An Infinite Series of Go Games? (Cross-posted from reddit, with minor edits.)

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Ran across a question in r/abstracts: “Do any Go variants exist where the gameboard starts out at a "toy size" and, based on certain conditions, can expand?

If not, how would you design a Go game with this property? What conditions would have to be met for the board to expand a step? (i.e. from 5x5 to 6x6, and so one, potentially to infinity sans a definitive endgame condition being met.)” - Duke Zhou .

This got me thinking, since I’d been looking at the Go variants GOmove and Double GOmove, and I’d been told that GOmove collapsed on 13x13 and smaller Go boards, but i saw did fine on a 19x19, although it seemed a little tight. So when I went to Double GOmove, the game went on a 23x23 board – I guessed the 19x19 or 21x21 board might be the ‘best’ size for GOmove, so jumped to the next odd size up. The game works well (not that I play it well) but quickly starts to use the entire extent of the board.

A thought experiment: Consider Hextuple GOmove. Players drop 6 stones/turn and may move up to 6 stones 1 intersection each. Is this game stable or unstable? If stable, is it recognizably Go or has it morphed into something different? I’m starting with a 31x31 board to give adequate room to play several turns anyway.

First, dropping 6 stones – they can form 2 tiny eyes in a corner; make 1 line 6 stones long needing 14 stones to surround it; surround and eliminate a 2 stone line in the interior of the board or a 4 stone line on the edge… With 6 drops and 6 moves, assuming 6 perfectly placed stones already on board, you can surround and eliminate a 5 stone line in the interior of the board or a 10 stone line along the edge.

So far, I haven’t seen anything that actually breaks Go. Tactics and strategy may change, but it’s still a recognizable form of Go. If the series works for Hextuple GOmove, then it clearly works for Triple, Quadruple, and Quintuple GOmove. And if this is true, then what breaks the series of games at larger sizes? So is there a break?

Let’s go back to the 2 tiny eyes in the corner as black’s first turn. White can completely surround the outside of that formation with the first 6 dropped white stones but can’t do it without leaving 2 isolated and surroundable lines of stones, a 2 and a 4 stone line. If white tries to pen black into the corner, then use a turn to drop enough stones in the corner to take it, you get a race where each side can break into a large area and each player’s twisty line, having 2 sides, isn’t really susceptible to being immediately surrounded. You don’t need invulnerable formations, just very very difficult to capture formations.

So what are difficult to capture formations? a few eyes connected by lines of stones make for a sprawling formation that is probably safe until maybe the end of the game. A 6 stone drop consisting of 2 'v' shaped 3 stone lines on opposite corners of a 3x3 board section cannot be captured or surrounded by the opponent's drop, although the 6 moves may allow captures. And with 4 more stones, you have 2 eyes, in a zig-zag pattern. This can be continued to wall off an area of the board...

I don't see any particular reasons now to think this approach won't work at higher levels, with more and more stones dropping and moving. If this does break down, where and how?