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Strong inferences induced by URs

 
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Sat Apr 04, 2009 11:39 am    Post subject: Strong inferences induced by URs Reply with quote

Strong inferences induced by the Unique Rectangle.

Unique rectangles, or the UR, present yet another but interesting way in forming a strong inference. As a demonstration and
hopefully a useful tool to rely on, this thread will provide examples of some inferences that are not commonly sought after
as steps along the path to a solution.
All of these examples rely on the core intuitive idea behind the UR. If the candidates that form the UR are locked into the
UR cells then the deadly pattern is forced to exist. the puzzle will have multiple solutions.
There are common moves (type 1...type 2...etc) that prevent the UR candidates from being the only candidates to occupy the UR cells.

this concept needs to be understood or this thread may not make sense.
-----------
the remaining text will provide examples of the following...

I. The type 3 UR provides the most common strong inference created by a UR.
II. Inferences associated with the Type 2 UR
III. Inferences associated with the Type 4 UR
IV An uncommon UR inference
V. A short cut method to some classic Type 3 URs
-----------


I.The type 3 UR is the most common example of a strong inference that can be formed in a UR
for example
this portion of a grid

Code:
| .  12 A123 |
--------------
| .  12 B124 |

the strong inference can be made between the 3 at A and the 4 at B
UR12[(3)B = (4)B]... this is because neither the 4 nor the 3 can both be false or the UR is left, which is a NO NO !!

the most common use of the Type 3 UR (and its original definition) is seeing the 3 and 4 as a single cell and forming a locked set with another {3,4} cell in the same box, row or column. AND!! seeing any candidates in the A and B cells and forming a locked "tuple" with other candidates in the same box, row or column.

the other use of the strong inference created is to use the 3 and the 4 in a chain.
this is a partial example
x cells cannot be 4
Code:
+----------+--------+--------+
| . 12 123 | . 37 . | . 47 . |
| . 12 124 | . .  . | x x  x |
| . .  .   | . .  . | . .  . |
+----------+--------+--------+
| . .  .   | . .  . | . .  . |
| . .  .   | . .  . | . .  . |
| . .  .   | . .  . | . .  . |
+----------+--------+--------+
| . .  .   | . .  . | . .  . |
| . .  .   | . .  . | . .  . |
| . .  .   | . .  . | . .  . |
+----------+--------+--------+
if the 4 in r2c3 is true, then r2c789 cannot be 4
if the 4 in r2c3 is not true, then r1c3 has to be a 3 (rules of a UR type 1)
leads to... r1c5 is 6... r1c8 is 4... r2c789 cannot be 4.
either way, the 4's in r2c789 can not be 4
or it can be viewed as this chain
UR12[(4)r2c3 = (3)r1c3] - (3=7)r1c5 - (7=4)r1c8; r2c789 <> 4

the type 3 UR provides a nice alternative strong inference that may be needed to keep the puzzle moving.

if the type 3 UR is a common sudoku tool then its no surprise that the next few examples in this thread should be recognized.
------------


II. Inferences associated with the type 2 UR

Part A. examples of strong inferences induced by the classic type 2 UR.
as unorthodox as it may be, explaining Type 2 UR inferences after the type 3 is necessary because of a simple fact. The Type 2 UR needs more candidates included in the cells of the UR. Naturally, the inferences are going to happen accross more cells. Its the nature of the beast.

what is a type 2 UR?
here it is
Code:
+--------+---------+-------+
| . 12 . | 123 . . | . . . |
| . 12 . | 123 . . | . . . |
| . .  . | .   . . | . . . |
+--------+---------+-------+

the example says that any other 3's in box 2 and any other 3's in column 4 can be eliminated.

adding some candidates to the grid...
Code:
+---------+---------+-------+
| . 124 . | 123 . . | . . . |
| . 124 . | 123 . . | . . . |
| . .   . | .   . . | . . . |
+---------+---------+-------+

the grid now says that there exists a strong inference on the 4's in r12c2 and the 3's in r12c4. Again, if both the 3's and the 4's are removed then the UR is forced to exist in the UR cells. neither can both be false.
this strong inference is made... UR12[(4)r12c2 = (3)r12c4]...
imagine removing the 4's, then by the rule of a type 2, the 3's must exist in r12c4. the opposite is also true.

the following are examples of chains that can come in handy using this type of strong inference.


example 1...
the x cells cannot be 5.
Code:
+---------+---------+-------+
| . 124 . | 123 . . | . . . |
| . 124 . | 123 . . | . . . |
| . .   . | .   . . | . . . |
+---------+---------+-------+
| . .   . | .   . . | . . . |
| . 45  . | x   x x | . . . |
| x x   x | 35  . . | . . . |
+---------+---------+-------+
(5=4)r5c2 - UR12[(4)r12c2 = (3r12c4] - (3=5)r6c4


example 2...
the x cells cannot be 5
Code:
+---------+----------+-------+
| . x   . | .   . 35 | . . . |
| . 124 . | 123 . .  | . . . |
| . 124 . | 123 . .  | . . . |
+---------+----------+-------+
| . .   . | .   . .  | . . . |
| . 45  . | .   . x  | . . . |
| . .   . | .   . .  | . . . |
+---------+----------+-------+

(5=4)r5c2 - UR12[(4)r23c2 = (3)r23c4] - (3=5)r1c6; r1c2 and r5c6 cannot be 5



Part B examples of strong inferences induced by the type 2B UR
the type 2B states that the roof cells happen in two separate boxes.
Code:
+---------+---------+-------+
| .   . . | .   . . | . . . |
| 12  . . | 12  . . | . . . |
| 123 . . | 123 . . | . . . |
+---------+---------+-------+
| .   . . | .   . . | . . . |
| .   . . | .   . . | . . . |
| .   . . | .   . . | . . . |
+---------+---------+-------+

the grid says that no other 3 in r3 can exist (this is not to be confused with the type 4 which will come later)
obviously the same rules apply as a type 2 except that since the roof cells are ONLY in the same row, then that row is where the eliminations happen. in this case row 3.

now adding a candidate into the mix and some interesting inferences can be made.


example 3...
the simplest example of a type 2B strong inference.
x cells cannot be 4
Code:
+---------+---------+--------+
| .   . . | .   . . | . . .  |
| 12  . . | 124 . . | x x x  |
| 123 . . | 123 x x | . . 34 |
+---------+---------+--------+

imagine removing the 3's in r3c14, then the 4 must exist in r2c4 (rules of a type 1 UR). if the 4 is removed from r2c4, then the 3's must exist in r3c14 (rules of a type 2B UR). if either is the case, then the 4's in r2c789 cannot exist.
the strong inference is between the 4 in r2c4 and the 3's in r3c14 because if both are false, the UR is left.
UR12[(4)r2c4 = (3)r2c14] - (3=4)


example 4... the sillyness starts to make its presence felt.
x cell cannot be 3
Code:
+---------+-----------+----------+
| .   . . | .   36 .  | 45 . .   |
| 12  . . | .   .  .  | .  . 124 |
| 123 . . | 379 39 37 | .  . 123 |
+---------+-----------+----------+
| .   . . | .   .  .  | .  . .   |
| .   . . | .   .  .  | .  . .   |
| .   . . | .   x  .  | 35 . .   |
+---------+-----------+----------+

again the strong inference is on the 4 in r2c9 and the 3's in r3c19...UR12[(4)r2c9 = (3)r3c19]...
remember that if the 4 is removed from r2c9 then a type 2B exists in r23c19 on {1,2} and the 3's in r3c19 must remain in order to break up the UR pattern.
(3=5)r6c7 - (5=4)r1c7 - UR12[(4)r2c9 = (3)r3c19] - (3)r3c456 = (3); r6c5 <> 3

although the inferences induced by a type 2 UR might be harder to find then a type 3, the usefulness is still nice to find.

-------------


III. Inferences associated with the type 4 UR

the type 4 UR provides the most interesting and sometimes outlandish step in reaching strong inferences through URs. Type 3 and Type 2 URs are very specific in nature because the move limits the number of candidates that can be present in the roof cells. The type 4 UR, on the other hand, allows multiple candidates to exist in the roof cells.

Part A... examples of strong inferences induced by the type 4 UR

what is a type 4 UR?
the type 4 says that the UR can be avoided if it is shown that one of the UR candidates is locked (or conjugate to the box, row or column that they are in) into the roof cells, the other candidate can be removed from the roof cells.
Code:
+---------------+----------+-----------+
| 12    . 12    | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
+---------------+----------+-----------+
| 12345 . 12345 | . 18 158 | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
+---------------+----------+-----------+

notice that r1c13 are the floor cells and only contain the UR candidates.
in the grid, the 2's are shown to be locked into (the only place in row 4 they can exist) r4c13. by applying the rule on type 4, we can safely remove the 1's from r4c13 and avoid the deadly pattern.
What might not be apparant is how this rule can add any information in finding a strong inference.

as before a candidate is added. the 9 in r1c3

Code:
+---------------+----------+-----------+
| 12    . 129   | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
+---------------+----------+-----------+
| 12345 . 12345 | . 18 158 | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
| .     . .     | . .  .   | .  .   .  |
+---------------+----------+-----------+

the inference in this case, isn't between the added 9 and the 1's in r4c13. instead the inference is between the 9 and the 1's in r4c56. if both the 9 and the 1's in r4c56 are false, then the UR candidates {1,2} are forced to exist in the UR cells. this is key. if this was the case, the puzzle would not have a unique solution. the 1's and the 2's could be switched and create another solution.
again, if it is shown that the UR candidates are locked into the UR cells, then a deadly pattern exists.

example...
this next grid contains an iconic and devastating move that fully exposes why a type 4 UR induced strong inference, or any move likes this is a gem to find.
Code:
.------------------.------------------.------------------.
| 5     9     16   | 14    24    7    | 26    8     3    |
| 4     7     68   | 3     5     28   | 1     26    9    |
| 38    2     138  | 189   6     89   | 4     5     7    |
:------------------+------------------+------------------:
| 1     8     2    | 45    7    *456  | 3     9    *56   |
| 6     5     7    | 2     9     3    | 8     1     4    |
| 39    4     39   | 58    1    *568  | 257   27  *A256  |
:------------------+------------------+------------------:
| 789   1     5    | 6    C24   D249  | 279   3   E-28   |
| 2     6     89   | 7     3     1    | 59    4     58   |
| 79    3     4    | 59    8    B259  | 2679  267   1    |
'------------------'------------------'------------------'

notice that the UR {56} cells are marked "*" in r46c69. the added candidate in this case is the 2 at A. removing that 2 will then expose a UR type 4 on {5,6} and the 5's in r46c6 can be removed, which forces 5 to exist in r9c6.
The strong inference is between the 2 in r6c9 and the 5 in r9c6 because if both are false, then the deadly pattern is forced to exist in the UR cells... this partial chain...
UR56[(2)A = (5)B]...
the elegance of this inference is the chain that ensues.
UR56[(2)A = (5)B] - (2)B = (2)CD; E cannot be 2. nothing but singles are left.

Part B... examples of strong inferences induced by the type 4B UR.

as expected, the type 4B UR deviates from a type 4 in that the roof cells are not contained in the same box. this means that the eliminations are ONLY made across the column or row that they occupy. many times it is mentioned how a type 4B and a type 2B can be the same, and in many cases they are the same. But, it is not safe to say that all type 2B URs are type 4Bs, or vice versa.

type 4B says the same as the type 4 except that one of the UR candidates has to be locked (or conjugate to the row or column) into the roof cells. this allows the safe removal of the opposite candidate from the roof cells.
Code:
+-----------+----------+-----------+
| .     . . | .  .   . | .     . . |
| 12    . . | .  .   . | 12    . . |
| 12345 . . | 18 158 . | 12345 . . |
+-----------+----------+-----------+

r2c17 are the floor cells and the 2 is locked into the roof cells, the 1's can be eliminated from the roof cells r3c17.

currently do not have a real example

obviously the type 4 and type 4B URs have a lot more going on and need careful inspection into which candidates are truly included in the inferences. Again, these are not common and probably not readily seen along a solution path.
-------------

IV. An uncommon UR inference.

This next type of inference isn't derived from a particular type of UR at all. Its derived straight from the concept behind avoiding the UR in the first place. The concept is that there are instances where none of the commonly known types are not going to make any eliminations but there is still a way to take advantage of the rules of URs and create a strong inference without the help of a common type. There are still strong inferences to be made if the right conditions are met.

Credit goes to Danny for expanding this type of UR in a earlier "other puzzle" thread into a contradiction which then helped me see the light once I got my head around it.

The trick is to notice what looks like a UR, but really can't take advantage of the common rules of URs to make any eliminations. AND its very important that the UR candidates are the ONLY candidates left in the floor cells. as in this grid
Code:
+--------+-----------+-------+
| 12 . . | 12345 . . | . . . |
| .  . . | .     . . | . . . |
| 12 . . | 12345 . . | . . . |
+--------+-----------+-------+
| .  . . | .     . . | . . . |
| .  . . | 13    . . | . . . |
| .  . . | 25    . . | . . . |
+--------+-----------+-------+
| .  . . | .     . . | . . . |
| .  . . | .     . . | . . . |
| .  . . | .     . . | . . . |
+--------+-----------+-------+

it just so happens that there is another 1 and another 2 in column 4. both cannot be false at the same time (that will force the UR to exist, NO NO !!) This provides yet another strong inference to exploit
UR12[(1)r5c4 = (2)r6c4]...


the same can be said in this example
Code:
+--------+-------------+-------+
| 12 . . | 12345 .  .  | . . . |
| .  . . | .     25 14 | . . . |
| 12 . . | 12345 .  .  | . . . |
+--------+-------------+-------+

the inference made is on the extra 1 and 2 in row 2


this is also true even if the roof cells don't occupy the same box (like in a type 2B or 4B) as in this example.
Code:
+-----------+--------+------------+
| 12    . . | .  . . | 12    . .  |
| .     . . | .  . . | .     . .  |
| 12345 . . | 25 . . | 12345 . 14 |
+-----------+--------+------------+

the inference is still on the 1 in r3c9 and the 2 in r3c4.


example 1...
this following grid contains the kind of example which shows off the concept nicely.
this rule can be implemented using the UR {1,8} in r78c17, marked *
Code:
.---------------------.---------------------.---------------------.
| 5     D68     4     | 7      1238   123   | 139    18    C13689 |
| 7      2      38    | 6      138    9     | 5      4      138   |
| 389    1      69    | 5      348    34    | 2      7      368   |
:---------------------+---------------------+---------------------:
| 38     9      1378  | 138    6      137   | 4      2      5     |
| 2      5      138   | 1348   9      134   | 6      18     7     |
| 6      4      178   | 128    5      127   | 13     9      138   |
:---------------------+---------------------+---------------------:
|*189   A78     5     | 24     24     6     |*1789   3     B19    |
|*18     3      2     | 9      7      5     |*18     6      4     |
| 4      67     69    | 13     13     8     | 79     5      2     |
'---------------------'---------------------'---------------------'

UR {1,8} r78c17 says that neither the 8 at A nor the 1 at B can both be false
or the deadly pattern on {1,8} is forced to exist in those UR cells.
creates this strong inference...UR18[{8}A = (1)B]...

can be extended into this chain...
UR18[(8}A = (1)B] - (9)B = (9-6)C = (6)D; r1c2 <> 8


example 2...
Code:
.---------------------.---------------------.---------------------.
| 2     *34589  158   | 18     57     6     |*35789  145   *489   |
|*145    6      158   | 3      57     9     | 578    1245   248   |
|U1359  U3589   7     | 4      2      18    |*3589   6     *89    |
:---------------------+---------------------+---------------------:
| 8      1      3     | 9      4      5     | 2      7      6     |
|U59    U59     2     | 6      1      7     | 4      8      3     |
| 6      7      4     | 2      8      3     | 1      9      5     |
:---------------------+---------------------+---------------------:
| 1345   3458   158   | 7      6      48    | 589    245    2489  |
| 7      458    9     | 58     3      2     | 6      45     1     |
|*45     2      6     | 158    9      148   |-58     3      7     |
'---------------------'---------------------'---------------------'

notice the UR cells marked on {5,9}. the UR says that in order to avoid the deadly pattern, both the 5 in r3c7 and the 9's in r3c79 cannot both be false.
in other words, UR59[(5)r3c7 = (9)r3c79]... which can be extended to eliminate the 5 in r9c7...
UR59[(5)r3c7 = (9)r3c79] - (9)r1c79 = (9-4)r1c2 = (4)r2c1 - (4=5)r9c1; r9c7 <> 5

When two bi-value cells in the same column or row stick out like a soar thumb, take notice to it and wonder if a UR can be formed. This type of move can be rewarding to the curious onlooker.

---------

V. A short cut method to SOME classic Type 3 URs

just a warning that this does not work with all type 3's. only in certain conditions.

This alternative view shows that when the extra UR candidates of the UR share a common cell, then by virtue of the rules of URs, those are the only two candidates that can exist in that cell.

this concept comes last because the type 3 is already employed by the solvers that see the locked "tuples" created by some type 3's. these can make some nice eliminations.
this section is here to provide an alternative view of seeing eliminations without knowing which "tuples" are locked and without realizing that its a type 3 at all.

In section IV, it was shown that a strong inference can be made if there are extra UR candidates in the same row, column or box, and none of the common UR rules apply. this will show what happens when those candidates share a common cell.

here is a type 3 again
Code:
+--------+---------+-------+
| 12 . . | 123 . . | . . . |
| .  . . | .   . . | . . . |
| 12 . . | 124 . . | . . . |
+--------+---------+-------+
| .  . . | 345 . . | . . . |
| .  . . | 346 . . | . . . |
| .  . . | 34  . . | . . . |
+--------+---------+-------+

the 3 in r1c4 and the 4 in r3c4 act like a single cell and thereby act like a locked set with the {3,4} cell in r6c4, eliminating any other 3's and 4's in column 4.

consider this grid and notice the UR {1,2} in r46c14
Code:
+--------+-------------+-------+
| .  . . | 3456789 . . | . . . |
| .  . . | 3456789 . . | . . . |
| .  . . | 12(3456). . | . . . |
+--------+-------------+-------+
| 12 . . | 12345   . . | . . . |
| .  . . | .       . . | . . . |
| 12 . . | 12345   . . | . . . |
+--------+-------------+-------+
| .  . . | 3456789 . . | . . . |
| .  . . | 3456789 . . | . . . |
| .  . . | 3456789 . . | . . . |
+--------+-------------+-------+

as stated before, the extra UR candidates 1 and 2 in r3c4 exist in the same cell. From the rules of URs we can now eliminate the {3,4,5,6} from r3c4 because neither the 1 nor the 2 can be false at the same time.
in other words, if r3c4 is anything other than 1 or 2, the deadly pattern is forced to exist in the UR cells.
having that knowledge can make it easier to remove {3,4,5,6} from r3c4 instead of looking for the locked sets.
this works when the "floor" cells contain only the UR candidates.
---------

I hope this provides some insight into some different yet available strong inferences derived from the rules of URs.
I welcome all feedback and corrections or suggestions that was not covered.


Last edited by storm_norm on Fri Apr 10, 2009 4:37 am; edited 2 times in total
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Thu Apr 09, 2009 8:45 am    Post subject: Reply with quote

Norm,

I haven't read all of this yet. However, your example grid in section III.PartB is not valid (it has no unique solution). Perhaps that doesn't matter for the point being illustrated, but I thought it curious.

As for that example, you have a continuous loop:
(7=9)r1c6 - 67UR[(9)r5c6=(7)r8c4] - (7)r7c5=(7)r7c2 - ER[(7)r123c2=(7)r1c12] - (7=9)r1c6...; r1c259|r8c2<>7

r1c2<>7 since it is the ERI of the b1 ER and the continuous loop renders this link conjugate; that is only possible if the ERI cell is not <7>. (The elimination at r8c2 could be seen the same way if the chain were propagated using the b7 ER rather than the r8 conjugate pair.) Of course, you can see it as being eliminated by the end pincers of your chain, as you have done. However, any time matching digit pincers eliminate an ERI candidate, you have a continous loop.

The <7> at r2c4 is also eliminated by the initial portion of your chain:
(7=9)r1c6 - 67UR[(9)r5c6=(7)r8c4]; r2c4<>7

I'll look at the remainder of your post another time.
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Thu Apr 09, 2009 9:24 am    Post subject: Reply with quote

Asellus wrote:
However, your example grid in section III.PartB is not valid (it has no unique solution).

Norm: Asellus' observation is a blow to this example.

You accidentally performed [r1c4]<>7 instead of [r1c5]<>7.

This appears to ruin your conclusion UR67[(9)r5c6 = (7)r8c4].

[Addendum: I have since learned.]

It's a good example of using an AUR in a network.

Code:
                (7)r1c4]
              /
UR67[(9)r5c6 =
              \
                (7)r8c4]
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Thu Apr 09, 2009 6:34 pm    Post subject: Reply with quote

Asellus and Danny,
I see that the example is not valid and I will be posting a better example.
I also realize now that not including real examples of all the strong inferences was gross negligence on my part.
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storm_norm



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Posts: 1741

PostPosted: Thu Apr 09, 2009 6:38 pm    Post subject: Reply with quote

here are some examples that I did not include in the first post.

UR Type 3 strong inference

Code:
.------------------.------------------.------------------.
| 1     37    8    | 479   479   5    | 367   3679  2    |
| 2     367   36   | 379   1     8    | 4     5     39   |
| 9     5     4    | 37    6     2    | 37    8     1    |
:------------------+------------------+------------------:
| 346   36    1    | 5     234   9    | 8     23    7    |
| 34    9     2    | 467   8     367  | 5     1     36   |
| 5     8     7    | 1     23    36   | 9     236   4    |
:------------------+------------------+------------------:
| 3678  4     36   | 2     5     1    | 367   3679  3689 |
| 378   2     5    | 679   379   367  | 1     4     38   |
| 367   1     9    | 8     37    4    | 2     367   5    |
'------------------'------------------'------------------'

UR {2,3}r46c58 says that neither the 4 in r4c5 or the 6 in r6c8 can both be false.
UR23[(4)r4c5 = (6)r6c8] - (6=3)r5c9 - (3=4)r5c1; r5c4 <> 4

Type 2 UR generated inference

real example.
this puzzle is easily solved SSTS. but it does contain an example of a type 2 based inference.
Code:
.---------------.---------------.---------------.
| 25   9    47  |U578 U578  1   | 3    24   6   |
| 6    3    47  | 2    9   *57  |*45   1    8   |
| 258  1    28  | 6    4    3   | 9    7    25  |
:---------------+---------------+---------------:
| 3    5    9   | 17   17   8   | 24   6    24  |
| 7    8    6   | 59   2    4   | 1    59   3   |
| 12   4    12  | 3    6    59  | 8    59   7   |
:---------------+---------------+---------------:
| 148  2    5   | 79   3    79  | 6    48   14  |
| 148  7    3   |U158 U158  6   |*245  248  9   |
| 9    6    18  | 4   -158  2   | 7    3   *15  |
'---------------'---------------'---------------'

the UR on {5,8} in r18c45 says that neither the 1's in r8c45 or the 5's in r1c45 can both be false.
creates this strong inference...
UR58[(1)r8c45 = (7)r1c45]...

can be extended into this chain...
UR58[(1)r8c45 = (7)r1c45] - (7=5)r2c6 - (5)r2c7 = (5)r8c7 - (5=1)r9c9; r9c5 <> 1

UR Type 2B generated strong inference
Code:
.---------------------.---------------------.---------------------.
| 9      6      25    | 245    14     125   | 3      7      8     |
| 4      1      27    | 8      37     237   | 5      6      9     |
| 37     35     8     | 6      9      57    | 1      4      2     |
:---------------------+---------------------+---------------------:
| 37     8      4     | 9      37     6     | 2      1      5     |
| 1      2      9     | 37     5      4     |U67     8     U367   |
| 6      35     57    | 1      2      8     |*47-9   39    *347   |
:---------------------+---------------------+---------------------:
| 2      7      36    | 345    146    1359  | 8      359   *34    |
| 5      4      1     | 237    8      2379  |U679    239   U367   |
| 8      9      36    | 23457  46     2357  | 47     235    1     |
'---------------------'---------------------'---------------------'

note the cells marked U contain the UR on {6,7}. also see that if the 9 is removed from r8c7 then the type 2B UR remains on {6,7} which says that the 3's in r58c9 cannot be removed.
this provides a strong inference on the 9 and the 3's because neither can both be false.
UR67[(9)r8c7 = (3)r58c9]... and this can be extended into this chain.

UR67[(9)r8c7 = (3)r58c9] - (3=4)r7c9 - (4)r6c9 = (4)r6c7; r6c7 <> 9
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storm_norm



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Posts: 1741

PostPosted: Fri Apr 10, 2009 1:51 am    Post subject: Reply with quote

this puzzle has replaced the example in the Type 4B strong inference section.
its a much better representation of the move.
Code:
.------------------.------------------.------------------.
| 2    H67    9    | 5   J4-68   3    | 68    1     47   |
| 4     57    157  | 9     68    17   | 2     68    3    |
|I3-6   8     137  | 17   A246  B24   | 5     9     47   |
:------------------+------------------+------------------:
| 5     4     8    | 6     3     9    | 7     2     1    |
| 7     3     2    | 4     1     5    | 68    68    9    |
| 9     1     6    | 2     7     8    | 4     3     5    |
:------------------+------------------+------------------:
| 8     9     4    | 3     5     6    | 1     7     2    |
| 36   G267   37   | 17   C24   D1247 | 9     5     8    |
| 1    F257   57   | 8     9    E27   | 3     4     6    |
'------------------'------------------'------------------'

The cells A,B,C,D contain a UR on {2,4}. If the 6 is removed from A then the resulting Type 4B UR says that the 2's can be removed from B and D because the 4's are conjugate with respect to column 6. This forces 2 into E.
What this means is that there exists a strong inference on the 6 at A and the 2 at E. neither can be false or the Deadly pattern is forced to exist. The strong inference can be made...
UR24[(6)A = (2)E]...

this chain can be formed to eliminate the 6 at I and J.
UR24[(6)A = (2)E] - (2)F = (2-6)G = (6)H; I and J is not 6.
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daj95376



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Posts: 3855

PostPosted: Fri Apr 10, 2009 2:48 am    Post subject: Reply with quote

storm_norm wrote:
this puzzle has replaced the example in the Type 4B strong inference section.
its a much better representation of the move.
Code:
.------------------.------------------.------------------.
| 2    H67    9    | 5   J4-68   3    | 68    1     47   |
| 4     57    157  | 9     68    17   | 2     68    3    |
|I3-6   8     137  | 17   A246  B24   | 5     9     47   |
:------------------+------------------+------------------:
| 5     4     8    | 6     3     9    | 7     2     1    |
| 7     3     2    | 4     1     5    | 68    68    9    |
| 9     1     6    | 2     7     8    | 4     3     5    |
:------------------+------------------+------------------:
| 8     9     4    | 3     5     6    | 1     7     2    |
| 36   G267   37   | 17   C24   D1247 | 9     5     8    |
| 1    F257   57   | 8     9    E27   | 3     4     6    |
'------------------'------------------'------------------'

This is not a good example for a UR Type 4B. If A<>6, then you have a UR Type 1 that combines with strong links in <2> & <4> to force:

Code:
D<>24, B=C=4, A=2   (for starters)

You need four candidates in cell A in order to keep it from degenerating into a UR Type 1.

What you can immediately deduce is A<>4.
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storm_norm



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PostPosted: Fri Apr 10, 2009 4:34 am    Post subject: Reply with quote

danny,
good point.
finding a useful type 4B strong inference may be more difficult than some of the other examples.
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Asellus



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Posts: 865
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PostPosted: Fri Apr 10, 2009 9:55 pm    Post subject: Reply with quote

Norm has requested that I post the following example (at the end of this post) from a previous thread. Before I do so, some comments on this thread.

First, I will remark that Norm's Section IV is one sort of example of a more general case. Anywhere that the UR digits external to the UR cells are arranged in such a way that if they are all false then the UR is true, those external UR digits (with any grouping necessary to contain them all) will have a strong inference between them. Norm's Section IV deals with URs in which the "floor" contains only the UR digits. However, the same general approach can be used in cases without such a restriction.

I don't have examples at hand for such a case. But, here is a schematic of one such case in which none of the UR cells must be a bivalue:
Code:

34A   34B   3R   3S

34C   34D   4T

A, B, C, D, R, S, and T can be any number of additional digits. If the cells 3R and 3S are the only cells in the top row outside the 34UR that contain <3> and the cell 4T is the only cell outside the 34UR in the bottom row that contains <4>, then we have the strong inference: 34UR[(3)RS=(4)T]. (Yes, I have used R, S and T to represent both additional candidate digits AND cell references.) Note that it doesn't matter how <3> and <4> are distributed in the columns external to the UR. (However, a similar strong inference might be possible in the columns.)

This notion of a strong inference in the UR digits external to a UR has been around for a long time. I first encountered the notion from ravel more than a year ago, and it wasn't new then. However, it isn't often used... at least in postings on this board.

Most often, people exploit the strong inferences induced among the extra digits internal to a UR, as in most of Norm's cases and examples above. The case of exploiting the strong inferences among the UR digits external to a UR (as I have illustrated here) is sort of complementary to this. It is possible to mix these two ideas:
Code:

34A   34B   3R   3S

347   347   4T

Let's suppose that this 34UR is located at r14c15. Then, we could exploit this strong inference: UR[(3)RS=(7)r4c15]. The external <3>s in r1 have a strong inference with the internal grouped <7>s in r4.

Variants of these basic ideas arise when considering conjugate links within the UR, etc. However, I believe all cases can be boiled down to these essential notions.


Now for the requested example:
Code:
.------------------.------------------.------------------.
| 5     4     29   |A13    289   89   |#1-3   6     7    |
| 19    8     3    | 7     6     5    | 4     19    2    |
| 129   6     7    | 4     239   19   | 5     189  B38   |
:------------------+------------------+------------------:
| 2368  12    126  | 5     38    7    | 9     18    4    |
| 389   5     19   |U136   4     189  | 7     2    U368  |
| 389   7     4    |U136   389   2    | 13    5    U368  |
:------------------+------------------+------------------:
| 26    12    126  | 9     7     3    | 8     4     5    |
| 7     9     8    | 2     5     4    | 6     3     1    |
| 4     3     5    | 8     1     6    | 2     7     9    |
'------------------'------------------'------------------'

The 36UR is marked "U". It is important to note that the <6>s in this UR form an X-Wing. This means that any external <3>s in the UR columns will destroy the UR and thus have a strong inference. In this case:
36UR[(3)r1c4=(3)r3c9]
This strong inference alone performs an elimination! <3> is removed from r1c7, marked #. You won't find an AIC much shorter than that!

In the original post, I pointed out that this has a Skyscraper-like structure, though with a UR "base". The grouped <3>s within the columns cells of the UR are weakly linked (we cannot have a true <3> within both columns of the UR cells), providing the Skyscraper form:
(3)r1c4=36UR[(3)r56c4 - (3)r56c9]=(3)r3c9

But, simply applying the external UR cells strong inference concept makes this explicit recognition of the internal weak link unnecessary.

[Edit to correct typo in AIC notation.]


Last edited by Asellus on Sun Apr 12, 2009 6:44 pm; edited 1 time in total
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Asellus



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Posts: 865
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PostPosted: Fri Apr 10, 2009 10:00 pm    Post subject: Reply with quote

PS:

In my previous post, I should point out that the same result can be obtained by exploiting the internal strong inference among the extra UR digits:
(3=1)r1c4 - 36UR[(1)r56c4=(8)r56c9] - (8=3)r3c9
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Sat Apr 11, 2009 12:16 am    Post subject: Reply with quote

Asellus,
Thank you.

Quote:
The 36UR is marked "U". It is important to note that the <6>s in this UR form an X-Wing. This means that any external <3>s in the UR columns will destroy the UR and thus have a strong inference. In this case:
36UR[(3)r1c4=(3)r3c9]


This is another useful strong inference created by a UR. In this case the Strong inference of the external 3's in column 4 and 9 is triggered by the fact that the 6's are a x-wing.
very nice.
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storm_norm



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Posts: 1741

PostPosted: Tue Apr 14, 2009 5:39 pm    Post subject: Reply with quote

Quote:
Variants of these basic ideas arise when considering conjugate links within the UR, etc. However, I believe all cases can be boiled down to these essential notions.


I think this is an important point.
The different UR types that the sudoku community knows or has familiarity with should be a base for realizing the potential for a broader understanding of what is going on with respect to the UR.
Those solvers who learn or figure out how and why the UR is a deadly pattern will hopefully take that step forward and realize that there can be inferences made from those familiar UR properties.
Obviously there are other ways to show inferences internal or external to a UR. Showing the inferences with respect to a particular type was just one approach to Asellus' point.
This thread could definitely have started with Asellus' words
all cases can be boiled down to these essential notions
then list the examples
etc,etc.
this would actually be a much broader and more generalized approach and would definitely include other inferences that have nothing to do with particular types of UR.

My experience with these inferences has been that they started to jump off the page which is BIG NEWS for a manual solver. My eyes seem to be drawn into them and thus started my fascination.
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