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Curious pattern
Koszmarny
Kwon-Tom Obsessive
Puzzles: 1087
Best Total: 21m 36s
Posted - 2013.11.27 18:04:33
During practicing I've made a certain trial and error and very quickly arrived at contradiction. I've reduced it to bare minimum and the below pattern appeared. I've never seen it before, and it seems quite useful.


[There is something to be marked here.]

Do you happen to see/know a way of thinking that would solve this and similar situations w/o trial and error.
mathmaniac
Kwon-Tom Obsessive
Puzzles: 1089
Best Total: 21m 12s
Posted - 2013.11.27 18:16:12
Nothing specific, but if you can think about how 2's work (possible patterns and propagation), then you'll be able to see fairly quickly the two negations to which you're referring. As for general rules, the only thing I can think of is the "lines in vs lines out" method. You know, making sure you don't have an odd number of ends in an enclosed area.
Koszmarny
Kwon-Tom Obsessive
Puzzles: 1087
Best Total: 21m 36s
Posted - 2013.11.27 18:42:12
I don't see what should propagate where here.
We don't have any definitely odd or even corner here.

As for cut-parity (even number of lines crossing in-out any area), I tried many areas here but nothing seems to work.

Only the intersection method seems to speed up things a little, but it still needs trial and error. (choose any binary choice in both ways and mark down the intersection).
Last edited by Koszmarny - 2013.11.27 18:44:27
astrokath
Kwon-Tom Obsessive
Puzzles: 3086
Best Total: 13m 42s
Posted - 2013.11.28 09:07:37
This is a highlander case, as far as I'm concerned.

If you add a horizontal line below the lower 2, you constrain the upper 2 to require a vertical line on the right, like this:

However, with that set-up the lines around the upper 2 could just as easily flow left/beneath the number as they could right/above, like this:

As there's only one correct answer, that horizontal line I placed beneath the lower 2 has to be wrong, which forces another cross to the right of the upper 2.

You can do it in reverse order just the same (starting with the line to the right of the upper 2), and it's certainly a pattern that I can 'see' without having to brute-force solve it like I've done here.
Last edited by astrokath - 2013.11.28 09:09:00
Koszmarny
Kwon-Tom Obsessive
Puzzles: 1087
Best Total: 21m 36s
Posted - 2013.11.28 11:21:20
Ah indeed this is a highlander after I stripped all the surroundings.
But in the original there were some neighbors that were affected.
qqwref
Kwon-Tom Obsessive
Puzzles: 2099
Best Total: 13m 28s
Posted - 2013.11.29 19:51:09
Cool pattern. My trial and error process would generally be something like this: The corner between the 2s is either even or odd. If it's even, contradiction. So the corner is odd (and we can mark two new X's).

Perhaps a more logical approach is as follows. The two X's you already have mean that any lines coming out of the top-left corners of the 2's must join with the line at the far top-left. So at most one of those lines exists, and anything that would make them both exist is wrong. Since these are 2's joined at a corner, their joining corner can't be even (since that would make those two lines exist), so it's odd (and we can mark two new X's).
LoopGuy
Kwon-Tom Obsessive
Puzzles: 761
Best Total: 45m 59s
Posted - 2013.11.30 04:09:16
I'm not sure this is better, but I sometimes look for possible patterns as opposed to contradictions.  For example, the line segment at the top left can only go three ways (marked by ?):



Each of these choices draws a simple pattern:





or



From those, I can see that two segments are never used.  If any other squares have numbers, it may further restrict the choices until there is one choice.

-Guy
LoopGuy
Kwon-Tom Obsessive
Puzzles: 761
Best Total: 45m 59s
Posted - 2013.11.30 04:17:57
Quote:
Originally Posted by qqwref
Cool pattern. My trial and error process would generally be something like this: The corner between the 2s is either even or odd. If it's even, contradiction. So the corner is odd (and we can mark two new X's).

I actually like this one.  Another way to think about it is that at least one of the top left corner of both 2's must be even, therefore, the corner which combines the 2's must be odd, which leads to the solution without trial and error.

-Guy
Koszmarny
Kwon-Tom Obsessive
Puzzles: 1087
Best Total: 21m 36s
Posted - 2013.12.05 15:40:00
Thanks for the insightful comments.
When I do trial and error I always try to start with a line that has only 2 options. (So that from contradiction gives me the other line instead of just cross).

One of my take-aways is that odd-even (instead of line-no line) might be a good exploration route. Especially if multiple numbers are connected by a corner.

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