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Can this simple forcing chain be "formalised"?

 
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George Woods



Joined: 28 Mar 2006
Posts: 231
Location: Dorset UK

PostPosted: Sat Jan 10, 2009 10:49 am    Post subject: Can this simple forcing chain be "formalised"? Reply with quote

Given that today's forcing chains often become tomorrow's standard technique, I have a simple focing chain here that I cannot formalise into a conventional logical technique. Can anyone help?

Code:

+------------+--------------------+---------------+
| 7   28  1  | 2689   24689 24689 | 3  2459 2459  |
| 5   238 4  | 123789 1289  23789 | 6  129  129   |
| 9   23  6  | 123    5     234   | 7  8    124   |
+------------+--------------------+---------------+
| 34  9   57 | 2578   28    1     | 58 6    23457 |
| 8   147 2  | 567    3     567   | 9  1457 1457  |
| 13  6   57 | 4      289   25789 | 58 1257 12357 |
+------------+--------------------+---------------+
| 146 5   9  | 16     7     46    | 2  3    8     |
| 126 127 8  | 123569 1269  23569 | 4  579  579   |
| 24  247 3  | 2589   2489  24589 | 1  579  6     |
+------------+--------------------+---------------+

Play this puzzle online at the Daily Sudoku site

The chain is one that shows that 4 cannot be at r1c5 and so r9c5 must be 4

if r1c5 =4 then r3c4=1, r7c4=6 But also r8c5 =6 so fails

I Know the grid above can be simplified eg 56 in box 5 - but this does not affect the argument!
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Jan 10, 2009 12:01 pm    Post subject: Reply with quote

Code:
(4)r1c5 - (4)r3c6 = (4-1)r3c9 = (1)r3c4 - (1=6)r7c4 - (6)r8c5 = (6)r1c5 => [r1c5]<>4

Your chain appears to be very generic in nature. I doubt if it has a name.
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Asellus



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

PostPosted: Sun Jan 11, 2009 11:41 am    Post subject: Reply with quote

I suspect that George probably saw the chain more in this manner, with the {23} locked set explicit:

(4)r1c5 - ALS[(4)r3c6={23}r3c26] - ({23}=1)r3c4 - (1=6)r7c4 - (6)r8c5=(6-4)r1c5; r1c5<>4

Viewed as such, it still has no name. However, there's another way to see this that can be named. First, note the {1234} ALS in r3c246. This contains the strong inference ALS[(1)r3c4=(4)r3c6], which is a 14 "pseudocell". This can be used to form an "XY-Wing" with <6> pincers:

(6=1)r7c4 - ALSr3c246[(1)r3c4=(4)r3c6] - (4=6)r7c6; r8c5<6> r1c5=6 (i.e. r1c5<>4)

What's more, this "XY-Wing with pseudocell" could be formed with <1> pincers or <4> pincers as well. That's because it's a continuous loop. So we could call it an "XY-Wing with pseudocell loop"! Here it is with a fully reduced basic grid:
Code:
+---------------+-------------------------+----------------+
| 7     28   1  | 2689     24689  #2689-4 | 3   2459  2459 |
| 5     238  4  |#23789-1  1289    23789  | 6   129   129  |
| 9     23   6  |b123      5      b234    | 7   8     124  |
+---------------+-------------------------+----------------+
| 34    9    57 | 278      28      1      | 58  6     34   |
| 8     14   2  | 56       3       56     | 9   147   147  |
| 13    6    57 | 4        89      789    | 58  12    123  |
+---------------+-------------------------+----------------+
|#14-6  5    9  |a16       7      c46     | 2   3     8    |
| 126   17   8  |#2359-16 #129-6  #2359-6 | 4   579   579  |
| 24    47   3  | 2589     2489   #2589-4 | 1   579   6    |
+---------------+-------------------------+----------------+

(6=1)r7c4 - ALSr3c246[(1)r3c4=(4)r3c6] - (4=6)r7c6 - Loop; r2c3<>1; r19c6<>4; r7c1|r8c456<>6
I have marked all those victims with "#".

This grid is actually quite interesting. There is also a Sue de Coq, marked "@" below:
Code:
+---------------+-------------------------+----------------+
| 7     28   1  | 2689    #46-289  24689  | 3   2459  2459 |
| 5     238  4  | 123789  #1-289   23789  | 6   129   129  |
| 9     23   6  | 123      5       234    | 7   8     124  |
+---------------+-------------------------+----------------+
| 34    9    57 | 278     @28      1      | 58  6     34   |
| 8     14   2  | 56       3       56     | 9   147   147  |
| 13    6    57 | 4       @89      789    | 58  12    123  |
+---------------+-------------------------+----------------+
| 146   5    9  |@16       7      @46     | 2   3     8    |
| 126   17   8  |#2359-16 @1269   #2359-6 | 4   579   579  |
| 24    47   3  | 2589    @2489   #2589-4 | 1   579   6    |
+---------------+-------------------------+----------------+

While not what George saw, it has the same overall effect.
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