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This article needs an example for Uniqueness Test 5, but I don't have any. Whoever have one, please add it in. Thanks. --Unkx80 04:27, 11 November 2006 (CET)

There should be something about the demand that the rectangle must be formed within 2 boxes, not 4! It's rather important.

The Unique Rectangle article already stresses this point, but I've repeated the definition in the introduction.

--Ruud 02:25, 13 November 2006 (CET)

Error in Uniqueness Test 6?

There appears to be a contradiction in the text: "one of these cells must be a 9 and the other coming from one of the extra candidates. So the digit 9 can be safely eliminated from these two cells." Shouldn't it say "one of the cells must be a 6"?

Good catch, corrected now.

Ruud 22:07, 17 December 2007 (CET)

Uniqueness Test 6 again

The text is correct now (I think), but the picture is misleading (the 6 are red, which in most examples stands for "possible elimination"). I provided a new one if you want to change it. -- hobiwan

Hobiwan, thanks for the replacement picture, but why is r5c5 shaded yellow? If Ruud can't revise the original one we can use it. As I've suggested to Professor Prune on his talk page, I think this test and much of what follows should be split into a subsidiary entry "Advanced UR Patterns" with additional new material that I'm slowly preparing. He doesn't appear to have seen it yet though. When I've finished running through the cases, I'll know better, but I think most of them will be illustrated just with the two boxes containing the UR. After this has been posted and everyone has had a chance to comment, we can edit this main entry again. Dpbobelisk 10:41, 25 January 2008 (CET)

The yellow cell is the cell with the focus. I didn't think about it, but I admit that it is confusing. I'll correct it. -- hobiwan

Upgraded the image. -- hobiwan

Uniqueness test 6 example still bad

The text currently reads like this:

An unique rectangle on the digits 6 and 9 plus two other candidates can be found in the intersections of rows 7 and 9, and the columns 3 and 5. In these rows and columns, the candidates for digit 9 can only be found in r7c3, r7c5, r9c5 and r9c3.

This text matches the example.

Assigning 9 to either r7c5 and r9c3 forces 6 in the other corner cells to give one of the two forbidden patterns. So the digit 9 can be safely eliminated from these two cells.

I think it is the digit 6 that can be eliminated. If it really is 9, then the rule is misstated:

Suppose exactly two cells in the rectangle contain extra candidates, and they are located diagonally across each other in the rectangle. Suppose the common candidates are U and V, and none of the other cells in the two rows and two columns containing the rectangle contain U. Then V can be eliminated from these two cells. This is also called a Type 6 Unique Rectangle.

Relating this to the example 9 is the U value and 6 is the V value. Hence 6 is the one that should be eliminated.

--Professor Prune 26 January 2008

Yes the rule definition is wrong. The recently corrected example leads to a valid full solution whereas the original version didn't. This is easy to check as the UR trick isn't needed to solve this particular puzzle. I've therefore edited the definition and changed over to Hobiwan's picture.

Dpbobelisk 23:27, 26 January 2008 (CET)

I have greatly expanded this article to include the information in Mike Barker's excellent treatise on Unique Rectangles in the Sudoku Players' Forum (see the external link in the article). I think we should encourage the more general notation suggested by Mike and Myth Jellies, vs. the older "Type X" classification.

I still have to add the "UR+4" series of patterns.

Many of the patterns need examples and further explanation of how the deductions work (although the reader can probably figure them out after reading the earlier examples that do have full explanations).

Example Eureka notation would be helpful (Eureka notation is one of my weak points).

--Professor Prune 00:31, 7 February 2008 (EST)

Comment on latest additions

Prof Prune, Your modifications have taken me by surprise, as I thought we'd agreed that the main page should major on the basic UR patterns 1 to 5 and the more advanced ones would be listed on an ancillary page. You've duplicated the work I've been doing on categorising the further patterns but keeping to Mike's classification codes, whereas I've been investigating other ways.

I've been working up a standard presentation to give:

• The cell diagram,
• A Eureka notation of the elimination(s) it provides
• A written description of the requirements
• A written justification of the eliminations
• Comments

I was hopeful that this could be displayed compactly in a table. I e-mailed Ruud asking for his Grid template so I could reduce it to two boxes and nest it in the new table, but he must be away or something, as he hasn't replied yet.

As you've pointed out, the order that URs were first discovered tends to obscure that the lower numbers are subtypes of later ones - eg UR type 3 is a subtype of UR type 5. Now that this is history I was aiming to present them more methodically.

I looked on the main page as an introduction for beginners with all the examples and the ancillary page, for those the more expert who wanted to include UR deductions into longer inference chains.

I would also like to avoid mixing in Killer and other Sudoku variant considerations in with the basic ones where these could be a cause of confusion. I'm not unhappy with how you've handled this so far, but as soon as we get any killer examples I think they deserve another sub-entry.

As I wrote earlier, when it was complete, I was planning to post my work as new pages for comment and possible correction before making any changes to the original entries. I'm therefore disappointed that you've taken a different approach. I want to avoid too many cooks spoiling the broth though, so I'll step back and see if anyone else wants to comment.

Dpbobelisk 11:52, 7 February 2008 (CET)

Dpobelisk,

Sorry about that. I only recently discovered Mike's extension and comprehensive classification of URs, which I think is very good, and I guess I got carried away in my enthusiasm.

You're right--it is an awful lot for the newcomer to digest at once, and a dual approach--with the more advanced UR versions on a separate set of pages--would be better than what I just did.

Tonight I'll back out my changes (but I will correct the error I found in my Hidden Unique Rectagle description). Please do keep working on your UR pages.

Professor Prune 12:01, 7 February 2008 (EST)

Uniqueness Test 6 & Hidden Unique Rectangle

Hi,

I'm implementing Uniqueness Test in my program JSudoku. So far I implemented type 1 to 5.

However I have problems understanding Uniqueness Test 6 & Hidden Unique Rectangle. For me, both reasonings are flawed.

Trying to understand the example for Uniqueness Test 6, as I see it, the reason r7c5 and r9c3 cannot have 9 is not in the UR and X-Wing themself, but because it would otherwize conflict with 6 at c7 locked in r79c7 (and also 6 at r8 locked in r8c25).

The example for Hidden Unique Rectangle is indeed almost the same pattern as Uniqueness Test 6, the UR matches an X-Wing on 4 except here only one corner cell has 2 candidates. Again, the reason r2c8 cannot be 3 is because it would conflict with 3 at c3 locked in r28c3, not because of the UR and X-Wing alone.

For both Uniqueness Test 6 & Hidden Unique Rectangle, it basically says something like "this cell (in the UR) cannot be X because it would give a dual solution". But if the cell is X, then there is no dual solution anymore, because it breaks the UR. In other words, I do not see any other solution with the cell = X

What did I miss ? Can someone show me the two solutions with the cell = X for both Uniqueness Test 6 & Hidden Unique Rectangle ?

TIA

PS: While investigating URs, I found a "UR hidden pair" which of course as its complementary naked "heptuplet" as a Uniqueness Test 3.

Although it's not really new, I believe it could be easier to spot a UR hidden pair than two ALS with seven candidates. Not sure it's very common, thought.

An example r1c12 = {12}, {12} at r4 locked in r4c125. Since r4c12 cannot have both 1 and 2 -> r4c5 = {12}, r4c12 must include one of {12}

Jean-Christophe 14:05, 16 April 2008 (CEST)

Regarding the logic behind Hidden Unique Rectangle:

The pattern for HUR is four cells A, B, C, D that share exactly two rows, columns, and boxes between them, where A=ab, B=abX, C=abY, D=abZ, and a is strongly linked between B and D and between C and D. We have postulated that the Sudoku has only one solution. The combinations A=a,B=b,C=b,D=a and A=b,B=a,C=a,D=b can't be true because they are interchangeable and the puzzle would have at least two solutions. But A=b,B=a,C=a,D=b is the only combination where D=b, because of the two strong links and because A must be either a or b. D thus cannot be b.

I think a similar line of reasoning applies to UR Type 6, but I haven't worked that one through.

Professor Prune 16:04, 16 April 2008 (EST)

Thanks for the reply,

I also asked explanations at sudoku.com

I didn't realized the deadly pattern could also be formed by solved cells with a single candidate, as long as none of the cells is given.

The Deadly pattern article here does not list this case, but only the case where each of the 4 cells have the two candidates. IMO it should also list that case.

Jean-Christophe 08:42, 17 April 2008 (CEST)

The diagram where each of the 4 cells has the two candidates is sort of a shorthand way of indicating the two interchangeable combinations. See also Avoidable Rectangle.

Professor Prune 10:26, 17 April 2008 (EST)

There seem to be some minor errors in the text for "Hidden Unique Rectangles":

"such that AD and BC are in opposite corners, and either AB or BC share a box": Since BC are in opposite corners, they can't share a box (that would place all four cells in one box) --- should be "BD share a box" instead of "BC share a box"

"Therefore, b can be eliminated as a candidate for B": should be "as a candidate for D"

Hobiwan 21:21, 19 September 2008 (MET)

Thank you, Hobiwan. You are correct. I have fixed the article.

Professor Prune 14:31, 17 September 2008 (EDT)

typo in Hidden Unique Rectangle docs?

"How this works: Suppose cell D has the value b. Then both C and D must have the value a, since these cells share a house with D where they are the only other place an a can go. This means that cell A must have the value b. But since none of the four cells in the rectangle is a given, this is precisely the deadly pattern that allows for two solutions to the Sudoku--we could swap a for b in the rectangle and still have a valid solution. Therefore, b can be eliminated as a candidate for D."

Unless I'm totally missing the boat, which I may be, where it says "Then both C and D must have the value a", I think it SHOULD read "Then both B and C must have the value a". For one thing, we just supposed cell D has the value b, and we immediately conclude from that that D "must have the value a".

re:

what I was looking for, thanks