stigant
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I ran into this puzzle on the demos page for Wolfram's Mathematica player:

http://demonstrations.wolfram.com/DevilishMazes/

You'll need to install the Mathematica player to view and interact with it, but I'll post screen shots and explain the puzzle here.

There are three puzzles. Each is played on a Go board. The board starts with a sort of maze of walls created from black stones and a white stone at the center. The white player is trying to connect his stone to the outside while black is trying to completely contain the white group (just like in Go). The catch is that white gets to play 2 stones each turn while black only gets to play 1. The question is, with best play, who wins in each situation?

Here are the initial positions:




Edit:
I should be more specific about the win conditions for white:
In maze 1, white wins if he connects his center stone to one of the white stones on the perimeter of the black wall.
In mazes 2 and 3, white wins if he connects his center stone to a stone clear of the black maze.

My answers (so far):


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[Last edited by stigant at 02-07-2014 03:17 PM]

stigant
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AH! I think I got it now:


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Someone Else
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stigant
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Actually:


Edit: no, I agree with you again. Argh.


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[Last edited by stigant at 02-07-2014 04:18 PM]

The spitemaster
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stigant
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Ok, I think you're correct, but I'll play black. Your move:


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TripleM
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Fascinating puzzle. I'm leaning towards black at this stage, but without any concrete proof.

Nuntar
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I haven't looked at #3 yet, but here's a simple proof for White win in #1:


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Someone Else
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For #3, labelling the lone white stone as the origin, I think a fairly complete solution is:

stigant
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Click here to view the secret text

×In its first two moves, white plays at (0,3)(0,-3)(-3,0),(3,0) and black plays at (0,1) and either (1,0) or (0,-1).

Ah, but


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TripleM
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Agreed - the (+-3,+-3) squares seem crucial squares for both sides to play on early.

The spitemaster
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My solution


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TripleM
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The spitemaster wrote:
My solution

Click here to view the secret text

×With the white square as the origin as follows. The board is rotationally symmetric so the solution is not a sole unique answer. White: (1,0) (0,1) Black (2,0) White (-1,0) (0,-1) Black (0,2) White (-2,0) (-3,0) Black (-4,0) White (0,-2) (0,-3) At this point White has four/five routes to the outside (Two of the routes coincide over a single point. However, if White plays there first then it really counts as five.) This means that if the board was infinite Black would never be able to cut it off as it has been unable to reduce the number of routes to the outside.

I'm not convinced at all even that position is a win for white, but change black's last move to (0,-3) (and you may as well change his early moves from (2,0) to (3,0) and (0,2) to (0,3)) and that's an easy win for black.

The spitemaster
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I will continue then while changing all of blacks responses to the intersections:


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The spitemaster
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I do however have to retract the bit about an infinite board. That is not true to my knowledge.

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The spitemaster
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And for the variant of black changing to (0,-3) at the indicated time.

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TripleM
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You're right, sorry, I was looking at the wrong thing. Still, as mentioned earlier, black playing in the center early doesn't seem to be a great strategy compared to some others, so that doesn't prove anything regarding who wins.

Tim
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I think that

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The spitemaster
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Here is an alternate

Here is the problem that I am having a hard time to convey; what the goal of white is. Really it is not too escape, first it is to have two stones on different 3 distance rows that are unblocked to the exit after blacks turn. This is because this will meant victory two turns later. So, how is white to achieve this? Well, having three different exits going into blacks turn. Then we can count that as White's victory as well. To pull it back yet another turn (or two depending on blacks choice), we can see that if white has one exit starting it's turn and then has a corner that it can get to on it's turn then it also wins.

To Tim's Variant:


As a side note: White keeps changing due to: perhaps an improvement on my part, or to an easier to demonstrate method.

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The spitemaster
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I've been thinking about this all day, and I am now convinced I am in error. I will be back when I can prove black wins regardless.

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I wish that man would stay away