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Something I've always been curious about https://www.calcudoku.org/forum/viewtopic.php?f=3&t=108 
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Author:  beaker [ Fri Dec 16, 2011 9:04 pm ] 
Post subject:  Re: Something I've always been curious about 
Once again, all I did was do the cells I was sure of and left the lower right hand corner to the end and then counted the numbers and found I was missing 2 fives and then the solution was obvious. 
Author:  clm [ Wed Feb 08, 2012 12:41 pm ] 
Post subject:  Re: Something I've always been curious about 
Hi, starling, sure you have noted that today's 6x6 difficult puzzle (Puzzle id: 417948) destroys all our previous hypothesis on this configuration, very interesting. 
Author:  starling [ Wed Feb 08, 2012 9:44 pm ] 
Post subject:  Re: Something I've always been curious about 
clm wrote: Hi, starling, sure you have noted that today's 6x6 difficult puzzle (Puzzle id: 417948) destroys all our previous hypothesis on this configuration, very interesting. That I did. Actually, I noticed that literally only 1 Lshaped xyx cage wasn't eliminated purely by the single digit cages, 515 in the 11+. So now I'm wondering if it's statistically more probably that an Lshaped xyx occurs, or if we just hit a really weird random stretch. 
Author:  sneaklyfox [ Thu Feb 09, 2012 6:19 pm ] 
Post subject:  Re: Something I've always been curious about 
Well, there is still an xyx cage in b1c1c2. All the other puzzles could be like that if you just flip them... 
Author:  clm [ Thu Feb 09, 2012 7:48 pm ] 
Post subject:  Re: Something I've always been curious about 
sneaklyfox wrote: Well, there is still an xyx cage in b1c1c2. All the other puzzles could be like that if you just flip them... No, no, sneaklyfox, that cage has nothing to do, if you make a central symmetry (or a rotation 180º degrees right), moving the upper left box to the bottom right box you obtain a totally different configuration but not this one. This was discussed in the past in this same thread, I reproduce the complete paragraph I included in my analysis on Nov 21, 2011: "Usually the top left area is solved first and quickly and the configuration in this area has "nothing to do" with the bottom right area since the puzzle is not symmetric (keeping the same aspect) in the sense that it would not be valid to rotate, for instance, 180 degrees to the right because we would obtain a different puzzle (only the cages themselves are symmetric, i.e., inside the blue areas, but not the full puzzle), we may observe that all six 3cell inline cages “look inside” the top left area providing, in some way, the required information, while what we could say of the bottom right area is that it is “blind” in this context. Then we will forget the symmetries of the full puzzle and work with the original configuration, the one of the graphic." 
Author:  sneaklyfox [ Thu Feb 09, 2012 8:58 pm ] 
Post subject:  Re: Something I've always been curious about 
clm wrote: sneaklyfox wrote: Well, there is still an xyx cage in b1c1c2. All the other puzzles could be like that if you just flip them... No, no, sneaklyfox, that cage has nothing to do, if you make a central symmetry (or a rotation 180º degrees right), moving the upper left box to the bottom right box you obtain a totally different configuration but not this one. This was discussed in the past in this same thread, I reproduce the complete paragraph I included in my analysis on Nov 21, 2011: "Usually the top left area is solved first and quickly and the configuration in this area has "nothing to do" with the bottom right area since the puzzle is not symmetric (keeping the same aspect) in the sense that it would not be valid to rotate, for instance, 180 degrees to the right because we would obtain a different puzzle (only the cages themselves are symmetric, i.e., inside the blue areas, but not the full puzzle), we may observe that all six 3cell inline cages “look inside” the top left area providing, in some way, the required information, while what we could say of the bottom right area is that it is “blind” in this context. Then we will forget the symmetries of the full puzzle and work with the original configuration, the one of the graphic." Sorry if I misunderstood. I guess I have not been reading in all the detail about this thread, but just thought that the discussion was based on the appearance of an xyx Lshaped cage which seemed to occur in every puzzle. Although the configuration is not the same because the 3cell inline cages would be oriented differently, I don't think that fact affects so very much. Perhaps every puzzle of this configuration has an xyx Lshaped cage somewhere whether it's in the "blind" bottom right area or the top left area. 
Author:  starling [ Fri Feb 10, 2012 1:03 am ] 
Post subject:  Re: Something I've always been curious about 
That said, I wonder if sneaklyfox is right that this puzzle broke that rule because of the other xyx L shaped cage. Maybe it's just that there has to be one. 
Author:  clm [ Fri Feb 10, 2012 1:20 pm ] 
Post subject:  Re: Something I've always been curious about 
starling wrote: That said, I wonder if sneaklyfox is right that this puzzle broke that rule because of the other xyx L shaped cage. Maybe it's just that there has to be one. But, since there is at least one puzzle (the one I provided yesterday, in my last reply to sneaklyfox, I believe with an unique solution) with that type of configuration and without any 3cell Lshape cages of the type xyx, we must then accept the possbility of finding a big amount of them. 
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