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3 x 4 Connect-The-Dots Game


Best Answer witzar, 08 July 2013 - 04:01 PM

 

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Good idea but does that guarantee a win?

 

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#11 BMAD

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Posted 08 July 2013 - 02:49 AM

If I understand correctly, the following pattern is not considered "crossing":

__Ω_Ω_Ω_Ω_Ω__

 

The loops are closed, but it is "touching" not "crossing".

yes! Witzar has the understanding.  There is a way to loop ( by touching a line at a single point but not going through the line) and guarantee victory.  but is very challenging to draw a long line that properly does this but there are shorter lines that can be drawn and succeed at doing this.


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#12 BMAD

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Posted 08 July 2013 - 03:21 AM

1 move.png
 
here is an example of a move.  Clearly this move doesn't guarantee victory but does illustrate the touching without crossing principle.  
 
So what is the best (e.g. shortest) path a player can draw to win the game?

Edited by BMAD, 08 July 2013 - 03:22 AM.

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#13 bonanova

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Posted 08 July 2013 - 05:52 AM

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.)

 

adO|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 lines
were drawn
, not how they end up.
a line cannot be drawn in a manner by
which 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 shares

the common touching point without crossing either the left part

or the right part of the green line.

 

Thus:

 

adO|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.


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#14 witzar

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Posted 08 July 2013 - 10:58 AM

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#15 BMAD

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Posted 08 July 2013 - 01:29 PM

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.)
 
adO|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 lines
were drawn
, not how they end up.
a line cannot be drawn in a manner by
which 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 shares
the common touching point without crossing either the left part
or the right part of the green line.
 
Thus:
 
adO|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.


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.
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#16 BMAD

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Posted 08 July 2013 - 01:30 PM

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Good idea but does that guarantee a win?
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#17 witzar

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Posted 08 July 2013 - 04:01 PM   Best Answer

 

Spoiler for


Good idea but does that guarantee a win?

 

Spoiler for


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#18 bonanova

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Posted 08 July 2013 - 05:55 PM

Spoiler for


You can't make a loop, without using one of the dots twice.
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- Bertrand Russell

#19 bonanova

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Posted 08 July 2013 - 05:57 PM

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.) adO|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: adO|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.

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.

The red line just stands on the doorstep as well.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#20 gavinksong

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Posted 09 July 2013 - 12:43 PM

 

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You can't make a loop, without using one of the dots twice.

 

 

*  *  *  *

*  *  *  *

*  *  *  *

 

You can start at the red dot, make the loop, and then end at the blue dot.


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