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McCarthy Stone competition

 
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alanr555



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

PostPosted: Fri Jan 06, 2006 1:19 am    Post subject: McCarthy Stone competition Reply with quote

Code:

The following puzzle was set with twentyeight initial clues
and I have been able to progress it as far as finding some
twentyfour additional values as shewn.

Using trial and error I have determined that the remaining
twentynine values are linked to a unique solution but I do
not know how to demonstrate this by logic.

There is a hundred pound prize and I can submit my entry
with the values derived by trial and error but I would prefer
to know the logic that confirms this result.

400 580 100
108 306 705
560 900 200

709 405 602
204 600 950
651 298 374

305 709 800
876 150 409
902 860 507

The twenty-nine missing values are
34123612417388138417394236279
but I have not laid out them as a grid
to avoid portraying the solution.

++
The original puzzle was set as
400 080 100
100 006 705
000 900 200

009 405 602
004 600 000
601 208 300

005 009 000
806 100 009
002 060 007
++
The candidates for unresolved cells are

- 29 37; - - 27; - 369 36
- 29 - ; - 24 - ; - 49 -
- - 37; - 17 147; - 348 38

- 38 - ; - 13 - ; - 18 -
- 38 - ; - 137 137; - - 18
- - -; - - -; - - -

- 14 - ; - 24 - ; - 126 16
- - - ; - - 23; - 23 -
- 14 -; - - 34; - 18 -

This gives quite a few "pairs" (three in column two!) and only
digits 1,3,7 appear more than twice in a row, column or region.

There is probably an 'implication' chain somewhere - but where
should one be looking to locate it? I am not interested very much
in the exact location but my interest is really in WHY one should be
looking for it in that very location (where, with hindsight, it exists).

Alan Rayner  BS23 2QT
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Fri Jan 06, 2006 9:47 am    Post subject: Reply with quote

Hi Alan,

Nice Sudoku puzle. You did it till the point where classic techniques can solve it.
From this point on, I apply the "double implication chain" technique, starting from the cells that contain a pair.

In our concrete case, even if there are a lot of pairs, I get a contradiction starting from the first one. So:

1. path: r1c2 = 2, r1c2 <> 9, r1c8 = 9, r1c8 <> 6, r7c8 = 6
2. path: r1c2 = 2, r1c6 <> 2, r8c6 = 2, c7c5 <> 2, r7c8 = 2

and in r7c8 can't hold both digits 6 and 2. This means that digit 2 can be excuded from r1c2.

The rest can be done with standard techniques.
Hope I could help.

see u, Alan,
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keith



Joined: 19 Sep 2005
Posts: 3179
Location: near Detroit, Michigan, USA

PostPosted: Sun Jan 15, 2006 3:20 pm    Post subject: BUG me again! Reply with quote

Let's take another look at this one. Using standard techniques, it can be solved to this stage:

Code:


  4  29  37   5  8   27    1  69  36   
  1  29  8    3  24  6     7  49  5   
  5  6   37   9  17  147   2  48  38   

  7  38  9    4  13  5     6  18  2   
  2  38  4    6  137 17    9  5   18   
  6  5   1    2  9   8     3  7   4   

  3  14  5    7  24  9     8  126 16   
  8  7   6    1  5   23    4  23  9   
  9  14  2    8  6   34    5  13  7   



Now, there is a forcing chain that involves the cells:

R5C6, R1C6, R1C2, R1C8, R1C9, R3C9, R5C9, and back to R5C6. If R5C6 = 7, the chain is in the order given. If R5C6 = 1, the chain is traced out in the reverse order. The starting cell does not matter.

Examining the "corners" of the chain, we see that:

One of R5C6 and R1C6 is 7, so R3C6 cannot contain 7.
One of R5C9 and R5C6 is 1, so R5C5 cannot contain 1.

Now, we have a BUG! Every square, except one, has only two possibilities. R7C8 can be <126>.

Take a look at the row, column, and block to which this square belongs. If the possibilities are 2 or 6, then each of these possibilities occurs twice (in the row, column or block), which is not allowed. So, R7C8 must be 1, and the rest is trivial.

Keith
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