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Something I've always been curious about http://www.calcudoku.org/forum/viewtopic.php?f=3&t=108 
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Author:  starling [ Fri Feb 10, 2012 5:31 pm ] 
Post subject:  Re: Something I've always been curious about 
clm wrote: 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. Ah missed that. I guess it's just a coincidence, then. Actually, it isn't. I think I just figured out why: Break the puzzle into the 4 3x3 quadrants. Each of those quadrants (We only care about the top left and bottom right) can contain either 0, 1, 2, or 3 of a number, which I believe are equally probable, though I can't actually prove that. If it contains 2 of the one placed in a2, b1, c2, b3, e4, d5, f5, or e6, there's a 1 in 4 chance that an xyx L shaped cage occurs at that site, with 8 sites. If it contains 3 of the one placed in a2, b1, c2, b3, e4, d5, f5, or e6, there's a 1 in 2 chance that an xyx L occurs involving that particular number. Also, each quadrant has to have at least 2 of these last two sets, since 3, 2, 1, 1, 1, 1, gives 9 and uses all 6 numbers. I think 3 cases of these two is the most common case, though. If it contains 1 in one of those spots, there's a 0 percent chance one occurs, but a 1 of a particular number only decreases the probability of an xyx L if it occurs in one of those 8 spots, meaning there's a 5/9 chance we don't care about a number that occurs once at all. And if it contains 0, this case doesn't trigger, so it doesn't matter. So basically, though I don't have exact numbers, because we don't have to care about a lot of the cases that are bad for this' likelyhood(55% of anything containing only 1 of a number, and 100% of anything containing 0), overall there's a higher probability of an xyx occurring than normal. Or maybe it's the same likelihood, and it just ended up getting noticed here, since these same figures should apply to any Lshaped cage in the same place, but this one has more Lshaped cages in it by structure than normal. and xyx's are more common than I thought. 
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