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Another "coloring" example

 
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Mon Feb 27, 2006 7:32 pm    Post subject: Another "coloring" example Reply with quote

Keith wrote:
Thank you. These colors, chains and cycles are coming into focus for me.

Today's "Nightmare" puzzle (27 Feb 2006) offers an excellent opportunity for "coloring" logic.
Code:
.1....6..
.8.2.4...
6...3...8
.5......1
..6.9.2..
7......8.
4...1...3
...8.3.5.
..9....7.

1. Naked pair {3, 4} in row 5.
2. Hidden pair {5, 8} in r7c3 & r9c1; r9c2 = 3.
3. Naked pair resolved -- r5c3 = 4, r5c8 = 3.
4. r9c7 = 1 (unique horizontal)
5. r9c1 = 8 (unique horizontal); r7c3 = 5 (sole candidate)
6. r5c1 = 1 (sole candidate); r7c1 = 2 (sole candidate); r7c3 = 1 (sole candidate)
7. {6, 7} pair in r7c2 & r8c2.
8. r4c3 = 8 (unique vertical); r7c7 = 8 (unique horizontal)
9. r1c5 = 8 (unique vertical); r5c6 = 8 (unique vertical)
10. r2c8 = 1 (unique horizontal)
11. The "2" in row 6 lies in middle left 3x3 box.
12. r2c7 = 3 (unique horizontal); r2c3 = 7 (sole candidate); r2c5 = 6 (sole candidate)
13. Naked triplet {2, 4, 9} in row 3.
14. The "9" in row 1 lies in top center 3x3 box.
15. Coloring on "5" in row 5 and column 7 reveals r3c4 <> 5
16. X-Wing on "9" in columns 4 & 6 -- r7c8 <> 9.

At this point we're ready for the final push -- the grid looks like this.
Code:
 35     1    234   579    8    579    6    24   2457
 59+    8     7     2     6     4     3     1    59-
  6    29-   24    17     3    157   57    249+   8
 39-    5     8   3467   247   276   479   469-   1
  1     4     6    57     9     8     2     3    57
  7    29+   23   13456  45    156   459(-) 8   4569*
  4    67     5    679    1   2679    8    26     3
  2    67     1     8    47     3    49(-)  5    469(+)
  8     3     9    456   245   256    1     7    246

We have a binary chain in the "9"s, marked +/- above. First we observe that the "9" at r6c9 can be eliminated because there's a "+" at r6c2 and a "-" at r2c9. With that "9" out of the way we can mark another "+" at r8c9, and two more "-9"s, at r6c7 and r8c7.

But now we have two "-9"s in the same column (7), and also two "-9"s in the same row (4). We conclude that "-" is not 9, and "+" is, solving for ten cells simultaneously and breaking the puzzle wide open. dcb
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