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Medusa Coloring: A Special Bivalue Extension

 
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Asellus



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

PostPosted: Tue Sep 25, 2007 10:15 pm    Post subject: Medusa Coloring: A Special Bivalue Extension Reply with quote

In normal Medusa Coloring, the coloring is extended by exploiting overt "strong" (conjugate) links. However, there is a special case involving a bivalue cell in which two "hidden" conjugate links can be exploited. This occurs if the two digits of a bivalue cell can each "see" a colored instance of themselves, even if only weakly linked.

If the colors are opposite, then eliminations will occur via Medusa "Trap" as shown in this example:
Code:
+------------------+------------------+------------------+
| 2     5     1    | 9     8     3    | 7     6     4    |
| 4r7g  6g7r  8    | 5     1g6r  1r4g | 3     2     9    |
| 4g9r  6r9g  3    | 4r6g  2     7    | 58    1     58   |
+------------------+------------------+------------------+
| 3     4     6    | 1     9     8    | 2     5     7    |
| 1g79 a1279  279  | 346r7 5     4r6g | 148   38    138  |
| 8    #17    5    | 347   37    2    | 149   39    6    |
+------------------+------------------+------------------+
| 6     3     279  | 78    4     5    | 189   789   128  |
| 1r79  8     4    | 2     36g7  1g6r | 59    379   35   |
| 5    b1g27  27   | 378   1r37  9    | 6     4     238  |
+------------------+------------------+------------------+

The affected bivalue is marked #. It "sees" the red <7> in R2C2 and the green <1> in R9C2. Thus, it can be colored: 1r7g. This results in elimination of <1> from "a" and elimination of <7> from "a" and "b".

The logic is fairly evident: Either red is true or green is true. If red is true, the bivalue cell is <1>. If green is true, it is <7>. Since there is no other possibility for the cell, the colors can be placed.

I'm not sure if it's possible (I haven't encountered an example), but, in principal, if the colors "seen" by the bivalue cell are matching, then a "Medusa Wrap" would occur eliminating that color. For instance, if the {17} bivalue could "see" a green <1> and a green <7>, then all green digits would be eliminated and all red values placed. [This "Wrap" scenario may not be logically possible. I would welcome either an example showing that it is or a rigorous proof that it is not.]
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