## AZ Republic 08-19-06

All about puzzles in newspapers, magazines, and books
daj95376, in classical logic theory, if a or b, there are 3 possible cases:

1. a=true and b=false
2. a=false and b=true
3. a=true and b=true

So in this particular instance, it turns out to be case (3). So the or relationship is not violated.

Basically, we just need to prove at least one of r5c6=9 or r6c8=9 must be true. That they are in fact both true doesn't violate our conditions...

Of course, an immediate result of r5c6=9 or r6c8=9 is that they must both be true (from the strong links on r5 and b6). But it was Carcul's choice to use that particular or relationship from the start and he's in the position to explain it... I'm as puzzled as you on why he chose to take that particular path...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk, if Carcul had provided a proof that at least one of the assignments was true, then we wouldn't need a refresher course on classical logic theory because an or statement would be unnecessary ... as should be this conversation! In your (1) and (2), the or relationship is just another way of saying that a conjugate relationship exists. Then, and only then, would an or statement be meaningful.
daj95376
2014 Supporter

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Joined: 15 May 2006

I just provided a proof (by contradiction) why r5c6=9 or r6c8=9 (NOT xor) must be true.

There is NO conjugate relationship, and we don't need one. Since at least one of r5c6 or r6c8 must be 9, r5c9 cannot be 9. That was what Carcul set out to do.

Whether the or relationship is meaningful/necessary is not important. As long as the logical steps are sound it is an acceptable solution path...

Once again I want to stress that it's not my choice to use that relationship to solve the puzzle. There are many ways to solve the puzzle (as I've listed out on the previous page). Carcul's way is just one of them. You might not like it because you think it's an overkill, but at least it's logical... Compare it to some others which for example uses the sums of the possible candidates with ambiguous logic which nobody understands, this one looks good enough to me...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:How Carcul could see this directly is beyond me (and many others I suspect)...

Code: Select all
` *-----------------------------------------------------------* | 5     89    48    | 16    49    16    | 3     7     2     | |#16    2     7     | 8     3     5     | 9     4    #16    | | 39    36    14    | 2     49    7     | 8    #16    5     | |-------------------+-------------------+-------------------| | 4     36    9     | 7     2     136   | 16    5     8     | |#16    5     2     | 4     8    #16(9) | 167   3     79    | | 38    7     18    |#16    5     1369  | 2    #16(9) 4     | |-------------------+-------------------+-------------------| | 2     89    5     | 3     16    4     | 167   1689  79    | | 7     4     3     | 9     16    8     | 5     2     16    | | 89    1     6     | 5     7     2     | 4     89    3     | *-----------------------------------------------------------*`

I saw it in a slightly different way. If r5c6 <> 9 and r6c8 <> 9, then we a contradiction in the following way: Pretend that you are looking for remote naked pair exclusions on cells with [16]. Start from r6c8. Above, one can see a chain of [16] cells of odd length (and > 1), where the initial and terminal cells can see each other, an impossibility.
re'born

Posts: 551
Joined: 31 May 2007

Rep'nA is right. In the grid, the ATILA in cells {r2c9|r2c1|r5c1|r5c6|
r6c4|r6c8|r3c8} allow us to conclude that we have the strong link
[r5c6]=9=[r6c8]*, and as the "9" in r5c9 can see both those cells, it
can be eliminated.

*In the sense that, if "9" is not in one of the cells then it must be in the
other.

Carcul
Carcul

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Joined: 04 November 2005

Thanks rep'nA & Carcul for the explanations...

Now that you've explained it, it seems much clearer why you made that move and hopefully daj95376 would not say "it doesn't make sense" anymore...
udosuk

Posts: 2698
Joined: 17 July 2005

Carcul wrote:the ATILA in cells {r2c9|r2c1|r5c1|r5c6|
r6c4|r6c8|r3c8} ...

As I recall, ATILA is an acronym for Almost Two Incompatible Loops (something). The listed cells -- {r2c9|r2c1|r5c1|r5c6|
r6c4|r6c8|r3c8} -- comprise one loop. What is the second?

Carcul wrote:... allow us to conclude that we have the strong link
[r5c6]=9=[r6c8]*, and as the "9" in r5c9 can see both those cells, it
can be eliminated.

*In the sense that, if "9" is not in one of the cells then it must be in the
other

"Strong inference" would be better than "strong link" IMO.
ronk
2012 Supporter

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Location: Southeastern USA

ATILA stands for "Almost Two Incompatible Loops Argument". The listed cells don't comprise any loop. However, we would have two (incompatible) loops in those cells if both r5c6/r6c8 were not "9": a "loop" of an odd number of cells all populated with only the same two candidates "x,y", for which we could then write, for any of those cells, two nice loops - one that forces that cell to be "x", and another that forces it to be "y", a contradiction.

Carcul
Carcul

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Joined: 04 November 2005

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