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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 x-y-x 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 3-cell L-shape cages of the type x-y-x, 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 x-y-x 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 x-y-x 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 x-y-x 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 x-y-x 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 L-shaped cage in the same place, but this one has more L-shaped cages in it by structure than normal.

and x-y-x's are more common than I thought.

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