Normal Go with an infinite board is rather hard to win as a new group can be started just by plonking another stone somewhere (possibly infinitely) far from the existing stones, then making it as big as you like. There isn't much point in fighting when you can get points elsewhere and ko's wandering off forever aren't very exciting to play.
My interest is in ways that the rules can be generalised into something that is playable in finite time. To do this, the rules will have to generate some reason why no further play is allowed.
[yeti_bob] No, you can't play infinitely far away. The board is infinite, but the distance between any two points on the board is finite.
davou What if a rule were applied wherein only 361 stones can be played (or present on the board at one time) otherwise, a finite limit on territory can be established, werein once a certain number of total territory were accumulated by both players the game is called for counting?
An infinite board could be infinite in both dimensions, semi-infinite (one edge), quarter-infinite (two perpendicular edges), strip-infinite (two parallel edges), semi-strip-infinite (three edges), or presumably lots of other semi-bounded areas.
- My first thought is to generalise the super-ko rule, such that the infinite goban is split into 19x19 areas which cannot be repeated anywhere else (but allowing an empty 19x19 board of course). As there are only finitely many legal go positions, eventually a position will arise where there are no legal moves for either player, so the game ends. Count territory in the fashion of Tromp-Taylor rules. The unused infinite area of the board will 'see' both colours and so be dame. Counting will take a while, but there will be a winner.
- After further thought, I think super-ko should be generalised to cover the 19x19 region centred on the last stone played. If a ladder runs far enough, it will get clear of all the other stones and so repeat in a super-ko sense. This makes ladders run into an edge, much like they do in normal go. It also means that if you play a stone a long way from all of the stones played so far, you can't do so again until you play another near it in a unique way. Notice that Black's second stone would have within the 19x19 area mapped by the first Black stone if White's first play is not sufficiently close. I fancy that a 5x5 limited game might even be playable by mortals.
- Another approach is to play with a finite number of moves or an absolute time limit (pass moves costing no time). In the given time, make the most moves you can so that some territory 'sees' only your colour. At first sight, both players will play far away from each other and wall off the biggest area possible with the number of stones in the time. As this will presumably be a draw (komi must surely be zero for an infinite board), perhaps an attack on or invasion inside the other player's territory is a better bet.
Anyone else got some bright ideas? ( I'm not volunteering to play test any of them!)
Davou: my idea was pretty good... infinate space to play, limited stones.... once you run on out stones to play from your reserve, you have to start removing them from solid shapes... thus weakening them.... the weaker shapes of course would be killed off first, and then larger more influential shapes come second.... winner is the one with the last stone on the board.... handycaps can be made by alloting some players more stones.. like?
mAsterdam: Would this work? The game is over as soon as one color's territory is n (say 5) points bigger than the other's.
Handicap would mean the placement of (infinitely many) stones on every m-th intersection in every direction. The smaller m, the bigger the handicap, of course.
DougRidgway: Another approach is to give a different weighting to different locations. So, e.g. area scoring, score for owning (x,y) = exp(-x^2-y^2), and there will be a finite total possible score. Play will stop eventually because the outer reaches are infinitesimally close to dame.
The real question is, do ladders work?
Davou: The would provided the ladder breaking points have strategic value
Jeremy Hankins: Another somewhat different option would be to only score the initial 19x19 area. Stones could be played outside this area in order to make life, or attack another group, but territory there would not contribute to the score. In fact, stones played outside the 19x19 area should probably count against territory. A better name for this might be "edgless go" rather than infinite go.
Or maybe score the 9x9 area, so games can go faster.
If the outer edges are infinitesimally close to 0 but there are infinitely many of them, tedomari will take a very long time.
Tas: At some point in time even the rest of the infinite goban will be worth less than one players lead, and the game will end. Nice idea with a gausian peak of valuable points.
Gronk: At least it is clear that tengen is the optimal first move on such a board. Or maybe that's the 1-1 point?
Tas: Do you mean to number tengen = the mean of the gausian as the 0-0 point, then? I guess so, naming the lower left corner 1-1 is not really possible...
A.T.: I actually have a new model to consider which keeps the boundaries and the four corners of the board in tact. I am also looking for people, especially mathematicians to work and correspond with on the general problem of solving go (recreationally for the time being) and you can email me at firstname.lastname@example.org. In any case, the infinite board basically consists of vertices established by the points in \[0,1\]x\[0,1\] over QxQ. Unfortunately this does not allow the players to actually play adjacent to any stone, making capture impossible. Then the net sum of the game becomes a draw.
In order to avoid this little conundrum we could define the game over a 4 dimensional space, which essentially creates a subset of ZxZ at every single point of the board. For further details you can email me. Even if you aren't great with math and have some cool insights, it's getting very boring working alone, so don't hesitate to email me if you are interested.
Anonymous: To address this right here: "Normal Go with an infinite board is rather hard to win as a new group can be started just by plonking another stone somewhere (possibly infinitely) far from the existing stones, then making it as big as you like. There isn't much point in fighting when you can get points elsewhere and ko's wandering off forever aren't very exciting to play." we could require the next stone to be placed be within a 19 by 19 square (or some other finite size n) of the previous stone placed. That way, people cannot plunk down stones far away from the board in order to avoid losing.
MakiOfSpring: What might work for ladders is to allow what I'd call "ω-moves": Once a ladder starts, the player who is put in atari can declare to "ω-reply", which means "If you put me in atari, I'll play the move to get out of it. The other player can then either confirm that they'd play out the ladder infinitely, or say after how many moves they'd stop. If they'd play it out infinitely, the ladder is extended into infinity. If they'd stop after a finite amount of moves, those stones are placed exactly and the playing giving the atari must then make a move that does not extend the ladder.
An open question is whether this infinite ladder should capture the stones inside or not. If it does, playing a ladder has a similar effect to finite go, but requires a different way of counting territory, as now infinitely many captures have been made on an infinite territory.
Alternatively, we could say that ω-moves never capture, but this makes playing out a ladder into infinity never a good idea for the side putting stones in atari, which would probably change a lot of intuition about good and bad moves.