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"4 star constellation"

 
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Sun Nov 20, 2005 6:47 pm    Post subject: "4 star constellation" Reply with quote

Hi,

Here the definition of a "4 stars constellation" similar to the "5 ...":

1. it involves 4 stars (meaning 4 cells)
2. it has 2 starting digits, in the so called starting cell. We always find this alpha star, the bright one to start with.
3. we follow 2 chains, or pathes, to other stars/cells, forcing a digit TO BE or NOT TO BE in this cell.
4. the overall result is a chain of 4 stars, with an ending in some result, as to set or exclude a digit in one of the stars, or a contradiction.

For example: some digit is to be excluded,
whatever digit we have choosen in the alpha star.

Lets examine the following 4 cells:


Code:

column      C        D
--------------------------
row A |   XY       XZ.....
      |
row B |   Y....    YZ.....



where the "." stand for some other digits.
Our alpha star is a double neutron star having digits X and Y.

(a) we will start with X. Looking at the row, we will exclude X from the XZ..... cell.
(b) now we take Y, on the column we will exclude Y from cell Y..... and continuing the path in the row, we will force Y in the YZ..... cell, which will exclude Z from the same cell, which will force in the column the Z in the XZ..... cell, excluding X from this cell.
from (a) and (b) we conclude that X can be excluded/eliminated from cell XZ.....

in short notation:

(a) rAcC = X rAcD <> x
(b) rAcC = Y rBcC <> Y rBcD = Y rBcD <> Z rAcD = Z rAcD <> X

Of course that col C has only this 2 occurences of Y,
row B has only 2 occurences of Y,
col D has only 2 occurences of Z.

P.S. Of course that we can define the "n star constellation" that are hard to detect even with the Hubble telescope.


Any feedback is wellcomed.
Example can follow, on request.
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alanr555



Joined: 01 Aug 2005
Posts: 193
Location: Bideford Devon EX39

PostPosted: Mon Nov 21, 2005 1:13 am    Post subject: Re: "4 star constellation" Reply with quote

Code:

column      C        D
--------------------------
row A |   XY       XZ.....
      |
row B |   Y....    YZ.....

> in short notation:
> (a) rAcC = X  rAcD <> x
> (b) rAcC = Y  rBcC <> Y  rBcD = Y 
        rBcD <> Z  rAcD = Z  rAcD <> X
> Of course that col C has only this 2 occurences of Y,
>               row B has only 2 occurences of Y,
>              col D has only 2 occurences of Z.

+++
Excellent work.
However the first constraint may be RELAXED.
Whilst it is essential that row B has only two of Y and col D only 2 of Z,
it does not matter how many Y exist in column C.
This is because the chain is considered only when the top left has
digit Y and so EVERY other cell in the C column MUST be "not-Y". The
binary force is not needed to require this to be so.

The rule then becomes:

When a row contains a cell with eactly two candidates and another cell
which contains just one of those two digits as a candidate (even if
with others), that candidate may be eliminated from the second cell - if there exists:

a) another row containing just two occurrences of the second original
    digit, each with a placement linked by column or region to one or the
    other of  the two cells in the original row
AND
b) a binary link using ANY digit between the second cell in the original
    row and one of the cells in the second row.

The rule applies also to columns mutatis mutandis.
++
Example:
12   149
257  246

This reduces to [1 and 49] or [2 and 4] depending on the value
actually present in the first position. either way the 1 is eliminated.

Alan Rayner BS23 2QT
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Sat Nov 26, 2005 6:54 am    Post subject: Reply with quote

Hi,

Let's see this 4 constellation in action:

with the example from Sunday Express 6-12 puzzle:

Code:
 Initial SuDoku Table
 
  - - - - 1 - - - -
  7 - - - - - 8 - -
  - - 3 - 6 - 4 - -
  3 1 - - 4 9 - 6 -
  - - 8 - 2 - 7 - -
  - 5 - 7 3 - - 1 8
  - - 9 - 8 - 3 - -
  - - 4 - - - - - 2
  - - - - 5 - - - -


After the following (warm-up):

Code:
 9 in r2c5  5 in r4c9  6 in r6c6  - Sole Candidate
 8 in r4c4  2 in r4c7  2 in r6c3  7 in r8c5  - Sole Candidate
 7 in r4c3  9 in r6c7  - Sole Candidate
 4 in r6c1  - Sole Candidate
 5 not in r1c1, it is in r1c3 or r2c3 (Column on 3x3 Block interaction)
 5 not in r3c1, it is in r1c3 or r2c3 (Column on 3x3 Block interaction)
 1 not in r7c9, it is in r8c7 or r9c7 (Column on 3x3 Block interaction)
 1 not in r9c9, it is in r8c7 or r9c7 (Column on 3x3 Block interaction)
 6 not in r1c1, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 6 not in r1c2, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 5 not in r1c4, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 5 not in r1c6, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 5 not in r1c8, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 6 not in r1c9, Nacked Pair 5 6 in r1c3 and r1c7 (same Row)
 1 not in r9c1, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 6 not in r9c1, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 6 not in r9c2, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 1 not in r9c4, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 6 not in r9c4, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 1 not in r9c6, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 6 not in r9c9, Nacked Pair 1 6 in r9c3 and r9c7 (same Row)
 2 not in r1c6, Hidden Pair 7 8 in r1c6 and r3c6 (in Column)
 2 not in r3c6, Hidden Pair 7 8 in r1c6 and r3c6 (in Column)
 3 not in r1c6, Hidden Pair 7 8 in r1c6 and r3c6 (in Column)
 4 not in r1c6, Hidden Pair 7 8 in r1c6 and r3c6 (in Column)
 5 not in r3c6, Hidden Pair 7 8 in r1c6 and r3c6 (in Column)
 6 not in r2c3, it is in r1c3 r1c7 r9c3 r9c7 (X-Wing on Column)
 6 not in r8c7, it is in r1c3 r1c7 r9c3 r9c7 (X-Wing on Column)


and now we have to deal with:
Code:
289  2489  56   234    1    78    56   2379   379   

7    246   15   2345   9    2345  8    235    136   

1289 289   3    25     6    78    4    2579   179   

3    1     7    8      4    9     2    6      5   

69   69    8    15     2    15    7    34     34   

4    5     2    7      3    6     9    1      8   

1256 267   9    1246   8    124   3    457    467   

1568 368   4    1369   7    13    15   589    2   

28   2378  16   2349   5    234   16   4789   479   


Here we can detect a 4 stars constellation:
alpha star is r1c3. The other 3 stars are in r2c2, r2c3 and r2c9.

one path is: 5 in r1c3, 1 in r2c3, 1 not in r2c9
second path: 6 in r1c3, 1 not in r2c2, 6 in r2c9, 1 not in r2c9

here only the first 2 rows:

Code:
289  2489  56   234    1    78    56   2379   379   
           A.
           .A
7    246   15   2345   9    2345  8    235    136   
           B.                                 b..
     ..a                                      b.B 


The rest is only an exercise:

Code:
 1 in r2c3  1 in r3c9  1 in r9c7  - Unique Horizontal
 6 in r9c3  - Unique Horizontal
 6 in r1c7  6 in r2c2  6 in r5c1  6 in r8c4  - Unique Horizontal
 5 in r1c3  9 in r5c2  6 in r7c9  9 in r8c8  9 in r9c4  - Unique Horizontal
 9 in r3c1  - Unique Horizontal
 9 in r1c9  - Unique Horizontal
 4 in r1c2  5 in r8c7  8 in r9c8  7 in r9c9  - Unique Vertical
 5 in r7c1  7 in r7c2  4 in r5c9  - Unique Vertical
 1 in r8c1  4 in r7c8  3 in r2c9  - Unique Vertical
 8 in r1c1  8 in r8c2  3 in r1c4  4 in r2c4  4 in r9c6  8 in r3c6  3 in r5c8  - Unique Vertical
 2 in r9c1  2 in r3c2  3 in r9c2  2 in r7c4  2 in r2c6  3 in r8c6  7 in r1c6  2 in r1c8  7 in r3c8  - Unique Vertical
 1 in r5c4  1 in r7c6  5 in r5c6  5 in r2c8  - Unique Vertical
 5 in r3c4  - Unique Vertical
 
  Final SuDoku Table
 
  8 4 5 3 1 7 6 2 9
  7 6 1 4 9 2 8 5 3
  9 2 3 5 6 8 4 7 1
  3 1 7 8 4 9 2 6 5
  6 9 8 1 2 5 7 3 4
  4 5 2 7 3 6 9 1 8
  5 7 9 2 8 1 3 4 6
  1 8 4 6 7 3 5 9 2
  2 3 6 9 5 4 1 8 7


see u,
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