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Ron Moore

Joined: 13 Aug 2006
Posts: 72
Location: New Mexico

Posted: Wed Dec 13, 2006 4:21 am    Post subject: 4 Dec 2006 Nightmare: 12-Cell Deadly Pattern

In the 4 Dec 2006 Nightmare, I came across a threatened 12-cell deadly pattern which could be used in the last major step in my solution. At the outset, I should acknowledge that there is a fairly short XY chain which gives the same elimination, so this is presented "just for grins and giggles" (to quote a well-known authority) rather than as a useful, practical step in the solution. However, there are some other points of interest as well.

The diagram below shows the position after initial eliminations from basic techniques, a naked triple of "189" in r8c238, a hidden pair of "14" in r1c6|r2c4, and a naked triple of "289" in r247c5:
 Code: .-------------------.-------------------.-------------------. | 389   2     3489  | 6       5    14   | 89     7    189   | | 7    *69    46-9  | 14     *29   8    | 3      5    126-9 | | 1     5     689   | 239     7    239  | 689    4    2689  | :-------------------+-------------------+-------------------: |*89   *689   56-89 | 125-89 *289  7    | 16-89  3    4     | | 2     7     35689 | 13589   4    139  | 1689   89   689   | | 389   4     1     | 389     36   369  | 7      2    5     | :-------------------+-------------------+-------------------: | 6     189   2     | 7       89   5    | 4      189  3     | | 4     189   89    | 23      36   236  | 5      189  7     | | 5     3     7     | 489     1    49   | 2      6    89    | '-------------------'-------------------'-------------------'

Let me say that I did see the XYZ wing in r2c2|r38c3 but I did not need to make use of it. I mention this to Ruud because the Sudocue solver evidently did not expect to see an XYZ wing with two eliminations (yes, this is another example). It gives the eliminations of (9)r1c3 and (9)r2c3 in separate steps, but referencing the same XYZ wing.

I also did not need to make use of the threatened "36" deadly pattern in r68c56.

In the diagram, the cells marked with "*" (r2c25, r4c125) form what I might call a simple ALS ring (a generalization of an XY ring), although I'm open to suggestions if there's a standard or better name.

(6=9)r2c2 - (9=2)r2c5 - (2=896)r4c125 - (6=9)r2c2

I have come across such ring structures occasionally in Nightmare solutions but have never commented on them before. The idea is the same as in an XY ring -- the common value(s) appearing in the two nodes forming any side of the ring can be removed from any cell which sees both nodes. This means that "9" can be removed from r2c39, and that 8 and 9 can be removed from r4c347. If there were any other "6" candidates in column 2, or "2" candidates in column 5, they could be removed also.

After these eliminations, we have the position below, in which there are two deductions from uniqueness arguments:
 Code: .----------------.----------------.------------------. | 389  2    3489 | 6     5    14  | 89     7     189 | | 7    69   46   | 14    29   8   | 3      5     126 | | 1    5    689  | 239   7    239 | 689    4     2689| :----------------+----------------+------------------: | 89   689 *56   |*15+2  289  7   |*1-6    3     4   | | 2    7   *56+3 |*15+3  4    13  |*16+89  89    689 | | 389  4    1    | 389   36   369 | 7      2     5   | :----------------+----------------+------------------: | 6   #189  2    | 7     89   5   | 4     #189   3   | | 4   #189 &89   | 23    36   236 | 5     #1-89  7   | | 5    3    7    | 489   1    49  | 2      6     &89 | '----------------'----------------'------------------'

The simpler argument first -- there are threatened deadly patterns in rectangle r78c28 (marked with #), and cells r8c3 and r9c9 (marked with "&") enter into the logic. Note that digit "1" is locked into all sides of the rectangle. To avoid a deadly pattern of "18" cells, either we must place the "8" in box 7 in r8c3, or the "8" in box 9 in r9c9. Either way, r8c8 sees an "8" so r8c8 <> 8. In AIC form:

(8=9)r9c9 - (9=18)r78c8 - UR - (18=9)r78c2 - (9=8)r8c3 => r8c8 <> 8

The pattern is symmetric with respect to 8 and 9 so a similar argument gives r8c8 <> 9.

The second elimination is based on the threatened 56-61-15 deadly pattern in r45c374 (marked with "*"). There is a way to avoid the deadly pattern in each of boxes 4, 5, and 6, but all escape routes lead to a common deduction, namely r4c7=1.

In box 4, digit "5" is locked into the pattern, so the only way to avoid the pair of "56" cells is to place "6" in r4c2.
(6)r4c2 => (1)r4c7.

In box 5, "5" is again locked into the pattern, so the only way to avoid the pair of "15" cells is to place "1" in r5c6.
(1)r5c6 - (1)r5c7 = (1)r4c7

In box 6, "1" is locked into the pattern, so the only way to avoid the pair of "16" cells is to place "6" in r5c9.
(6)r5c9 => (1)r4c7.

Following up these deductions, we have the following position. Here there is a threatened 12 cell deadly pattern with base candidates 8 and 9, marked with "*" in the diagram. There is a surplus candidate in six of the 12 cells, but all lead to a common deduction, namely that r2c4=1.

 Code: .----------------.-----------------.-----------------. | 389   2    3489| 6     5     14  |*89     7   *89+1| | 7     69   46  | 1-4   29    8   | 3      5    126 | | 1     5    689 | 239   7     239 | 689    4    2689| :----------------+-----------------+-----------------: |*89    689  56  | 25   *89+2  7   | 1      3    4   | | 2     7    35  | 135   4     13  |*89+6  *89   689 | |*89+3  4    1   |*89+3  36    369 | 7      2    5   | :----------------+-----------------+-----------------: | 6     1    2   | 7    *89    5   | 4     *89   3   | | 4     89   89  | 23    36    236 | 5      1    7   | | 5     3    7   |*89+4  1     49  | 2      6   *89  | '----------------'-----------------'-----------------'

((8or9)=1)r1c9 - (1=4)r1c6 - (4=1)r2c4

((8or9)=6)r5c7 - (6=891)r1c79|r3c7 - (1=4)r1c6 - (4=1)r2c4

((8or9)=4)r9c4 - (4=1)r2c4

((8or9)=2)r4c5 - (2=9)r2c5 - (9=6)r2c2 - (6=4)r2c3 - (4=1)r2c4

((8or9)=3)r6c4 - (3=1)r5c6 - (1=4)r1c6 - (4=1)r2c4

((8or9)=3)r6c1 - (3=1894)r1c1679 - (4=1)r2c4

After placing r2c4=1 the rest is straightforward.

An XY chain which gives this same conclusion is:

(4=1)r1c6 - (1=3)r5c6 - (3=5)r5c3 - (5=6)r4c3 - (6=4)r2c3 => r2c4 <> 4.
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