3 x 4 Connect-The-Dots Game

[BMAD]

There is a simple game that I used to play as a kid in my home country. Here is a simplified version. There is a game board with 4 dots on three levels setup as follows:

* * * *

* * * *

* * * *

One player moves first (usually the youngest) and both follow the same rules. Connect one dot to any other dot. When connecting two dots you may not cross over another dot, you may not cross a line made from a prior connection, nor can you cross over your own line. Dots that are already connected are 'off limits' from reuse. Dots may be connected using any type of line (curve, straight, etc.) as long as it is continuous. The player that cannot make a move is the loser.

Assuming that the youngest person goes first, which player has the winning strategy to guarantee victory?

[witzar]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

Good idea but does that guarantee a win?

The playing field is separated into two equivalent areas before my opponent's move.

The above statement is invariant in my strategy (it's true after every move I make).

Therefore after every my move either there is no move for my opponent (I win) or there is one.

In later case, no matter what he does, the invariant guarantees, that I can duplicate his move in the other area.

This way (1) I always have a move after my opponent, (2) the invariant is conserved after my move.

Number of moves in the game is obviously finite, therefore this strategy guarantees a win.

[bonanova]

There are 12 points and each connection removes two.

If players connect neighboring points each player get three moves

Younger player loses for lack of move.

Suppose first move connects dots A and B with a serpentine line that weaves in out and around all the other dots.

The playing area is not divided into disjoint subsets by so doing.

Any other pair of dots may be connected on the following turn.

Topologically, dot B can be slid back along the first connecting line until it is adjacent to dot A.

Same for the next turn that connects dots C and D.

And so forth until each has made three connections, and Younger player has no dots left to connect.

Obviously I am missing something.

[witzar]

Obviously I am missing something.

Me too.

[Nins_Leprechaun]

Obviously I am missing something.

Me too.

If we are playing this game, me three.

[BMAD]

The trick that kids in my homeland soon discover is the rule of crossing. As long as the line runs tangent to another line (or itself) then it did not cross but did effectively cut off some of the board.

This distinction may not exist in English, and I am sorry if it confused anyone. I once got in a heated discussion with someone over what the word cross means. As my friend meant touching is the same as crossing, but for my people, crossing requires the passage over (beyond) the line.

[BMAD]

I think you will discover many such winning lines, but now consider the line (or summation of line lengths), and try to find it/them that is/are needed to create the shortest guranteed win

[witzar]

So you can win instantly in first move connecting two dots and cutting off each of the remaining dots with a loop around it?

[BMAD]

Yes, if you have the skill. Which is hard for young kids cause violating the rules is an immediate lost. Now, given that it is hard to make such a line for a child, what is the shortest possible line they would have to make to gurantee victory?

[bonanova]

To divde the plane requires a line segment of infinite length or a closed loop.

A closed loop would require starting and ending on the same point, or a line segment that crosses itself.

Both cases seem prohibited by OP.

So, somewhere in one of those statements is what I am missing?

[witzar]

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".

[BMAD]

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.

[BMAD]

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 July 8, 2013 by BMAD

[bonanova]

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 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:

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.

[witzar]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

[BMAD]

[BMAD]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

Good idea but does that guarantee a win?

[bonanova]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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

[bonanova]

[gavinksong]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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.

[gavinksong]

Bonanova is right. A third line can run tangent in between two lines that touch without crossing either.

[BMAD]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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

. That is true but you can loop in your line as long you do not go over your line

* * * *

* * * *

* * * *

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

[BMAD]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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

. That is true but you can loop in your line as long you do not go over your line

* * * *

* * * *

* * * *

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

[BMAD]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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

. That is true but you can loop in your line as long you do not go over your line

* * * *

* * * *

* * * *

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

what you say is true but the rules allow you to loop within lines as long as you do not go over them

[bonanova]

How about making just one big loop that splits the playing field into two equal parts?

After this just repeat copying your opponent moves (what he does in one of the parts, copy in another),

until he runs out of moves.

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

. That is true but you can loop in your line as long you do not go over your line

* * * *

* * * *

* * * *

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

what you say is true but the rules allow you to loop within lines as long as you do not go over them

Think of 100 circles of radii 1, 2, 3, ..., 100 centered at x = 1, 2, 3, ..., 100.

They are mutually tangent at the origin.

Any number - a countable infinity of them - could be mutually tangent.

And none of them "go over" any of the others.

Between circles 1 and 100 there are 98 circles that do not violate the OP.

You would have to add a statement that once two curves are made to touch,

a third such line is prohibited, in order to limit the curves that touch to only two.

So, mentally I will add that statement, and completely agree.