from this point. r2c7 can be either a 1 or a 5, and the other 3 cells in the rectangle will work out and there will be a 1 and a 5 in row 2, row 5, column 7, coulmn 9, box 3 and box 6 either way.

If you ever arrive at a deadly pattern like that through legitimate eliminations, then the puzzle is invalid.

Code: Select all

 
 *-----------------------------------------------------------*
 | 16    3     5     | 2     8     7     | 146   69    149   |
 | 18    2     9     | 4     3     6     | 1578* 78    158*  |
 | 7     4     68    | 1     9     5     | 68    3     2     |
 |-------------------+-------------------+-------------------|
 | 249   1     247   | 5     6     48    | 278   789   3     |
 | 49    678   3467  | 39    2     48    | 15*   6789  15*   |
 | 5     68    236   | 39    7     1     | 268   4     89    |
 |-------------------+-------------------+-------------------|
 | 3     9     48    | 6     1     2     | 48    5     7     |
 | 24    67    2467  | 8     5     3     | 9     1     46    |
 | 68    5     1     | 7     4     9     | 3     2     68    |
 *-----------------------------------------------------------*


We're going to make an assumption, though. We are going to assume that the puzzle has one and only one solution. Given that assumption, we know that r2c7 and r2c9 cannot both be 1,5. If they were, then we'd have a deadly pattern and the puzzle would have more than one solution.

So.. r2c7 and r2c9 cannot both be 1,5.. how does that help us? Here's how: of r2c7 and r2c9, at least one of the cells has to be a 7 or an 8. It's required, otherwise we get the deadly pattern. Think of those 2 cells as a single virtual (7,8). There's an actual (7,8) in r2c8. Together they make a naked pair knocking the 8 out of r2c1. The puzzle is solved.

If the idea of a "virtual pair" doesn't sit right, look at it this way: of r2c7 and r2c9, at least one of the cells has to be a 7 or an 8. If one is an 8, then the 8 is knocked out of r2c1. If one is a 7, then r2c8 is made an 8, and the 8 is knocked out of r2c1. r2c1 loses its 8 either way.

Unique rectangles are easy to spot with a little practice, and they turn up more often than you might think.

I picked up the book : "Black Belt Sudoku" by Frank Longo. It advertises 300 Super Tough Puzzles.

Unique rectangles have turned up in more than half of the puzzles I've solved in it so far.

#295 is a good example. The candidate grid shows where Simple Sudoku gets stuck:

Code: Select all

 *-----------*
 |...|3.5|.47|
 |2..|..4|..8|
 |...|9..|6..|
 |---+---+---|
 |...|..9|.5.|
 |8..|.6.|..1|
 |.4.|7..|...|
 |---+---+---|
 |..8|..3|...|
 |9..|4..|..5|
 |52.|1.6|...|
 *-----------*


 *-----------*
 |186|325|947|
 |2..|6.4|5.8|
 |.5.|9..|6..|
 |---+---+---|
 |.12|849|.56|
 |8..|562|4.1|
 |645|731|...|
 |---+---+---|
 |..8|253|1..|
 |9.1|4..|..5|
 |52.|196|...|
 *-----------*

 
 *--------------------------------------------------*
 | 1    8    6    | 3    2    5    | 9    4    7    |
 | 2    379  379  | 6    17   4    | 5    13   8    |
 | 347  5    47   | 9    178* 78*  | 6    12   23   |
 |----------------+----------------+----------------|
 | 37   1    2    | 8    4    9    | 37   5    6    |
 | 8    79   379  | 5    6    2    | 4    37   1    |
 | 6    4    5    | 7    3    1    | 28   289  29   |
 |----------------+----------------+----------------|
 | 47   67   8    | 2    5    3    | 1    679  49   |
 | 9    36   1    | 4    78*  78*  | 23   26   5    |
 | 5    2    347  | 1    9    6    | 378  78   34   |
 *--------------------------------------------------*


The Unique Rectangle is marked with asterisks. If the 1 is knocked out of r3c5 we get a deadly pattern. Since we assume the puzzle has just one solution, r3c5 must be a 1.

Set r3c5 to 1 and the puzzle solves itself.

I'd hate to think of the tortuous solution I'd have to find if I hadn't spotted that Unique Rectangle.

Unique Rectangles.. Learn to use them. You'll be glad you did.

[RW]

Code: Select all

 *-----------------------------------------------------------*
 | 16    3     5     | 2     8     7     | 146   69    149   |
 | 18    2     9     | 4     3     6     | 1578* 78    158*  |
 | 7     4     68    | 1     9     5     | 68    3     2     |
 |-------------------+-------------------+-------------------|
 | 249   1     247   | 5     6     48    | 278   789   3     |
 | 49    678   3467  | 39    2     48    | 15*   6789  15*   |
 | 5     68    236   | 39    7     1     | 268   4     89    |
 |-------------------+-------------------+-------------------|
 | 3     9     48    | 6     1     2     | 48    5     7     |
 | 24    67    2467  | 8     5     3     | 9     1     46    |
 | 68    5     1     | 7     4     9     | 3     2     68    |
 *-----------------------------------------------------------*


The thing about Unique Rectangles is this.. There can't ba a "deadly pattern"

Nice description Sped. In this particular case, however, I wouldn't consider any quantum cells and other complicated things. Not only must there be another number in at least one of the cells in the potential deadly pattern, one of the numbers forming this deadly pattern must also be outside the pattern. The only possibility to place a number outside the pattern in either boxes is 1 in r1c7 or r1c9 => eliminate 1 from r2c7 and r2c9. Alternatively you could immediately solve a cell by noticing that the only way to place a number outside the pattern in either row is 1 in r2c1 (which of course is a hidden single if you first made the box reduction).

RW

[Sped]

Nice description Sped. In this particular case, however, I wouldn't consider any quantum cells and other complicated things. Not only must there be another number in at least one of the cells in the potential deadly pattern, one of the numbers forming this deadly pattern must also be outside the pattern. The only possibility to place a number outside the pattern in either boxes is 1 in r1c7 or r1c9 => eliminate 1 from r2c7 and r2c9. Alternatively you could immediately solve a cell by noticing that the only way to place a number outside the pattern in either row is 1 in r2c1 (which of course is a hidden single if you first made the box reduction).

RW

Good point RW. When it's understood that the idea is to avoid the deadly pattern, lots of things follow:

The 78 thing in row 2 of box 3.

There has to a 1 or 5 outside of r2c7 and r2c9 in box 3. The only possibility is a 1 r1c7 or r1c9. That eliminates the 1 in r1c1.

There has to be a 1 or 5 in row 2 outside of r2c7 and r2c9. The only possibility is a 1 in r2c1.

There are a lot of ways to look at it. The "virtual pair" idea is probably not the best way to think of it, but it may click with some people. It's the first thing I saw, anyway.

[ravel]

Thanks Sped,

this was the best explanation for UR's from pencilmarks, i read so far.

And thanks RW,

you brought us another look at the UR thing. I now can also see the "no pm's" view:

Code: Select all

+-------+-------+-------+
| . 3 5 | 2 8 7 | . . . |
| . 2 9 | 4 3 6 | . . . |
| 7 4 . | 1 9 5 | . 3 2 |
+-------+-------+-------+
| . 1 . | 5 6 . | . . 3 |
| . . . | . 2 . | . . . |
| 5 . . | . 7 1 | . 4 . |
+-------+-------+-------+
| 3 9 . | 6 1 2 | . 5 7 |
| . . . | 8 5 3 | 9 1 . |
| . 5 1 | 7 4 9 | 3 2 . |
+-------+-------+-------+


the 15 pair in r5c79 is easy to spot. When i concentrate on box 3/row 2 then, column 8 is blocked again for both, but c79 are possible (-> AUR). So only r2c1 is remaining for one of 1 and 5 in row 2, but 5 is blocked there => r2c1=1.
Its easy, but i am sure, i would have counted a lot of candidates in this situation

[ronk]

In this particular case, however, I wouldn't consider any quantum cells and other complicated things. Not only must there be another number in at least one of the cells in the potential deadly pattern, one of the numbers forming this deadly pattern must also be outside the pattern.

I think most of use have learned to look for UR Type 3 before UR Type 4, because the latter destroys the former. If and when we become accustomed to seeing ...

Code: Select all

 b+X  b+Y

 ab   ab

... as a UR Type 3, even though UR digits are missing, that might change.

Here's a bit of trivia ... useless trivia AFAIK. When the quantum cell of a UR Type 3 and another cell form a naked pair, exactly one of the UR cells ultimately contains one of the extra (non-UR) digits. The same is true for larger N-tuples. Hmm, I wonder if that's true generally ... even if no disjoint subsets exist.

[TKiel]

Sped,

So how come you can explain a UR but you don't generally look for them in puzzles?

Tracy

[Sped]

Sped,

So how come you can explain a UR but you don't generally look for them in puzzles?

Tracy

I look for them now. Harvard showed me the light at the top of this thread. Some puzzles that would otherwise require difficult solving paths crack easily with URs, and the thing is.. they're easy to spot.

[keith]

Coincidentally, I wrote this guide to UR's last week.

http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=744

I think it is a little different, in that it tries to progess through the logic; it does not categorize the UR's into "Types". Also, the more newly discovered diagonal forms (Types 5 and 6) are integrated as variant patterns when the discussed logic applies.

Keith

[RW]

I think most of use have learned to look for UR Type 3 before UR Type 4, because the latter destroys the former.

I never look for any specific pattern before looking for another, I also never categorize URs as different types. To me any number that causes a deadly pattern is false and that's all I care about. I can see your point though, if you use pms the type 3 would destroy the type 4. I just wonder if anybody has seen an actual situation where the type 3 solves the puzzle but type 4 doesn't? As I don't use pms I wouldn't have a problem with this. When there's no pms to remove I can still spot the reduction based on type 3 and make the reduction, even if I earlier had spotted the type 4 and looked for possible solved cells based on that.

Spotting type 3 reductions in a non pm grid is however not very easy, I can actually remember only a few cases when I would have used that pattern. But the type 4, as Ravel mentioned, is a lot easier to see in the non pm grid.

Here's a bit of trivia ... useless trivia AFAIK. When the quantum cell of a UR Type 3 and another cell form a naked pair, exactly one of the UR cells ultimately contains one of the extra (non-UR) digits. The same is true for larger N-tuples. Hmm, I wonder if that's true generally ... even if no disjoint subsets exist.

That is true, nice observation. If you can form a N-tuplet and two of the UR cells would contain a non-UR digit, you would end up with an empty cell in you N-tuplet.

RW