The New Sudoku Players' Forum

by **Havard** » Sat Apr 15, 2006 3:39 pm

keith wrote:Question 2:

Havard wrote:
Type 3: UR with two extra candidates in addition to the UR-pair. Can create a locked set with other cells, as long as all the UR-extra-candidates-cells are concidered one cell in the locked set, and of course all cells in the locked set can see eachother...

Would it not be more accurate to say that the UR-pair is one cell in the locked set? Since the UR-pair can occupy only one of two cells in the UR. Then, the locked set of 2+N possibilities will only occupy 1+N cells?

To say it another way: The extra candidates will be fully represented in the solution of the locked set. Only one (of two) of the UR candidates will be there.

Thanks to all,

Keith

Hi Keith!

I guess I was thinking in very general terms, and that it is possible for the type 3 to have extra candidates in three cells:

Code: Select all: * xy abxy---abxy | | | | | | ab-----abxy

Code: Select all: . . . | . . * | . . . . . . | . . xy| . . . . . . | xy--xy| . . . ------+-|---|-+------ . . . | | . | | . . . . . . |ab---xy| . . . . . . | . . . | . . . ------+-------+------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

As you can see from these illustrations, the * can not be either x or y, because that would with the xy cell eliminate all xy's in the UR, and make a deadly pattern! As you can see, this would only apply for this special case, and would only ever be able to eliminate in one cell, but I still think it is worth mentioning, and that is why I generalised my description of the type 3 to not just include two cells

havard

by **Mike Barker** » Sat Apr 15, 2006 5:16 pm

Type 9 is also a continuous nice loop. If the nodes are numbered starting with the bottom left then

Code: Select all: xabx====xabx || a || ||b b|| || || xabx----xabx N1=b=N2=a=N3=b=N4-b-N1 => N2,N3={ab}

Also b can be eliminated from the cells in the line containing N1 and N4 excluding N1 and N4. The UR is now type 6. I wonder if this is why Ron didn't see any Type 9 URs or maybe they are just rare?

by **keith** » Sat Apr 15, 2006 8:34 pm

Found in the wild, another Type-6 Unique X-wing.

http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=669

I would be interested in how to classify the other UR in this puzzle.

By the way, in the tests of frequency of Type-6, do we know how many have just one conjugate link, and are not X-wings?

Keith

by **Myth Jellies** » Sat Apr 15, 2006 8:56 pm

Keith, you could have performed either a type 3 or a type 4 (what you did) reduction on that efgh rectangle. The type 3 reduction would note the virtual 24 naked pairing in ghi and eliminate the 24's in all other cells in the box/row.

by **ronk** » Sat Apr 15, 2006 9:21 pm

Myth Jellies wrote:Keith, you could have performed either a type 3 or a type 4 (what you did) reduction on that efgh rectangle. The type 3 reduction would note the virtual 24 naked pairing in ghi and eliminate the 24's in all other cells in the box/row.

I agree, and since exclusions due to type 4 "destroy" the type 3 UR, most programmed solvers first perform exclusions due to type 3, if any.

keith wrote:By the way, in the tests of frequency of Type-6, do we know how many have just one conjugate link, and are not X-wings?

If you're asking about URs where the only extra candidates are on a diagonal, I earlier reported 28 candidates excluded (in the top1465) due to "type 7" versus 8 candidates for type 6. Note that type 6 had priority over type 7, and if the type 6 UR technique had been disabled, 4 of the 8 type 6 exclusions might have been reported as type 7.

I should also point out that almost every other logical technique had a crack at the puzzles BEFORE the uniqueness technique.

by **keith** » Mon Apr 17, 2006 2:12 am

There is a UR in Ruud's Nightmare of April 17 that is both Type-1 and Type-6.

http://www.sudocue.net/forum/viewtopic.php?t=114

Of course, the Type-1 reduction forces the placement of the values <3>, whether you recognize the Type-6 or not.

Best wishes,

Keith

by **ronk** » Mon Apr 17, 2006 4:00 am

keith wrote:There is a UR in Ruud's Nightmare of April 17 that is both Type-1 and Type-6.

Type 1 has extra candidates in only one cell, and type 6 has extra candidates in diagonal cells ... so how can that statement be correct?

by **keith** » Mon Apr 17, 2006 5:00 am

Ron,

The Type-6 pattern is, to me, an overlay of an X-wing on a UR. It is useful if the UR does not have extra possibilities on one diagonal. It does not require extra possibilities on both the other diagonal squares.

In this situation:

Code: Select all: +----------------+----------------+----------------+ | 2589 78 3 | 1 2678 2678 | 256 569 4 | | 25 14e 14f | 9 236 26 | 2356 7 8 | | 289 6 79 | 4 2378 5 | 123 39 129 | +----------------+----------------+----------------+ | 89 78 5 | 27 4 3 | 267 1 2679 | | 34a 34b 6 | 278 1 2789 | 257 59 2579 | | 1 2 79 | 6 5 79 | 8 4 3 | +----------------+----------------+----------------+ | 34c 134d 148g | 5 9 4678 | 1467 2 167 | | 6 9 248 | 2378 278 1 | 3457 358 57 | | 7 5 1248 | 238 268 2468 | 9 368 16 | +----------------+----------------+----------------+

abcd is an X-wing on <3>, and a UR on <34>.

a = d = <3> would force the Deadly Pattern, (b = c = <4>), so we must have b = c = <3>.

The Type-1 reduction is, of course, d = <1>.

Which sort of brings up my previous question: Are these UR classifications lists of the possibility patterns, or of the reductions?

And now, for me, to bed.

Keith

by **ronk** » Mon Apr 17, 2006 12:09 pm

keith wrote:The Type-6 pattern is, to me, an overlay of an X-wing on a UR. It is useful if the UR does not have extra possibilities on one diagonal. It does not require extra possibilities on both the other diagonal squares.

I think I see your point. We only need the UR pair without any extra candidates on one diagonal.

Code: Select all: Type 1 UR: . ab . | . ab+(Y) . . a|. . | . a|. . . ab+X . | . ab . Lower case indicates exactly one candidate Upper case indicates one or more candidates Parenthesis indicates "optional" candidate(s)

Rereading ...

Ruud wrote:When a rectangle can be found in a single floor or tower, with candidates a and b as the only possibilities in 2 of its corners, and candidates a and b are also present in the other 2 corners along with extra candidates, and the candidates for digit a form an X-Wing pattern, then candidate a can be removed from the 2 corners with extra candidates.

... it seems that's what Ruud said too. My bad!

by **Mike Barker** » Sat Apr 22, 2006 4:30 am

Based on Havard's and Keith's works I've a couple of more unique rectangles to throw into the pool. The first combines types 2 and 5:

Code: Select all: . ab | abx . . . abx | abx * * . . | . . . where x is a single digit common to the three "abx" and eliminations of "x" can occur at "*"

The second may have already appeared, but I'm not finding it. It uses the strong link of type 7 with the configuration of type 4:

Code: Select all: . ab | ab . . | | . a| | . . . | | . abX | abY . . where X and Y are possibly different with possibly more than one element and the strong link on "a" allows elimination of "b" in "abY"

Also it seems that the eliminations created by a type 6 (diagonal form with an X-wing) are equivalent to those of type 7 (one strong link) where type 7 is applied twice, so it is somewhat redundant.

Based on these additions I'd recommend the following definitions for UR types:

There are lots of other alternatives. The argument for the history of type 5 being the diagonal version is compelling, but with the added types (2C, 3C, 3D, and 4C) and redundancy of strong links and X-wing a contraction of the types seems appropriate.

by **Myth Jellies** » Sat Apr 22, 2006 7:57 am

I don't know that catagorizing dozens of different kinds of UR reductions as different "types" really buys us anything. The thing to note is that you have to avoid the deadly pattern and that allows us to make various nifty reductions. It's cool that strong links can be used as an aid in this, but I don't know that I care to discern between a type 2D vs new type 5 vs old type 5.

How about just refering to these reductions as what they are: things like A UR+1; A UR+2 assisted naked pair; a UR+2 w/a locked candidate; or a UR+2 X-Wing. A little extra verbiage might avoid some future arguments and save some wear and tear on the old memory cells.

by **Havard** » Sat Apr 22, 2006 12:59 pm

Mike Barker wrote:The second may have already appeared, but I'm not finding it. It uses the strong link of type 7 with the configuration of type 4:
Code: Select all
. ab | ab . . | | . a| | . . . | | . abX | abY . . where X and Y are possibly different with possibly more than one element and the strong link on "a" allows elimination of "b" in "abY"

Also it seems that the eliminations created by a type 6 (diagonal form with an X-wing) are equivalent to those of type 7 (one strong link) where type 7 is applied twice, so it is somewhat redundant.

Great find Mike!

Here is a puzzle that jumped out when I was fooling around a bit with the "new rules"!

Code: Select all: Unique Rectangle type 3: 247 1 8 | 5 379U# 6 | 3479U# 2479- 34# 9 257 256 | 47 8 34 | 1 267 356 3 457 56 | 2 79U 1 | 479U# 8 456 ---------------------+----------------------+--------------------- 5 37 4 | 9 137 8 | 6 17 2 27 6 9 | 147 1257 245 | 478 3 148 8 237 1 | 467 2367 234 | 5 47 9 ---------------------+----------------------+--------------------- 1246 9 25 | 8 1256 7 | 34 146 1346 126 258 3 | 16 4 25 | 89 169 7 146 48 7 | 3 16 9 | 2 5 1468 Unique Rectangle with 2 strong links: 247 1 8 | 5 379 6 | 3479 279 34 9 257 256U-6 | 47 8 34 | 1 267 356U 3 457 56U | 2 79 1 | 479 8 456U ---------------------+----------------------+--------------------- 5 37 4 | 9 137 8 | 6 17 2 27 6 9 | 147 1257 245 | 478 3 148 8 237 1 | 467 2367 234 | 5 47 9 ---------------------+----------------------+--------------------- 1246 9 25 | 8 1256 7 | 34 146 1346 126 258 3 | 16 4 25 | 89 169 7 146 48 7 | 3 16 9 | 2 5 1468 Unique Rectangle with 1 Strong Link and diagonal extras: 247 1 8 | 5 379 6 | 3479U 279 34U 9 257 25 | 47 8 34 | 1 267 356 3 457 6 | 2 79 1 | 479 8 45 ------------------------+-------------------------+------------------------ 5 37 4 | 9 137 8 | 6 17 2 27 6 9 | 147 1257 245 | 478 3 148 8 237 1 | 467 2367 234 | 5 47 9 ------------------------+-------------------------+------------------------ 1246 9 25 | 8 1256 7 | 34U 146 1346U-3 126 258 3 | 16 4 25 | 89 169 7 146 48 7 | 3 16 9 | 2 5 1468

None of the steps is really that useful, but I thought it would be good to see som "live" examples of all this theory!

Havard

by **Havard** » Sat Apr 22, 2006 1:03 pm

here is a few more:

Code: Select all: Unique Rectangle with 1 Strong Link not connecting the in-line extras: (Mike Barkers discovery) 8 4 39 | 13 5 6 | 7 2 19 7 1 2 | 4 9 8 | 6 5 3 69 36 5 | 7 12 23 | 89 48 149 ---------------------------+----------------------------+--------------------------- 13469U-6 5 8 | 1369 146 349 | 2 16U 7 169 67 79 | 1569 126 259 | 4 3 8 1346U 2 34 | 8 7 34 | 59 16U 59 ---------------------------+----------------------------+--------------------------- 2 78 47 | 56 468 1 | 3 9 45 345 38 6 | 59 48 459 | 1 7 2 45 9 1 | 2 3 7 | 58 48 6 Unique Rectangle with 2 strong links: 8 4 39 | 13 5 6 | 7 2 19 7 1 2 | 4 9 8 | 6 5 3 69 36 5 | 7 12 23 | 89 48 149 ------------------------+-------------------------+------------------------ 1349 5 8 | 1369 146 349 | 2 16 7 169 67 79 | 1569U-9 126 259U | 4 3 8 1346 2 34 | 8 7 34 | 59 16 59 ------------------------+-------------------------+------------------------ 2 78 47 | 56 468 1 | 3 9 45 345 38 6 | 59U 48 459U | 1 7 2 45 9 1 | 2 3 7 | 58 48 6

by **Havard** » Sat Apr 22, 2006 4:45 pm

heh, I also found this one...

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

Now I have put UR's way up front in my solver, but it came out with these steps (all in this puzzle, only UR-eliminations shown)

Code: Select all: Unique Rectangle type 2: 2 479 3 | 469 8 1 | 4579-9 45679 5679 8 49 1 | 7 4569 4569 | 2 469 3 5679 479 4567 | 2349 249 2349 | 1 479 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 278 57U | 2689 269 2689 | 579U 3 1 357 2378 57U | 1 249 2489 | 579U 5679-9 5679-9 ------------------------+-------------------------+------------------------ 3679 3479 467 | 3469 1 34569 | 34579-9 8 2 379 5 2478 | 23489 2479 23489 | 6 1 79 3679 1 24678 | 234689 245679 2345689 | 34579-9 4579 579 Unique Rectangle with 1 Strong Link not connecting the in-line extras: 2 479 3 | 469 8 1 | 457 45679 5679 8 49 1 | 7 4569 4569 | 2 469 3 5679 479 4567 | 2349 249 2349 | 1 479 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 278 57U | 2689 269 2689 | 579U 3 1 357 2378 57U | 1 249 2489 | 579U-7 567 567 ------------------------+-------------------------+------------------------ 3679 3479 467 | 3469 1 34569 | 3457 8 2 379 5 2478 | 23489 2479 23489 | 6 1 79 3679 1 24678 | 234689 245679 2345689 | 3457 4579 579 Unique Rectangle with 1 Strong Link and diagonal extras: 2 79U 3 | 69 8 1 | 4 5679U-7 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479U 6 | 2349 249 2349 | 1 79U 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 2689 269 2689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ------------------------+-------------------------+------------------------ 679 3 4 | 69 1 569 | 57 8 2 79 5 28 | 23489 2479 23489 | 6 1 79 679 1 28 | 2689 25679 25689 | 3 4 579 Unique Rectangle with 1 Strong Link not connecting the in-line extras: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 2349 249 2349 | 1 79 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 28U 57 | 2689 269 2689U-2 | 579 3 1 3 28U 57 | 1 249 2489U | 59 567 567 ------------------------+-------------------------+------------------------ 679 3 4 | 69 1 569 | 57 8 2 79 5 28 | 23489 2479 23489 | 6 1 79 679 1 28 | 2689 25679 25689 | 3 4 579 Unique Rectangle with 1 Strong Link not connecting the in-line extras: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 2349 249 2349 | 1 79 8 ---------------------+----------------------+--------------------- 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 2689 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ---------------------+----------------------+--------------------- 679 3 4 | 69 1 569 | 57 8 2 79U 5 28 | 23489 2479 23489 | 6 1 79U 679U-7 1 28 | 2689 25679 25689 | 3 4 579U Unique Rectangle with 1 Strong Link and diagonal extras: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 2349 249 2349 | 1 79 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 2689 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ------------------------+-------------------------+------------------------ 679U 3 4 | 69U 1 569 | 57 8 2 79 5 28 | 23489 2479 23489 | 6 1 79 69U 1 28 | 2689U-6 25679 25689 | 3 4 579 Unique Rectangle with 2 strong links: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 2349 249 2349 | 1 79 8 ------------------------+-------------------------+------------------------ 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 2689 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ------------------------+-------------------------+------------------------ 679 3 4 | 69 1 569 | 57 8 2 79 5 28 | 23489 2479U-9 23489 | 6 1 79U 69 1 28 | 289 25679U 25689 | 3 4 579U Unique Rectangle with 3 strong links: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 2349U 249 2349U-4 | 1 79 8 ---------------------------+----------------------------+--------------------------- 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 2689 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ---------------------------+----------------------------+--------------------------- 679 3 4 | 69 1 569 | 57 8 2 79 5 28 | 23489U 247 23489U-4 | 6 1 79 69 1 28 | 289 25679 25689 | 3 4 579 Unique Rectangle type 1: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 234 249 239 | 1 79 8 ---------------------------+----------------------------+--------------------------- 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 28 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ---------------------------+----------------------------+--------------------------- 679 3 4 | 69 1 569 | 57 8 2 79 5 28U | 2348U-28 247 2389 | 6 1 79 69 1 28U | 28U 25679 25689 | 3 4 579 Unique Rectangle with 1 Strong Link not connecting the in-line extras: 2 79 3 | 69 8 1 | 4 569 567 8 49 1 | 7 4569 4569 | 2 69 3 5 479 6 | 234 249 239 | 1 79 8 ---------------------------+----------------------------+--------------------------- 1 6 9 | 5 3 7 | 8 2 4 4 28 57 | 28 269 689 | 579 3 1 3 28 57 | 1 249 2489 | 59 567 567 ---------------------------+----------------------------+--------------------------- 679 3 4 | 69 1 569 | 57 8 2 79 5 28U | 34 247 2389U | 6 1 79 69 1 28U | 28 25679 25689U-2 | 3 4 579

I think it pretty much covers all the "new" UR discoveries!

Havard

by **ronk** » Sat Apr 22, 2006 4:50 pm

Havard wrote:Unique Rectangle with 1 Strong Link not connecting the in-line extras: (Mike Barkers discovery)

Your example has a conjugate link between the same two cells for each UR digit.

Code: Select all: . ab . | . ab . || | . a||b . | . . . || | . ab+X . | . ab+Y . Excludes UR digits a and b from the ab+Y cell

Therefore, both the 1 and 6 may be excluded from r4c1.

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