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Short, Two-Value Chain #3

 
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Tue Aug 18, 2009 9:50 pm    Post subject: Short, Two-Value Chain #3 Reply with quote

[Reposted]

My solver reports a W-Wing as a short, two-value chain. It also reports a typical (generalized) M-Wing as a short, two-value chain. Occasionally, it'll report other short, two-value chains. Here's one that popped up recently.

I don't know what to call this pattern (if anything), but it is simple and usable. Here's the general case.

Consider strong links on <4> and <9> where opposing endpoints see each other. Also, "vwxy" represent other candidates. Now, also assume that a bivalue cell of <49> exists and that it sees one pair of opposing endpoints. The other opposing endpoints can have the complementary candidate eliminated.

Code:
 [r5c3]=4 [r5c5]<>4 [r2c5]=4(<>9) [r2c8]<>4 => (x)<>9, (y)<>4
 [r5c3]=9 [r5c8]<>9 [r2c8]=9(<>4) [r2c5]<>9 => (x)<>9, (y)<>4
 +-----------------------------------+
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |  .  .  .  |  .  4x .  |  .  9y .  | <-- opposing endpoints w/eliminations
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |  .  49 .  |  .  4v .  |  .  9w .  | <-- opposing endpoints + <49> cell
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |  .  .  .  |  .  /  .  |  .  /  .  |
 |  .  .  .  |  .  /  .  |  .  /  .  |
 +-----------------------------------+

Ironically, I need two AICs to get both eliminations.

Code:
          (9)r2c8 = r5c8 - (9=4)r5c2 - r5c5 = (4  )r2c5 => r2c5 <> 9
(9)r2c5 -    r2c8 = r5c8 - (9=4)r5c2 - r5c5 = (4-9)r2c5 => r2c5 <> 9

          (4)r2c5 = r5c5 - (4=9)r5c2 - r5c8 = (9  )r2c8 => r2c8 <> 4
(4)r2c8 -    r2c5 = r5c5 - (4=9)r5c2 - r5c8 = (9-4)r2c8 => r2c8 <> 4

In this real example, the strong link on <9> is in [b2].

Code:
 Set XY_03 Puzzle 036: [r3c1]=49 => [r1c5]<>4
 +-----------------------------------------------------+
 |  8    39   1    |  37   349* 6    | *47   2    5    |
 |  34   6    25   |  17   234  45   |  17   8    9    |
 | *49   25   7    |  12   8    459* |  14*  3    6    |
 |-----------------+-----------------+-----------------|
 |  69   58   58   |  4    1    3    |  69   7    2    |
 |  367  27   26   |  9    5    8    |  36   4    1    |
 |  1    39   4    |  6    7    2    |  39   5    8    |
 |-----------------+-----------------+-----------------|
 |  67   78   68   |  25   49   49   |  25   1    3    |
 |  25   4    9    |  325  23   1    |  8    6    7    |
 |  25   1    3    |  8    6    7    |  25   9    4    |
 +-----------------------------------------------------+
 # 42 eliminations remain


[Addendum]: 999_Springs in Players' Forums

The even more general case: only one pair of opposing endpoints on the bilocations need to see each other.

Code:
 +-----------------------------------+
 |  /  /  /  |  .  .  .  |  .  /  .  |
 |  /  /  4x |  .  .  .  |  .  9y .  | <-- opposing endpoints w/eliminations
 |  /  4v /  |  .  .  .  |  .  /  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  .  /  .  |
 |  .  49 .  |  .  .  .  |  .  9w .  |
 |  .  .  .  |  .  .  .  |  .  /  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  .  /  .  |
 |  .  .  .  |  .  .  .  |  .  /  .  |
 |  .  .  .  |  .  .  .  |  .  /  .  |
 +-----------------------------------+
cell x can't be 9
cell y can't be 4
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Wed Aug 19, 2009 12:10 am    Post subject: Reply with quote

Examples of 999_Springs' version.

Set XY_03 Puzzle unposted:

Code:
 strong link (2): "v" = "x"
 strong link (8): "w" = "y"
 [r1c3]=28 sees endpoints "v" and "w"
 opposing endpoints "x" and "y"
 => "x"<>8 (redundant because of strong link on <8>), "y"<>2
 +-----------------------------------------------------+
 |  1    6   *28   |  259  3    259  |  7    89   4    |
 |  5    9    7    |  8    1    4    |  3    6    2    |
 | v24   248  3    |  279  27   6    |  1    89   5    |
 |-----------------+-----------------+-----------------|
 | x23   7    258w |  1    258y 235  |  9    4    6    |
 |  9    1    245  |  245  6    25   |  8    3    7    |
 |  34   48   6    |  479  78   39   |  5    2    1    |
 |-----------------+-----------------+-----------------|
 |  8    3    1    |  6    4    7    |  2    5    9    |
 |  7    245  24   |  25   9    8    |  6    1    3    |
 |  6    25   9    |  3    25   1    |  4    7    8    |
 +-----------------------------------------------------+
 # 37 eliminations remain


Set XY_03 Puzzle unposted:

Code:
 strong link (3): "v" = "x"
 strong link (8): "w" = "y"
 [r2c4]=38 sees endpoints "v" and "w"
 opposing endpoints "x" and "y"
 => "x"<>8 (redundant because of strong link on <8>), "y"<>3
 +--------------------------------------------------------------+
 |  7     2368  368   |  18    12    9     |  5     23    4     |
 |  238w  4     5     | *38    236   7     |  1     236   9     |
 |  23x   9     1     |  4     5    v36    |  26    7     8     |
 |--------------------+--------------------+--------------------|
 |  4     138   7     |  136   13    5     |  689   69    2     |
 |  389y  138   389   |  136   7     2     |  68    4     5     |
 |  6     5     2     |  9     4     8     |  7     1     3     |
 |--------------------+--------------------+--------------------|
 |  1     28    89    |  5     36    36    |  4     29    7     |
 |  239   236   369   |  7     8     4     |  29    5     1     |
 |  5     7     4     |  2     9     1     |  3     8     6     |
 +--------------------------------------------------------------+
 # 47 eliminations remain
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keith



Joined: 19 Sep 2005
Posts: 3163
Location: near Detroit, Michigan, USA

PostPosted: Wed Aug 19, 2009 1:06 am    Post subject: Reply with quote

Danny,

The way I see it is this:

Code:

49 - 4v =4x
  \       |
    9w = 9y
x <> 9, y <> 4.

Still thinking ...

Keith
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keith



Joined: 19 Sep 2005
Posts: 3163
Location: near Detroit, Michigan, USA

PostPosted: Wed Aug 19, 2009 2:59 am    Post subject: Reply with quote

Methinks:

1. 999_Springs has observed that v and w do not have to "see" each other. daj did not say they do, although his example has them in the same row. My cartoon similarly requires no direct relationship between v and w, other than their places in the chain.

2. In daj's example, there is a DP <25> that solves R8C4 as <3>, and the puzzle.

edit: I think the surprising thing about this pattern is that, indeed, there are no conditions relating v and w, other than their positions in the chain

Keith


Last edited by keith on Wed Aug 19, 2009 3:33 am; edited 1 time in total
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keith



Joined: 19 Sep 2005
Posts: 3163
Location: near Detroit, Michigan, USA

PostPosted: Wed Aug 19, 2009 3:26 am    Post subject: Reply with quote

keith wrote:
Danny,

The way I see it is this:

Code:

49 - 4v =4x
  \       |
    9w = 9y
x <> 9, y <> 4.

Still thinking ...

Keith

I can think of at least four ways my cartoon can be laid out:

1. 49 v w are in the same row or column (daj's diagram).

2. 49 v are in the same row (column) and 49 w are in the same column (row). This is the first of 999_Springs' examples.

3. 49 v w are in the same box, none share a row or column.

4. 49 v are in the same box, 49 w are in a row or column.

etc.

This is the same slightly silly argument about the distinctions between a kite, a skyscraper, and a Turbot fish.

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Wed Aug 19, 2009 8:21 am    Post subject: Reply with quote

keith wrote:
Methinks:

1. 999_Springs has observed that v and w do not have to "see" each other. daj did not say they do, although his example has them in the same row. My cartoon similarly requires no direct relationship between v and w, other than their places in the chain.

2. In daj's example, there is a DP <25> that solves R8C4 as <3>, and the puzzle.

edit: I think the surprising thing about this pattern is that, indeed, there are no conditions relating v and w, other than their positions in the chain

This statement was meant to imply that the endpoints of both strong links see each other. It was a key point in why I thought the pattern might be easy for manual solvers to locate. I don't think the more general pattern is as obvious.

I wrote:
where opposing endpoints see each other
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strmckr



Joined: 18 Aug 2009
Posts: 61

PostPosted: Thu Aug 20, 2009 5:29 am    Post subject: Reply with quote

Hey danny this post was the reason i joined this site how about

"split - wing" as the 2 strong links are split with one common node?


this thread and pattern is one that Ive made a note in my binder a few months back haven't had a lot of time to investigate its occurrences fully

I'll work some out fully and provide some details on it soon enough Smile nice work by the way.

the one noted by springs is the one that caught my attention ive notice the same thing when i was coding w-wings type e and grouped variations.

here

i also like m-wings how ever there isn;t any detail on it in the players forum only found traces back to here.

Smile
cheers chris.

Quote:
This is the same slightly silly argument about the distinctions between a kite, a skyscraper, and a Turbot fish.


or a 2 string kytes/grouped box nodes that fall in a box that also meets the Empty rectangle requirements...
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