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Abstract Games» Forums » General

Subject: New Game: Enclose rss

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Russ Williams
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kocbftn wrote:
Enclose is played on a 19by19 toroidal Go board with Go stones. if a player encloses an area with their stones they capture all the pieces in that area and their opponent may not place their stones onto intersections in the enclosed area. The game ends when there are no intersections that have not been enclosed or occupied. Once the game ends, the winner of the game is the player that has the most empty intersections enclosed. Unlike Go, there are no such thing as cycles and so there are no rules needed to combat them. Enclose is simpler than Go, but what remains to be seen is the level of depth Enclose has compared to Go.

Some refinement is needed in the notion of "enclosing"; otherwise the first stone placed "encloses" the remaining 360 points (it's a maximally connected group of empty spaces touching only one color, after all), and the first player wins. Maybe like in Cathedral the solution is simply making an exception for the first move (i.e. that as an exception, the first move does not enclose).

Even then, though, the wraparound board makes it tricky. How are you defining "enclose"? E.g. in the following diagram (on a smaller board), does X "enclose" the 1 point region, or the 11 point region, or both regions?

. X . .
X . X .
. X . .
. . . .
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kocbftn wrote:
I thought this would be self-explanatory, but the enclosed area is the area within the stone formation, and at least 4 stones are required to enclose an area.

But you specified that it's a toroidal map, i.e. wraparound, right? So there is no "within" and "without", and the edges of the map have no meaning.

E.g. shift the 4 stones "north":
X e X .
. X . .
. . . .
. X . .

Your intent is still that the single point "e" is "enclosed", right?

Or expand the original stones to "enclose" a 2x2 array of points:
. X X .
X e e X
X e e X
. X X .

Your intent is that the 4 points "e" are enclosed, right?


Now shift the shape up:
X e e X
X e e X
. X X .
. X X .

This is an identical game state on a toroidal/wraparound board.

And then shift left:
e e X X
e e X X
X X . .
X X . .

Same game state. But your intent is still that the 4 points "e" are "enclosed", and that the others are not, even though that's clearly arbitrary here, as there's no evident different between the "upper left" 4 empty points and the "lower right" 4 empty points?
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Russ Williams
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kocbftn wrote:
the enclosed area is always the smaller of the two areas on either side of the enclosing stone formation. If this is a tie, then the enclosed area is the area which on the non-imagined portion of the board looks to be of the same shape as the enclosing stone formation. So in those examples the e's are enclosed but the dots are not.

The e's and the dots have identical 2x2 square shapes, so I don't understand the distinction you are making.
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Toroidal Go
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russ wrote:
Or expand the original stones to "enclose" a 2x2 array of points:
. X X .
X e e X
X e e X
. X X .

Your intent is that the 4 points "e" are enclosed, right?

You have cheated a little bit here, because your group of stones wraps around the whole board and consequently no longer has the topology of a simple closed curve. So it's not surprising that it encloses more than one region. To have an entirely analogous situation on 19x19 you'd need a huge and carefully constructed group.

We could clarify the originally proposed rule by saying that only a contractible area can be enclosed. This rule wouldn't work at all on a sphere, because a nonintersecting closed curve divides a sphere into two contractible regions, but it works fine on a torus.
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My rule question, though, is what if one player encloses the other's enclosure? For example:
. . . . . . .
. . X X X . .
. X . O . X .
. X O . O X .
. X . O . X .
. . X X X . .
. . . . . . .

If the X stones are played first, then it's illegal for O to play those stones, so no problem. But what if the O stones are played first and the Xs are played afterwards? Are the Os removed from the board? Do they remain on the board but not score? Or do both players score for enclosing the central point?
 
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milomilo122 wrote:

Seems more like Toroidal Rin.
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kocbftn wrote:
the enclosed area is always the smaller of the two areas on either side of the enclosing stone formation. If this is a tie, then the enclosed area is the area which on the non-imagined portion of the board looks to be of the same shape as the enclosing stone formation. So in those examples the e's are enclosed but the dots are not.

What is a "non-imagined portion of the board"?
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Russ Williams
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aquoiboniste wrote:
russ wrote:
Or expand the original stones to "enclose" a 2x2 array of points:
. X X .
X e e X
X e e X
. X X .

Your intent is that the 4 points "e" are enclosed, right?

You have cheated a little bit here, because your group of stones wraps around the whole board and consequently no longer has the topology of a simple closed curve. So it's not surprising that it encloses more than one region. To have an entirely analogous situation on 19x19 you'd need a huge and carefully constructed group.


I'm not "cheating": enclosure does not depend on how big or small a group is, or how big it is relative to the entire board. In a game like this, the normal way I'd expect to see "enclose" formalized is that it's a contiguous region of points touching only one color. But on a wraparound map that is very problematic and breaks down. Even on a normal map that breaks down if you have the notion of surrounding and removing enemy stones, but at least on a normal map, normal geometrical intuition helps grok the intention. Here, on a wraparound board, the intention seems genuinely unclear and undefined to me.


Quote:
We could clarify the originally proposed rule by saying that only a contractible area can be enclosed. This rule wouldn't work at all on a sphere, because a nonintersecting closed curve divides a sphere into two contractible regions, but it works fine on a torus.


Hmm? I'm not sure I grok what you mean, but how would you define "contractible area" in the game rules (i.e. not presupposing that the reader has technical topological knowledge)?

And how would that work in my previous example:
e e X X
e e X X
X X . .
X X . .
Why is the 2x2 area of e "contractible" but the 2x2 area of dots not "contractible"? To me, they seem clearly equivalent and isomorphic to each other. Am I overlooking something?


On a large toroidal board, a small group also "encloses" 2 regions. E.g. on a 9x9 board, this group "encloses" one region of 9 points and another region of 60 points:

. X X X . . . . .
X . . . X . . . .
X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .


I'm beginning to wonder if we're not all on the same page here about how "enclosing" works on a wraparound board.

E.g. do we all agree that the board state I just showed is completely equivalent to:

X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. X X X . . . . .
X . . . X . . . .


and completely equivalent to:

. . . X . . . . X
. . . X . . . . X
X X X . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
X X X . . . . . .
. . . X . . . . X


Right?
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russ wrote:
Hmm? I'm not sure I grok what you mean, but how would you define "contractible area" in the game rules (i.e. not presupposing that the reader has technical topological knowledge)?

And how would that work in my previous example:
e e X X
e e X X
X X . .
X X . .
Why is the 2x2 area of e "contractible" but the 2x2 area of dots not "contractible"? To me, they seem clearly equivalent and isomorphic to each other. Am I overlooking something?

Sorry, I was unclear here. Both regions are contractible and both are (in my opinion, which differs from the game author's) enclosed by that set of Xs. I'm saying that your example is a nonrepresentative example and that a set of stones which encloses the entire board in two regions like this is hard to replicate on a much larger board. With my "contractible" definition of enclosure, you'd have to do it something like this:
. X X X . . . . .
X . . . X . . . .
X . . . X X X X X
X . . . X . . . .
. X X X . . . . .
. . X . . . . . .
. . X . . . . . .
. . X . . . . . .
. . X . . . . . .


Quote:
I'm beginning to wonder if we're not all on the same page here about how "enclosing" works on a wraparound board.

E.g. do we all agree that the board state I just showed is completely equivalent to: [snip examples]

Right?
Yes, they are all equivalent. But the inside is not equivalent to the outside, not even topologically — so in all of your examples, the 9-point region is still clearly the inside and the other region is clearly not. A topological distinction can be based around the observation that the outside contains nonbounding loops, such as:
. X X X . . . . .
X . . . X . . . .
X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
* * * * * * * * *
. . . . . . . . .
. . . . . . . . .

whereas the inside does not. (By nonbounding, I mean that the loop of *s doesn't even divide the torus into two regions: if you start colouring in the region on one side of it, you'll end up colouring in the other side too.)
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aquoiboniste wrote:
Yes, they are all equivalent. But the inside is not equivalent to the outside, not even topologically — so in all of your examples, the 9-point region is still clearly the inside and the other region is clearly not. A topological distinction can be based around the observation that the outside contains nonbounding loops, such as:
. X X X . . . . .
X . . . X . . . .
X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
* * * * * * * * *
. . . . . . . . .
. . . . . . . . .

whereas the inside does not. (By nonbounding, I mean that the loop of *s doesn't even divide the torus into two regions: if you start colouring in the region on one side of it, you'll end up colouring in the other side too.)

Which seems to confirm my original point: that "enclosure" is a tricky non-obvious concept on a wraparound board, so the rules need to be more explicitly clear about what it means. (And the typical player will not immediately grasp such subtleties as whether a region contains nonbounding loops...)
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Sounds like a lot of fun so far.

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kocbftn wrote:

. X . .
. X . .
. X . .
. X . .

Which of the two empty regions is considered enclosed here?


Neither one, that is a straight line, you need a loop to enclose.

But it's a wrap-around board, so that "straight line" does indeed form a loop.

Quote:
On a large toroidal board, a small group also "encloses" 2 regions. E.g. on a 9x9 board, this group "encloses" one region of 9 points and another region of 60 points:

. X X X . . . . .
X . . . X . . . .
X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .


In this case, only the region of 9 points is enclosed.

Why? Both regions are enclosed.

How about this one, on a 11x11 board:

...........
..xxxxxxx..
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
..xxxxxxx..
...........


Now we have a region of 49 spaces and a region of 44. Which region is "enclosed"?
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kocbftn wrote:
Enclosed area must have the same shape as the enclosing stone formation. In this case the enclosed region is the region which appears to be inside the stone formation.

Hmm??? An enclosed area's shape will normally not at all be the same shape as the enclosing stone formation.


X X X
X . X
X X X


The stone formation is a rectangular ring of 8 stones. It has a single empty point inside. That empty point is not at all the same shape as the 8 stones. Yet surely you intend it to be considered "enclosed", right?
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kocbftn wrote:
Enclosed area must have the same shape as the enclosing stone formation. In this case the enclosed region is the region which appears to be inside the stone formation.

'Same shape' seems to me totally unclear, sorry. I suppose, though, that you mean what aquoiboniste wrote earlier about 'contractible' areas:
aquoiboniste wrote:
(...) the inside is not equivalent to the outside, not even topologically — so in all of your examples, the 9-point region is still clearly the inside and the other region is clearly not. A topological distinction can be based around the observation that the outside contains nonbounding loops, such as:
. X X X . . . . .
X . . . X . . . .
X . . . X . . . .
X . . . X . . . .
. X X X . . . . .
. . . . . . . . .
* * * * * * * * *
. . . . . . . . .
. . . . . . . . .

whereas the inside does not. (By nonbounding, I mean that the loop of *s doesn't even divide the torus into two regions: if you start colouring in the region on one side of it, you'll end up colouring in the other side too.)
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luigi87 wrote:
milomilo122 wrote:

Seems more like Toroidal Rin.

Yes, except in the fact that Rin is played with move order 12* (and on a smaller board, though that hardly matters).
 
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I suggest definition of enclosed area:

a. ring area
b. side area
c. corner area

a.

.......
..OO...
.OeeO..
.OeeO..
..OO...
.......
.......


b.

.OeeeO.
.OeeO..
..OO...
.......
.......
.......
.......


c.

.Oeeeee
.Oeeeee
..OOeee
....OOO
.......
.......
.......


Very nice, simple and natural version of Go.
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Malaschitz wrote:
I suggest definition of enclosed area:

a. ring area
b. side area
c. corner area

a.

.......
..OO...
.OeeO..
.OeeO..
..OO...
.......
.......


b.

.OeeeO.
.OeeO..
..OO...
.......
.......
.......
.......


c.

.Oeeeee
.Oeeeee
..OOeee
....OOO
.......
.......
.......


Very nice, simple and natural version of Go.

But the OP is proposing a wraparound (toroidal) map, so there are no sides or corners.
 
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I've always found the interest in toroidal boards surprising, but leaving it for the moment and following Rio's suggestion, I see a territory game in the making (in a way that has been explored before, so I'm not sure anything new will emerge) but I wonder what exactly the relation with Go would be.
 
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I think what John is aiming for might be something which I would call a finite enclosure, i.e. where only a finite amount of wraparounds form a full enclosure. In the example


...........
..xxxxxxx..
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
.x.......x.
..xxxxxxx..
...........


this would only be the inner part of that circle visible at first glance. The outer part is only enclosed for infinitely many copies of the game board next to each other. Similar case for


. X . .
. X . .
. X . .
. X . .


If I am not mistaken, then 8 copies of a board should be enough to see any such finite enclosures, i.e. if the game is played on board A, and B are copies of A each, then all loops which are closed on


B B B
B A B
B B B


should be considered.

Mathematically this can be succinctly stated by defining a point to be enclosed if and only if the line encircling it is homotopy equivalent to that point (on the 2-torus, which is the game board). But I guess this is way too mathematical for any board game rule. What it means in non-mathematician language would be that the string, which the enclosing stones form, could be contracted into any point in the centre of it, but not in the ones around it (on a 2-torus, which is the shape of a donut and also of a square game board with periodic boundary conditions like this one here).
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kocbftn wrote:
In Rio's suggestion, it is much easier to enclose an area using the sides of the board as help than it is to form a loop using only stones to enclose an area. This is why the game is more balanced when played on toroidal board.

I'm not sure of the causality here. Why is the game 'more balanced' after a rule change that does affect both players the same way? For now the presentation of the rules is arguably in want of more balance.
 
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christianF wrote:
kocbftn wrote:
In Rio's suggestion, it is much easier to enclose an area using the sides of the board as help than it is to form a loop using only stones to enclose an area. This is why the game is more balanced when played on toroidal board.

I'm not sure of the causality here. Why is the game 'more balanced' after a rule change that does affect both players the same way? For now the presentation of the rules is arguably in want of more balance.


Anything that can be proven to expand the game tree in a practical way will provide balance as a matter of proportion of one move to the entire tree. Each move in a 10 move game will have more impact than each move in a 100 move game, so on a completely hypothetical level the 100 move game is more balanced if we have no other information about the games.
 
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charrm wrote:
If I am not mistaken, then 8 copies of a board should be enough to see any such finite enclosures, i.e. if the game is played on board A, and B are copies of A each, then all loops which are closed on


B B B
B A B
B B B


should be considered.


Unfortunately this is not enough, you can always (provided the board is large enough) create an area that 'spirals' a sufficient number of times to require an arbitrary number of board copies to aligned to appear enclosed. Here is example on a 13-by-13 board: the area with the question marks is enclosed but does not appear as such with only nine copies of the board placed in a square (indeed, it requires a height of four board to do so).

· # ? # · # ? # · # ? # ·
· # ? # · · # · · # ? # ·
· # ? # · · · · · # ? # ·
· # ? # · · · · · # ? # ·
· # ? # · · · · · # ? # ·
· # ? # · · · · · # ? # ·
· # ? # · · · · · # ? # ·
# ? # · · · · · # ? # · ·
? # · · · · · # ? # · · #
# · · · · · # ? # · · # ?
· · # · · # ? # · · # ? #
· # ? # · # ? # · # ? # ·


Here is the same idea, in a more minimalistic style:

? # # # ? #
? # · # ? #
? # · # ? #
? # · # ? #
# · # ? # ?
# # ? # ? #
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CoreyClark wrote:
christianF wrote:
kocbftn wrote:
In Rio's suggestion, it is much easier to enclose an area using the sides of the board as help than it is to form a loop using only stones to enclose an area. This is why the game is more balanced when played on toroidal board.

I'm not sure of the causality here. Why is the game 'more balanced' after a rule change that does affect both players the same way? For now the presentation of the rules is arguably in want of more balance.


Anything that can be proven to expand the game tree in a practical way will provide balance as a matter of proportion of one move to the entire tree. Each move in a 10 move game will have more impact than each move in a 100 move game, so on a completely hypothetical level the 100 move game is more balanced if we have no other information about the games.

Doesn't that make Tic Tac Toe 'highly unbalanced'?
 
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christianF wrote:

Doesn't that make Tic Tac Toe 'highly unbalanced'?


compared to what
 
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