The New Sudoku Players' Forum

[Havard]

In trying to find puzzles that would solve with the Empty Rectangle stuff, I have not been very successful... However, i did find this one, where the solution is a "tiny" bit shorter by using it:

concider this one:


now you can apply the ER rule here:


and the puzzle solves instantly!

However, there are also a 2-string kite there, but then you need another few single candidates (1 or 2) before the puzzle is solved, so the ER is slightly better...

the thing I like best about this technique is that is actually much easier NOT to look for candidates, then it is looking for them

havard

[ronk]

if X does NOT have a candidate in it you can go:

Code: Select all

. . . | . . . | X . X
. . . | . . . | X . X
. a . | . . . | . . .
--|------------------
. | . | . . . | . . .
. | . | . . . | . . .
. | . | . . . | . . .
--|------------------
. b . | . . . | . * .
. . . | . . . | . . .
. . . | . . . | . . .

Notice how the fact that the X's never are in the same row as the "a" is self-explanatory, since if there WERE one X and one or two candidates there, that would eliminate the "a". So in other words, as long as where the X's are are NOT candidates, it does not matter what the distribution of candidates are in the remanining cells in that box.

Are you sure a candidate can exist at the intersection of the "Empty Rectangle Lines"?

Ron

[Havard]

Are you sure a candidate can exist at the intersection of the "Empty Rectangle Lines"?
Ron

I'm not sure of anything, but I think so because:

example with all candidates present at all location:

Code: Select all

. . . | . . . | X c X
. . . | . . . | X c X
. a . | . . . | c c c
--|------------------
. | . | . . . | . . .
. | . | . . . | . . .
. | . | . . . | . . .
--|------------------
. b . | . . . | . * .
. . . | . . . | . . .
. . . | . . . | . . .


No the logic would go: If "b" is a number, then the elimintation of * is obvious!
If "b" is NOT a number, then "a" would have to be a number.

now if "a" is a number you would be left with:

Code: Select all

. . . | . . . | X c X
. . . | . . . | X c X
. a . | . . . | X X X
--|------------------
. | . | . . . | . . .
. | . | . . . | . . .
. | . | . . . | . . .
--|------------------
. b . | . . . | . * .
. . . | . . . | . . .
. . . | . . . | . . .


and the * would be eliminated as a box-line intersection because of the two remaining c's.

does that make sense?

havard

[Havard]

here is another puzzle:

Code: Select all

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


that you can make a tiny bit of progress with using:


it's not much, but at least no other simpler tech can do this one much better...

[Carcul]

Hi Havard.

Just a comment: your "ER" rule is simply a different name for the already existing Grouped Turbot Fishes. All your examples can be written in terms of discontinuous grouped nice loops (X-cycles). This is because the existence of the ER forces the candidates "x" in his respective box to be distributed by only two units, and so in the box where the ER on candidate "x" lyes we have a grouped strong link on "x". For example:

Code: Select all

. a . | . . . | . . .
. | . | . . . | . X X
. | . | . . . | . X X
--|------------------
. | . | . . . | . . .
. | . | . . . | . . .
. | . | . . . | . . .
--|------------------
. b . | . . . | * . .
. . . | . . . | . . .
. . . | . . . | . . .


[r7c7]-x-[r7c2]=x=[r1c2]-x-[r1c7|r1r8|r1c9]=x=[r2c7|r3c7]-x-[r7c7], => r7c7<>x.

it's not much, but at least no other simpler tech can do this one much better...

The two"real" examples that you have posted can also be expressed in terms of a grouped X-cycle, a concept that already exists for some time.

Regards, Carcul

[ronk]

does that make sense?

With a candidate at the intersection of the "ERLs", the eventual outcome could be an x-wing ... and I see nothing in your argument to rule that out.

Ron

[Havard]

With a candidate at the intersection of the "ERLs", the eventual outcome could be an x-wing ... and I see nothing in your argument to rule that out.
Ron

sorry Ronk, I don't understand what you mean? In my second example of applied ER there is a candidate where the two ERL intersect. Can you explain it a bit better for my small brains?

[ronk]

I don't understand what you mean? In my second example of applied ER there is a candidate where the two ERL intersect.

Your deduction is correct, but I didn't think your argument proved it. However, consider the following ...

Code: Select all

. a . | . . . | c e e
. | . | . . . | d X X
. | . | . . . | d X X
--|------------------
. | . | . . . | . . .
. | . | . . . | . . .
. | . | . . . | . . .
--|------------------
. b . | . . . | * . .
. . . | . . . | . . .
. . . | . . . | . . .

... where c may be considered a member of either set D or set E. Because of the empty rectangle, one of the following must be true: c, or one of set D, or one of set E.

1. If a d is true, * is eliminated.
2. If an e is true, a is false, b is true, and * is eliminated.
3. Whether c is considered a member of set D or set E, * is eliminated.

Ron

[re'born]

Harvard,

I think that your empty rectangle rule is equivalent to the "Hinge" rule of Rod Hagglund (see http://www.sudoku.org.uk/discus/messages/29/414.html?1135572720). The only difference is that he identifies the hinge made by the remaining 5 candidates instead of the empty rectangle.

Keep up the good work, though, of finding easily recognized patterns inside the morass of grouped x-cycles and the like.

[Havard]

hi again!

In playing around with my new toy, the Empty Rectangle, I constructed this theoretical example:


all candidates with a circle (red or blue) can be eliminated!
cool, eh? Candidate Carnage!

I also have found that instead of talking about the ERL's (empty rectangle lines) it makes more sense talking about the ER-intersection(ERI) where the two ER lines meet. (the two little blue "aim" looking things on the board) This is the only point you need to think about in relation to other strong links (or other ERI's)

havard

[Havard]

Hi tarek, remember this one:

Code: Select all

*-----------------------------------------------------------------*
| 5      9      68    | 1     *48     7     | 346   *348    2     |
| 3      2      68    | 568    9      456   | 46     1      7     |
| 4      1      7     | 268    3      26    | 5      9      68    |
|---------------------+---------------------+---------------------|
| 2      367    19    | 35689  58     13569 | 367    3578   4     |
| 16    #346    5     | 2368   7      12346 | 9      38     68    |
| 8     *3467  -49    | 3569  *45     34569 | 2      357    1     |
|---------------------+---------------------+---------------------|
| 69    *46     2     | 379    1      39    | 8     *47     5     |
| 19     8      14    | 579    6      59    | 47     2      3     |
| 7      5      3     | 4      2      8     | 1      6      9     |
*-----------------------------------------------------------------*
Eliminating 4 From r6c3 (Finned Swordfish in Columns 258)


unfortunately, there is no need for the fin here either:


havard

[ronk]

In playing around with my new toy, the Empty Rectangle, I constructed this theoretical example:


all candidates with a circle (red or blue) can be eliminated!

I think of the remaining candidates in a box containing the "Empty Rectangle" (ER) as a grouped strong link. Since they may appear anywhere, many different X-cycle patterns are possible. Here is another theoretical one:



The blue and green pairings in box 3, box 5, row 9 and col 2 highlight the strong links. Together they form a continuous X-cycle of length 8 ... (I think) yielding the eliminations shown in red. This particular pattern has the look of a 'skewed swordfish'.

Ron

[tarek]

Another strong link

Code: Select all

*--------------------------------------------------------*
| 3     4     257  | 79    78    279  | 58    1     6    |
|*19    59    6    | 3    *148   24   | 7     58    29   |
| 8    *179   27   | 6    *17    5    | 4     3     29   |
|------------------+------------------+------------------|
|*5    *6    #147  |*147   3     478  | 2     9    *148  |
|-1479 -1379  1347 | 5     2     468  | 18    67    148  |
|*147   2     8    |*147   9     46   | 3     67    5    |
|------------------+------------------+------------------|
| 2     35    345  | 8     6     1    | 9     45    7    |
|*147  *17    9    | 2     457   3    | 6     458  *18   |
| 6     8     1457 | 479   457   479  | 15    2     3    |
*--------------------------------------------------------*
Eliminating 1 From r5c1 (Finned Squirmbag in rows 23468)
Eliminating 1 From r5c2 (Finned Squirmbag in rows 23468)

Tarek

[ronk]

Another strong link

To what are you referring? TIA, Ron

[tarek]

To what are you referring?

I was referring to this:

I have not yet dealt with 3 links patterns in this post. One extra (or more) strong link can be added to any of the patterns belonging to the Turbot Fish in this manner, as you know:

.......

so instead of proposing a "virtual" (finned) swordfish, I would just call it an extended skyscraper with one extra strong link:

Tarek