I don't see how a 1-stone group could possibly be immortal. Am I missing something here?

In general, a borderless goban that locally still behaves "like usual" (i. e. like the center area of a finite, not too small, rectangular board) is toroidal. The infinite board described in the problem is a special case of this, but "a go board without edges" isn't necessarily infinite.

Also, I don't believe the person is trolling. The exact length of such a shortest sequence might be quite challenging to determine, so asking for bounds seems reasonable.

Here are my thoughts on the matter. Somebody more enlightened than I can refute these if they would like.

If this topic is not relevant to this site and no work will be done toward establishing this page, please indicate this, so I can move these thoughts to Pillbox The Player.

copied to PillboxThePlayer/BoundlessBoardThoughts -- pillbox [2008-03-27]

Observations

1. EDIT: Concept of Territory is not present in this scenario. This cannot really be called Go. This is more akin to Atari Go.
   1. The concept of Territory hinges strongly upon establishing a pass alive group. We can see this challenge as an academic study of fights between two or more floating groups (whatever value that has)
   2. This simplifies Strategy. In fact, in the typical sense of Strategy as it is applied Go, it does not apply.
   3. Some components of Strategy remain, however, and could lead to a winning strategy.
      • Strategic concepts that (probably) don't work
        • Sense Of Direction: territory (perhaps framework is more descriptive) is not a concern
        • Positional Judgement: same as above
        • Tenuki:
          • Finite tenuki: A tenuki played within a finite distance of an existing group. This is not a tenuki in the general sense, except that if a player who is not forced to respond to a position immediately can play a tenuki that establishes a winning sequence, provided that it is played so there is a forced sequence that prevents the other player from winning. In this context, a tenuki becomes a Vital Point play, but for sequences with deep reading. I haven't stumbled upon a different terminology for this, but it might be seen as a sort of Ear Reddening Move. This probably belongs in Strategic concepts that work
          • Infinite tenuki: A tenuki played at a point infinitely distant from an existing group. If a winning strategy exists for either side, Black infinite-tenuki prolongs the game indefinitely (which is a victory for White). The new group and any existing groups can never be connected. If White has no play within a finite distance that will establish indefinite play without a Black victory, White has lost. White should respond to a Black infinite-tenuki as long as White has a strategy to prevent Black for meeting victory conditions; else... why did Black infinite-tenuki?
      • Strategic concepts that work
        • Influence: I think this concept is the core fundamental to this challenge.
        • Pivotal Stones: I believe these are important as well. The drawback is that the fight that ensues in response will generate more weak groups; groups that, according to the thoughts expressed here, will continue to be weak, but never dead.
        • Miai: Applies as in usual play.
        • Aji: Most likely favors Black. Good aji for White could be difficult, as throwing away any stones can lead to successful victory conditions for Black. This changes the application for honte.
        • Efficiency: Always at work.
        • Double Purpose Plays: Seems to be a component of optimal play.
      • Strategic concepts that work in different ways
        • Junk Stones: There is probably a point at which junk stones are just that, junk stones. For White to be successful, she must be careful of this concept, as junk stones can easily be turned into eyes for the opponent. It is important for White to realize when a group can no longer be saved, and at that point play to force a false eye. This is an important factor in actual play, but the burden is on White here to be mindful of junk stones, as they cannot be played as they might in regular play.
        • Thickness: Black should play in a way that flows near or toward his thickness. White should play in a way that prevents this. This is common in regular play, but common ideas about thickness and playing near thickness probably don't apply, since there is no strong concept of territory.
          • While researching this line of thinking, I found Thick Territory very interesting. While territory (as it applies to regular play) does not exist in this context, a thick group that has strong influence on one side while impeding the influence of the opposing groups on the other would be a powerful weapon in this scenario. Thickness Attenuation is a decent response in normal play, and should be applicable here as well.
        • Weak Group: Weak groups, by definition, will occur and often. Local fights will create weak groups.
2. Black's first move cannot ever be considered a poor move on an infinite board. anywhere is always perfect play. (see Possible Refutations #3)
3. White must eventually make contact. There is no winning strategy for White if contact is not made, assuming optimal play for Black.
4. If White does not ever make contact, we can assume that the answer to the shortest scenario that satisfies the question would be however many moves it takes to make a group with 2 eyes without the use of the corner or sides, which I believe is ten.
   • see SmallestGroupWithTwoEyes
5. Until perfect play is possible, I'm not sure there is a definite answer.
6. It is possible, however, there exists a sequence of optimal plays that denies Black a pass alive group. The question does not require that White never obtains a pass alive group.

just my thoughts...

-- pillbox

EDIT: (multiple edits over time)

Additional thoughts

1. "Winning strategy" in my above thoughts (observation #3) for White refers to a strategy that prevents Black from attaining a pass alive group. That is, White wins in the case that play continues forever.
2. After reviewing my thoughts, white does not have to play a contact play to disrupt eyespace for Black (contrast to observation #3). White can play moves, specifically, that prevent eyes.
   • see ShapePoint
   1. Is there a definite sequence where this is possible?
   2. Black can opt to make first contact in this scenario, assuming there is a sequence that refutes this.

Assumptions

1. Most tactics do not work on infinite boards.
2. Black must rely on forcing moves to create eyes. (please refute if possible)
   • as long as White is able to take VitalPoints
3. Playing Atari, in optimal play on an infinite board, should never work.
   • That is, playing Atari should never be able to capture a group, unless that group has no effect on the outcome of the game, e.g. Atari-Capture to gain no eyes, a False Eye
     • strong and potentially incorrect assumption: If Black where able to setup a Tesuji to create an eye by atari, I will assume White is not playing optimally.
4. Playing DoubleAtari may be an exception to Assumption #3.
5. As mentioned before, ladders favor white in most (all?) scenarios.
   • Spiral Ladder?
6. A net should not work, as White should take the VitalPoint before Black
   • refutation?
7. Black should need to capture a White group to create an eye.
   • How is this done if Atari does not work and White plays optimally to not allow DoubleAtari?
8. White can allow capture of one of her groups as long as the resulting eye is false.

At this point, I am going to posit that the criteria for Black to win can never be met. White should always be able to take Vital Points. White should never allow capture or potential for DoubleAtari unless the resulting eye is false.

Possible refutations to my own ideas

1. A sequence for Black or White that confines play to within a certain distance, thus creating a perimeter group of either Black or White in which standard tactics would apply. Note that if this perimeter exists, it would necessarily have to be force-able without refutation.
   1. If a perimeter is established (the opposing group/groups inside the perimeter cannot escape), and the opposing group/groups inside the perimeter cannot become pass alive, this should be successful.
   2. If there exists a sequence for Black (specifically first-to-play) to accomplish this, the criteria can be met
   3. If there exists a sequence for White (specifically, second-to-play) to accomplish this, Black may pass if necessary to take White's strategy.
      • If White passes and both Black and White continue to pass, Black will have failed to meet the conditions required for success.
      • If White's strategy is the only viable solution: If Black's winning strategy is to pass to negate White's winning strategy, Black will always pass. White then needs to pass in order to utilize the winning strategy. This is a failure for Black.
        • This introduces a winning strategy for White that encompasses both branches of play. That is, if White has a winning strategy for Black first-to-play, and passes are not restricted, the winning strategy also indicates that White can pass after the first and each successive pass.
        • In order for Black to succeed, Black must eventually play a move on the board first.