Another Unique Rectangle question

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Another Unique Rectangle question

Postby stumble » Sun Jan 27, 2008 2:42 am

I have a Unique Rectangle question on the bb1-26-08SuperHard puzzle. Here is the original:
Code: Select all
.---------------------.---------------------.---------------------.
| 36     7      2     | 368    4568   3458  | 9      1      5     |
| 5      9      13    | 1238   7      238   | 28     6      4     |
| 8      46     146   | 1269   1456   245   | 257    25     3     |
:---------------------+---------------------+---------------------:
| 19     45     1489  | 7      2      6     | 1358   358    59    |
| 126    3      1678  | 5      8      9     | 12678  4      267   |
| 269    256    6789  | 4      3      1     | 25678  258    25679 |
:---------------------+---------------------+---------------------:
| 7      26     369   | 2368   4568   23458 | 2356   235    1     |
| 4      1      36    | 236    9      235   | 2356   7      8     |
| 236    8      5     | 1236   16     237   | 4      9      26    |
'---------------------'---------------------'---------------------'

After the ‘easy’ cancellations I have:
Code: Select all
.---------------.---------------.---------------.
| 36   7    2   | 36   4    8   | 9    1    5   |
| 5    9    13  | 123  7    23  | 8    6    4   |
| 8    46   146 | 9    16   5   | 7    2    3   |
:---------------+---------------+---------------:
| 1    45   48  | 7    2    6   | 3    58   9   |
| 26   3    67  | 5    8    9   | 1    4    267 |
| 9    256  678 | 4    3    1   | 26   58   267 |
:---------------+---------------+---------------:
| 7    26   9   | 8    5    4   | 26   3    1   |
| 4    1    36  | 236  9    23  | 5    7    8   |
| 236  8    5   | 136  16   7   | 4    9    26  |
'---------------'---------------'---------------'

Now I think I spot a Unique Rectangle in r7c2,r7c7,r6c2,r6c7[26].
I want to cancel the 2 and 6 in r6c2, leaving a 5. Unfortunately the correct answer for that cell happens to be 6.
Code: Select all
.---------------.---------------.---------------.
| 36   7    2   | 36   4    8   | 9    1    5   |
| 5    9    13  | 123  7    23  | 8    6    4   |
| 8    46   146 | 9    16   5   | 7    2    3   |
:---------------+---------------+---------------:
| 1    45   48  | 7    2    6   | 3    58   9   |
| 26   3    67  | 5    8    9   | 1    4    267 |
| 9    5-26 678 | 4    3    1   | 26*  58   267 |
:---------------+---------------+---------------:
| 7    26*  9   | 8    5    4   | 26*  3    1   |
| 4    1    36  | 236  9    23  | 5    7    8   |
| 236  8    5   | 136  16   7   | 4    9    26  |
'---------------'---------------'---------------'

These are my notes on my subsequent thrashings to solve the puzzle, I’m sure I made several unneeded steps, but that’s my usual MO.

bb1-26-08SuperHard
[26]UniqueRectangle WRONG! R7c2,r7c7,r6c2,r6c7=>r6c2<>2or6 WRONG!!!
XwingOn2 r5c1,r5c9,r9c1,r9c9=>r6c9<>2
XwingOn3=>r1c1,r1c4,r9c1,r9c4=>r2c4,r8c4<>3
AICOn6 r1c1=r1c4-r8c4=r8c3=>r3c3<>6
XychainOn2 r5c1-r1c1-r1c4-r8c4-r8c6-r8c3-r7c2=>r6c2<>2
XychainOn6 r6c2-r4c2-r3c2=>r7c2<>6
XychainOn3 r1c4-r8c4-r8c6-r8c3-r9c1=>r1c1<>6

So… why isn’t this a Unique Rectangle?
stumble
 
Posts: 52
Joined: 29 October 2007

Postby daj95376 » Sun Jan 27, 2008 3:31 am

A Unique Rectangle must follow the Rule of 2's: Two rows, Two columns, and Two boxes.

Your pattern occupies: Two rows, Two columns, and Four boxes.

BTW #1: The Rule of 2's is not my creation. I just can't remember where I read it ... and they might not have been the first anyway! I do remember Carcul telling me that a deadly pattern can occur in N rows, N columns, and N boxes. My brain went TILT!

BTW #2: You are not alone. I don't remember how many times I've seen others make this mistake after I made it!
daj95376
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Postby stumble » Sun Jan 27, 2008 6:45 pm

daj95376 wrote:A Unique Rectangle must follow the Rule of 2's: Two rows, Two columns, and Two boxes.

Your pattern occupies: Two rows, Two columns, and Four boxes.

BTW #1: The Rule of 2's is not my creation. I just can't remember where I read it ... and they might not have been the first anyway! I do remember Carcul telling me that a deadly pattern can occur in N rows, N columns, and N boxes. My brain went TILT!

BTW #2: You are not alone. I don't remember how many times I've seen others make this mistake after I made it!

Thanks, daj. I was not aware of that rule and I read thru the Unique info I'd printed out and didn't spot my error. What I was really afraid of was that I'd erred in the creation of my 2nd grid, and I was too lazy to recreate it to find out.
stumble
 
Posts: 52
Joined: 29 October 2007

Postby Steve K » Mon Jan 28, 2008 11:03 am

Although this puzzle solves without any uniqueness techniques, one does exist. Note candidates 26 at r5c19, r6c27, r7c27, r9c19. To prevent a 4x4x4 column,row,box deadly pattern (I prefer forbidden pattern, in my head, anyway) one needs at least one of the following:
(5)r6c2,(7)r5c9,(3)r9c1. Each of these forces, respectively by box, row, column (2)r5c1. Thus one can safely conclude (2)r5c1.
Unfortunately, the puzzle is still not reduced to singles.
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Posts: 98
Joined: 18 January 2007

Postby Sudtyro » Mon Jan 28, 2008 6:21 pm

Code: Select all
.---------------.---------------.---------------.
| 36   7    2   | 36   4    8   | 9    1    5   |
| 5    9    13  | 123  7    23  | 8    6    4   |
| 8    46   146 | 9    16   5   | 7    2    3   |
:---------------+---------------+---------------:
| 1    45   48  | 7    2    6   | 3    58   9   |
|*26   3    67  | 5    8    9   | 1    4   *267 |
| 9   *256  678 | 4    3    1   |*26   58   267 |
:---------------+---------------+---------------:
| 7   *26   9   | 8    5    4   |*26   3    1   |
| 4    1    36  | 236  9    23  | 5    7    8   |
|*236  8    5   | 136  16   7   | 4    9   *26  |
'---------------'---------------'---------------'

Nice find, Steve K. This seems to be a really good example of the 5th 8-cell configuration (slightly transformed) in Sudopedia's Deadly Pattern article.

[Edit: AIC question withdrawn.]
Last edited by Sudtyro on Mon Jan 28, 2008 7:55 pm, edited 1 time in total.
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Posts: 68
Joined: 21 December 2006

Postby Steve R » Mon Jan 28, 2008 7:01 pm

Code: Select all
.---------------.----------------.---------------.
| 36   7    2   | 36    4    8   | 9    1    5   |
| 5    9    13  | 23+1  7    23  | 8    6    4   |
| 8    46   146 | 9     16   5   | 7    2    3   |
:---------------+----------------+---------------:
| 1    45   48  | 7     2    6   | 3    58   9   |
| 26   3    67  | 5     8    9   | 1    4    267 |
| 9    256  678 | 4     3    1   | 26   58   267 |
:---------------+----------------+---------------:
| 7    26   9   | 8     5    4   | 26   3    1   |
| 4    1    36  | 23+6  9    23  | 5    7    8   |
| 236  8    5   | 136   16   7   | 4    9    26  |
'---------------'----------------'---------------'


The complicated uniqueness patterns are more interesting but the UR based on (23) in r28c46 implies that r3c4 contains 1 or r8c4 contains 6.

The (16) pair in column 5 is thus resolved. So is the puzzle.

Steve
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