dale walton
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I have been thinking about how to create a Go without the grid, and this is my best shot so far.

Ok, it wont play the same, as friendly pieces can consume liberties without being connected, and placements may force the removal of both players pieces in some cases, or in a choice of opponent's groups to remove.

I would suggest trying with clear discs with circles stuck to them, or using a drawing application.

I am particularly interested in how Go players feelabout this, but would be happy to hear from anyone who likes to play new games as well.
======================

Go 4 Real
Go without the Grid)

Go 4 Real is intended to be similar to go, but played without angle and distance restrictions.

Equipment:
Go 4 Real is played on an arbitrary region of a flat surface with a sufficient supply of cylindrical "stones" surronded by "connection" rings, in two colors corresponding to "Black" and "White". The board region is agreed upon prior to play (normally a circle) and is fixed for the duration of the game.

Goal:
The goal of the game is to maximize territory. A player's territory is the total area of the board that is closer to any of his pieces than to any of the opponent's pieces.

Units of measure:
The cylinder and its ring determine a minimum stone center to center distance, which is defined as the unit of linear measure. To ensure there are no crossing connections the diameter of the ring is 2.141421 times the diameter of the cylinder (maximum) This is equal to 1.41421 units in diameter.

Parameters: (things to play test, or to ad variety)
Total board area and shape
Ring diameter: Reducing the outer ring diameter could have significant effect on play.

Play
One player places any number of black stones on the board. Then the opponent either plays a white stone or claims black and passes.
Play continues with White and then Black placing a stone in alternation, with passing allowed. In certain situations "dead" stones are removed following a stone placement.
The game ends with 3 consecutive passes. The third pass is normally negotiated with an agreement of which stones on the board are dead (or mortal) and are to be removed before scoring.

Scoring
Each player's score is the measure of their total territory after the agreed stones are removed.
Black's territory consists of the regions of the board that are closer to a black stone than a white stone, and White's territory is the area closer to a white stone.

Stone placement
The cylindar part of the stones must be fully inside the board area, and may not overlap the any partof any other piece. The ring part of the pieces may overlap the rings of other pieces, and freindly pieces are considered as directly connected if their rings orverlap by any amount (but not if they mearly touch)
Stones may not be placed in a Koh (i.e. a repeated board position, as described below)

Groups of stones
A group of connected stones are two directly connected stones and all the stones (and only the stones) that are directly connected to them or to another stone already in the group.
A connected group of stones is "alive" if and only if it can be added to without first removing other pieces. (Even if adding to it would kill it, or if adding to it is not allowed on the current turn)
Discrete non-overlapping regions where a stone may be added to a group are called the liberties of that group.

Removing dead stone and/or dead groups of stones
A placement may create one or more dead groups of stones.
The player who made the placement chooses the dead group removal sequence.
Dead groups of stones are removed, a group at a time, until no dead groups remain.
Removal of a dead group may restore life to a previously dead group.
(Unlike in Go, there can be cases where a placement forces the removal of both enemy and friendly groups, and/or also cases where a dead enemy group may survive after the removal of a companion enemy group.)

Koh:
A situation in which a stone, when played, would recreate the same functional position as previously existed.
Two placements are functionally equivalent if the number of liberties at every stone of every group are the same number and general arrangement.

Example of a game position in which black has just placed a stone to kill a white group of two stones. The example also shows the areas that currently belong to Black and White once the 2 stones are removed.


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Russ Williams
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There are obvious practical problems with this kind of gridless/pointless continuous space game, i.e. that the exact position of each piece matters, so the slightest table bump can mess up the position, and some positions will be difficult or practically impossible to measure sufficiently precisely to judge whether a given placement is legal or illegal. These problems make me not a fan of continuous space games, at least not for "serious" games (as opposed to quick light filler games like Light Speed); they seem interesting more as a theoretical design exercise than as a real game I'd play.


Of possible interest: the game Calculus, which uses Go stones and continuous space (but is a connection game rather than territory game).
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Several people have tried similar ideas for Go on continuous boards. See for example http://www.di.fc.ul.pt/~jpn/gv/boards.htm and https://senseis.xmp.net/?GoOnABoardWithoutLines. It might be instructive to compare your idea with theirs.

I always found the capturing rules for these games a bit unsatisfactory, and your example position shows why. Looking at the white stone that's below the captured group, it seems to me that if that stone were not on the board at all, then the white group would not be captured.

It seems counterintuitive that a white stone can take away a white group's liberty in this way (ETA: without being part of the group). If you can come up with a capturing rule for continuous Go that doesn't have this weird effect, I would like to hear about it.
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aquoiboniste wrote:
It seems counterintuitive that a white stone can take away a white group's liberty in this way.
But in real Go a white stone can take away a white group's liberty, so it seems consistent.
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Eric Brosius wrote:
aquoiboniste wrote:
It seems counterintuitive that a white stone can take away a white group's liberty in this way.
But in real Go a white stone can take away a white group's liberty, so it seems consistent.
Good point, I didn't speak precisely. In real Go a white stone can only take away a white group's liberty by being connected to the group and sharing its fate. In this game as proposed, a white stone which is free/connected to the outside can take away the liberty of a white group it's not connected to.
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dale walton
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Thanks for bringing this issue up.

The game needs to be bump-proofed. -And close fits need a method of adjudication fair to both players.

I was hoping overlapping ring/discs would help on these two points somewhat, but one cannot guarantee knife-edge conditions won't occur.
Possibly a clay or Velcro surface could also help?

On obvious work around is to implement it on computer by going to a fine lattice (zoomed pixels), rather than true continuous.positioning. On a computer the pieces won't move, it could be designed so that no pixel-pixel distance falls at the exact edge of the cylinder or ring.

While the exact position indeed matters, mostly one would not be able to see why until much later, which means that there is some risk and uncertainty that much be considered in trying to over-optimize moves. Granted that keeps it from being purely deterministic abstract strategy. It would be interesting to know how important playing on the brink is to winning/losing, and in what percent of moves it is critical to play in that manner...

It sounds like you have experience in playing such games and could comment on the last point.


 
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I'll look into those.

I have noted the capture implications in my intro; but, as a rather poor player of go, I don't know if this would destroy it as a game or not - I am not sure of the ramifications. It would, as noted make it significantly different.
 
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It looks similar to "continuous go" except for connection definition.

Pekka would consider the lower 4 white stones connected, but this means long connections can get broken by the opponent. If the "shortest of all crossing potential connections is the connection" is limited to the areas in which a new piece cannot be placed, it might work better...

However, what is the implication of the rule as I have it on game play?
 
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dale walton
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Looking at the links provided, I'll take this discussion there...

Thanks
=============
However, I'm coming back here, because those discussions, although quite relevant, are old/dead
 
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aquoiboniste wrote:
Eric Brosius wrote:
aquoiboniste wrote:
It seems counterintuitive that a white stone can take away a white group's liberty in this way.
But in real Go a white stone can take away a white group's liberty, so it seems consistent.
Good point, I didn't speak precisely. In real Go a white stone can only take away a white group's liberty by being connected to the group and sharing its fate. In this game as proposed, a white stone which is free/connected to the outside can take away the liberty of a white group it's not connected to.

I thought about this and came up with 3 ways to solve it, but a bigger issue is that in go one doesn't need to connect all the pieces surrounding a group, which gives it it's tension. So I present here a solution for both:

1) Define a connection as a center-to-center separation of 2 friendly pieces by less than the square root of 2 diameters.

2) define a boundary as the separation of 2 enemy pieces by less than 2 diameters, except where crossed by a connection.

3) Define liberty for a group of pieces fully surrounded by an enemy boundary as a location within the boundary where the center of a piece can be placed.

Group(s) are dead if they are surrounded with no liberties.
Probably all dead enemy groups should be removed after a placement.

I believe that these rules, when played on a grid, are the same as go.
Note that in both games, boundaries can cross, but connections can't.

Does this solve your main issue?
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I think your definitions lead to a wrong, unintended definition of Koh. Any placement of a stone, not connected to any previously existing group, creates a Koh, as you've defined it. I think what you meant is this:

"Two placements are functionally equivalent if the number of liberties at every stone of every group are the same number and general arrangement."

Since you defined a group as being two or more stones, single stones positions and liberties aren't part of Koh-evaluation.
 
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dale walton wrote:
Does this solve your main issue?
I like the idea. I'm not sure about the numerical calculations (actually I didn't understand them in the original post, where does the number 2.141421 come from and is it a typo for 2.41421?)

I would have said that if the minimal possible separation between stones is 1 unit, then the greatest separation that forms a boundary should be sqrt(3) units. That forces every white stone to be entirely on one side or entirely on the other side of (the line joining the midpoints of stones in) a black boundary. And that's what prevents a white stone "outside" from interfering with white liberties "inside".

But it's quite likely that I'm not correctly allowing for the "ring" and "cylinder" parts of the piece.
 
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For ko, you could just have a rule which explicitly defines the usual ko-situation (but potentially leaves other infinite situations possible).

For example, say a "ko-move" is a move which captures the stone which has just been played by the opponent and captures no other stones. Then the ko-rule could be, if your opponent has just played a ko-move then you may not reply immediately with another ko-move.
 
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aquoiboniste wrote:
dale walton wrote:
Does this solve your main issue?
I like the idea. I'm not sure about the numerical calculations (actually I didn't understand them in the original post, where does the number 2.141421 come from and is it a typo for 2.41421?)

I would have said that if the minimal possible separation between stones is 1 unit, then the greatest separation that forms a boundary should be sqrt(3) units. That forces every white stone to be entirely on one side or entirely on the other side of (the line joining the midpoints of stones in) a black boundary. And that's what prevents a white stone "outside" from interfering with white liberties "inside".

But it's quite likely that I'm not correctly allowing for the "ring" and "cylinder" parts of the piece.
I'm pretty sure he means sqrt(2), which is 1.4142... This prevents "cross connections" - stones placed like this:
coffeesugar
sugarcoffee
If they're exactly at the corners of a square, and as close together as possible, the two white stones will not be connected, and neither will the two black stones. If they're at the corners of a rhombus, then either the two white, or two black stones could be connected, but never both pairs.
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Thanks, the picture helped. Dale's numbers are right, though, now that I understand exactly what he's saying:

The radius of the (inner) cylinder is 1 - 1/√2.
The thickness of the (outer) ring is √2 - 1, and thus the total radius of the piece is (1 - 1/√2) + (√2 - 1) = 1/√2.

This comes from the unique solution to the pair of linear equations:

2 × r_cylinder + r_ring = 1
2 × r_cylinder + 2 × r_ring = √2

And it means that, just as Dale said:

The diameter of the whole piece is 2 × (1/√2) = √2.
The ratio of piece diameter to cylinder diameter is √2 / (2 - √2) = 1 + √2.
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For physical implementation, I could ad a connection marker to be used to record intent for hard to see connections.

Here are some pictures. Thanks for the Koh suggestions, given the new solution, I probably need a careful rethink of the needed wording...

Text says: 1) Not a boundary, 2) Marked boundary, 3) Obvious boundary, 4) Connection and boundary.

Text Says: When an overlap is possible, Black is theatening
to cut the white boundary with a connection.


Text says:
1) These crossing boundaries cannot be
broken (converted to connections)
without a piece capture, first.
2) If white captures a black, either player
can connect here, however, if black captures a
white, white has no threat to reconnect across
the boundary.

Note on above: White's pieces are further apart than Black's - More specifically, neither of the centers of the white pieces fall within a circle whose diameter is the segment between the centers of the black pieces (this is a requirement to be able to form a connection), In addition, if one was placed as close as possible in that region later, the other is still out of reach in this case.


Text says:
1st illustration) These boundaries cannot be cut without a capture first.
2nd illustration) Either player can cut the other's boundaries.

(that is by playing to the open area in the center...)
(updated connection markers for better clarity - now it measures the distance from cylinder to ring.)
 
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The boundary limit of 2 diameters is saying that beyond that it is not a boundary because another piece can be placed between them.

The connection limit is because below that the other player's pieces cannot make a shorter crossing.

There is a detail rule that needs to say that if the distance is exactly square root of 2 it is not a connection, or perhaps say that connections are first-come-first served and players could make their own special ad-hoc marker for it. I do not think this would arise frequently except maybe for very tactical play.

The advantage of this system over continuous go is comparatively large regions of roughly equivalent results, so less need for precision in play and adjudication. (Cannot eliminate entirely)

These boundary definition solves the fratricidal capture effect of the outside piece white piece by ensuring that either there is already a boundary there or a white connection, or else there is still a place to place a piece. Thus the side-effect is isolated.

 
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dale walton wrote:


Text says:
...
Note on above: ... - More specifically, neither of the centers of the white pieces fall within a circle whose diameter is the segment between the centers of the black pieces (this is a requirement to be able to form a connection),...

This was wrong. Here is a template aid for placing pieces to attack boundaries:



If a piece is placed totally inside when the template is tangent to the boundary forming pieces, and there is space for a piece touching those pieces on the other side, then the piece is threatening the boundary.

Here black would succeed but is currently blocking himself, while white fails in any case.

This should make play easier.
 
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One more point on play:

In the below position the white piece on the left is loosely connected to the group on the right, meaning that despite no direct connection the pieces all together behave as a connected group because black cannot play to sever them. - They live and die together.



=====================

I also had a curious brainstorm on scoring to capture the idea of efficiency in a new way - how would this affect the game?
I fear any departure from go might make the game less of interest but,...

what if the winner is the most efficient player by the measure of (area /(stones played - stones captured)) -- would it make any difference at all to the nature of play
1) on a regular board?
2) in this game in which ultimate stone density depends somewhat on placement?
 
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aquoiboniste wrote:

For example, say a "ko-move" is a move which captures the stone which has just been played by the opponent and captures no other stones. Then the ko-rule could be, if your opponent has just played a ko-move then you may not reply immediately with another ko-move.

An exception to this simple case is needed (for a situation which is not possible in standard go) one must add:

"unless your placement also makes a new connection."

I would say it like this:

A "ko-move" is a move which captures the stone which has just been placed by the opponent, captures no other stones, and leaves your own stone exposed to recapture.

If your opponent has just played a ko-move then you may not reply immediately with another ko-move.
===================

I am not sure how common this exception would be: Normally, if available, it would be a choice between playing the ko or securing both players pieces. Only in special circumstances might it be possible to secure your piece and take the opponent's, or secure your piece and threaten to take the opponents on another turn.

There will certainly still be cases where the ko will occur without giving this choice.

I do not know if the implication is ko might be less important in this game. Boundary breaking, on the other hand, is an emergent tactical consideration.
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Here is a weirder idea to reduce the occurrence of ko situations:

Removing dead stones: After a placement, if both players have dead stones, then remove the stones of the player with the fewest dead stones. However, if the number of stones are equal replace all the dead stones with gray stones. Grey stones become permanent neutral territory and are not liberties for any group.

How would such a rule affect the game of Go?
 
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dale walton wrote:
Here is a weirder idea to reduce the occurrence of ko situations:

Removing dead stones: After a placement, if both players have dead stones, then remove the stones of the player with the fewest dead stones. However, if the number of stones are equal replace all the dead stones with gray stones. Grey stones become permanent neutral territory and are not liberties for any group.

How would such a rule affect the game of Go?


I meant remove the stones of the player with more dead stones, of course...
I had been thinking about counting the lengths of the boundary, but thought better of it, but forgot to clean up the edit.
 
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dale walton wrote:
Here is a weirder idea to reduce the occurrence of ko situations:

... Grey stones become permanent neutral territory and are not liberties for any group.

How would such a rule affect the game of Go?

Perhaps better if the grey stones are not permanent, but remain until (and are removed when) a later capture gives them a liberty.

I am hoping this is of interest to a few of you on the list. I apologize for so many posts, if not.
 
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aquoiboniste wrote:
Eric Brosius wrote:
aquoiboniste wrote:
It seems counterintuitive that a white stone can take away a white group's liberty in this way.
But in real Go a white stone can take away a white group's liberty, so it seems consistent.
Good point, I didn't speak precisely. In real Go a white stone can only take away a white group's liberty by being connected to the group and sharing its fate. In this game as proposed, a white stone which is free/connected to the outside can take away the liberty of a white group it's not connected to.

I have been looking for other solutions that might allow getting a feeling for the game on a regular board.

I found setting specific sets of grid distances for connection and weak connection (pinches) on a triangular grid comes close to modelling the behavior of the weak connections (i.e. that connections can often be formed across them, to break them)

However the square grid, in general, prevents this until the numbers of different connection patterns becomes quite unwieldy.

This led to a somewhat different approach, which is possibly similar to "Continuous Go" referred to on the other links you sent.

This new game (Proximity Go) is fairly simply defined, so I include it here....
(It avoids simple repeat moves by leaving equal pairs of dead stones on the board temporarily until the next capture. - How much does that affect the game?)

Try with stones 2 to 3 times the grid size...
===============

Goal:
The goal of the game is to maximize territory. A player's territory is the total number of grid triangles that are closer to the center of any of his pieces than to the center of the opponent's pieces.

Parameters
Players agree on the grid size, stone size, and which relative grid points are too close for stones to be placed on both points.

Play
Players agree on the placement of handicap stones, then Black begins by placing a black stone on any open grid intersection that is not too close to another stone.
Turns then alternate between White and Black, in the same fashion.
Under conditions described in the section “Capture” stones are removed at the end of each turn.
Passing is allowed. A consecutive pass must be accompanied by an offer to end the game that designates which insecure pieces to remove before scoring.
The other player either accepts or moves.
Other than by passing, repeating a board position is a voidable move, and knowingly repeating a board position is a draw.

Scoring
Remove the agreed stones.
Count the number of squares whose centers are closest to your stones (ignore those equidistant to your opponent's stones) and compare the tallies. The larger score wins.

Capture:
Capture is based on boundaries and confinement.

Boundaries
Your stones can be bounded by straight lines between the centers of your opponent’s stones.
A line between the centers of your opponent’s stones is part of a boundary unless
• You have a piece exactly on the line, or
• There is a line between a pair of your own stones that crosses it and is shorter.
If the lines crossing are exactly the same length, both boundaries are effective there.

A normal line from the edge of the board to an opponent’s stone is part of a boundary if you have no line crossing it that is shorter than twice its length.
(Imagine a boundary between the opponent’s piece and its reflection beyond the board’s edge.)

Boundaries divide the board into multiple disjoint regions.

Confinement
A stone is confined if no other stone can be placed on the same side of an opponent’s boundary, as it exists.

Removal of confined stones:
After every placement action the number of White and Black confined stones are counted, and the confined stones with the higher tally are removed. If the tally is equal all the stones remain.

The confined stone tally can change after 4 kinds of stone placements: Those that…

1. Subdivide a region, leaving no room for the opponent’s further placement within.
2. Fill in the last available placement points for an opponent’s stones within region that you bound.
3. Fill in the last available placement points for your own stones within a region that your opponent bounds (suicide)
4. Break a boundary around your previously confined stones (rescue), causing the opponent’s remaining confined stones to be removed.

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