Japanese Rules on tiny boards.

General Go Discussion
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I was curious about how the Japanese Rules apply to tiny boards, of sizes 2x2 and under.

While technically they don’t apply, since Article 3 stipulates a 19x19 board,
the commentary on that article at [1] explains that Article 3 is formulated for professional players and that “The players may of course agree to use other boards, such as a 9 x 9 beginners’ board, a 13 x 13 board, or (in the future) a 21 x 21 board.”

Let’s see what games and what exceptional outcomes are possible on tiny boards like 1x1, 2x1, and 2x2 under these rules.

A 1x1 board forces both sides to pass, stopping the game and yielding a jigo.

On a 2x1 board, an initial black move creates a basic ko, which can not be immediately retaken. So White must pass, and Black must pass too, stopping the game. White can then claim that the black stone is dead. Indeed, Black can not prevent White from capturing it, and while that allows Black to play a new stone, that one can also not avoid capture.
So White wins by 1 prisoner. (Black’s initial move was a mistake and the optimal game would have both players pass).
What if either player requests a resumption? Could the ko then be retaken, followed by another game stop and another resumption, ad infinitum? The rules seem not to prevent this.
The players could also agree to apply Article 12 giving a No result due to whole-board repetition. The rules do seem to prevent both players from losing, since at any stopping point, there is no effective move that yields a better result for the moving player.

On 2x2, many possibilities arise. (In fact, 386356909593 games are possible obeying PSK [2]). A simple and optimal game proceeds as Black A1, White B2, pass, pass, leaving two one-eyed dead groups, and hence jigo. Besides the normal outcomes of jigo, white win, and black win, we can have the exceptional outcomes of no result, infinite play, and both players lose (since after Black A1, pass, pass, White may notice the effective move B2).
Only the possibility of infinite play in the confirmation phase [3] seems to be ruled out.

Comments and feedback welcome.

[1] Article 1. The game of go

[2] Solving 2x2 go

[3] Odd Cases 🤔 in the Japanese Rules - #12 by yebellz

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Article 6 is worded as

A shape in which the players can alternately capture and recapture one opposing stone is called a “ko.” A player whose stone has been captured in a ko cannot recapture in that ko on the next move.

The ko prohibition is not created by the shape alone, but also requires an immediately preceding capture. Thus, on a 2x1 board, if Black plays an initial stone, White can then capture it, but then Black must pass.

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For the infinite play version, if white decides to play like B1, can’t black pass instead and only capture snapback positions.

If black answers every move the position cycles (or a rotated version) and the net captures are 0. Black captures three stones and then white captures three in a snapback.

If black passes when white plays B1, the scoring is the same, all stones are dead, but left on the board.

If white keeps playing this time white only captures one stone and black three when the position cycles. Do that a couple more times and you can even agree that all your stones are dead, the opponent’s are alive with territory and you win because of the captures.

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Oh, right you are. So then White must pass as well, and now Black can claim the white stone to be dead, resulting in 1 prisoner each, and jigo. Playing and passing are equally good in the initial position.

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If both players enjoy playing an infinite game, they can do so without a net change in prisoners by always capturing 2 stones. What you showed is that If either player wants to avoid an infinite game, they can do so without losing points.

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Oh I see what you mean, you are just exploring it in the limit of whether infinite play becomes possible in general, not with any restriction on one or both players playing optimally etc.