This debate about touching vs crossing was the heated debate for which I spoke of earlier :)The difference in this problem's contest is subtle. Think of a drawn line as a physical object with defined spaced (which is key to understanding how my homeland uses the words). If one draws a line and touches a line they simply stood in the other line's space like standing on a doorstep (the drawn line) to knock on a door and left if no one answers, while If one draws a line that crosses the line they knocked on the door and went inside.So I think key to solving this problem is understanding that it treats a line segment as an object with physical properties, area, length, etc. and not as a cutting tool.I believe, one of my friends best described this idea as an old Tron game where two people cut off the board drawing segments in motorbikes.
I don't think that works. We are saying that the green curve below, connecting two green dots,by virtue of sharing a single point (touching but not crossing)isolates the lower red dot from the upper red dot. (The green characters do not actually touch, they can't, but they represent touching lines.) aO dO|fsO a\| fs/ sa\e| / asd\|/ asd/|\ as/d|f\ s/ad|fa\ /cxv|cxz\ \cxvO|cz/ a\s/|f\/ s/\a|f/ /cx\_/ Touching and crossing cannot be distinguished, once the line has been drawn.It's exactly the same locus of points. But let's say it's all about how the lineswere drawn, not how they end up. a line cannot be drawn in a manner bywhich an existing line is crossed by the pencil. The "touching but not crossing" line still does not isolate the red dots. The red dots can still be connected by a red line that also sharesthe common touching point without crossing either the left part or the right part of the green line. Thus: aO dO|fsO a\ds|fs/ sa\e| / asd\|/ asd/|\ as/d|f\ s/ad|fa\ /cxv|cxz\ \cxvO|cz/ a\s/|f\/ s/\a|f/ /cx\_/ So I still contend you can isolate a portion of the plane, for the purposes of this game,but only by using a line segment of infinite extent or by crossing an existing line.
The red line just stands on the doorstep as well.