This is based on two posts lazy made.

1.  lguser, the swap can be generalized in a way that I’ve discovered several years ago. First player chooses a number N0 and places N0 stones of any color, second player chooses if he wants to play black or white or if he wants to choose another number N1 such that N1<N0 and to place N1 stones and then first player has to choose and the process repeats with N2, N3, etc etc.

This should be the optimal swap rule, making the full use of the possibilities of the whole board.

2. David, It’s mother of all kludge because, depending on what you want, it can gives birth to different babies! You say chess960 has 480 distinct starts? In my game, there are ~2^{128 distinct starts for 9x9 and ~2}191 for 11x11! And the branching factor remains almost as high for the next moves unless both players agree on a starting position and they start playing conventional hex from there. So you’ve plenty of choices, indeed, you’ve ALL the choices possible while ensuring termination! :)

\\\______

So, nothing is to prevent the first player from choosing N0=1 which is the pie rule as we play it. I suppose a white stone could be played under your rules, producing a position with white to move, but that would be silly.

In general, the more stones are on the board, the simpler the position is. I cannot conceive of any position created by the first player in which it would be desirable, from the second player's point of view, to add more stones to the board and hand the swap choice over to the opponent, instead of making the swap choice their self.

The numbers you provide are upper limits, much larger than the the truly interesting initial positions which have not been considered when analyzing more complicated positions with fewer stones on the board. Keep in mind the branching factor drops significantly faster with each move in Hex than it usually does in chess.

Memorizing an opening repertoire for chess 960 seems to me very difficult but not impossible, and I believe the same is true for Hex, regardless of what sort of kludge we munge onto its face. Of course, a larger grid is likely a better way to go., up to a point.

I wonder what would be the point that would limit how far up in board size that it is worth it to go. I'd say that with the right client, even board sizes over 50x50 are feasible for online play. Then you could have a format where the board size of the game is randomized before the game, drawing inspiration from chess960.

Heh not on MY smart phone. I have enough trouble with 30x30 Twixt. And I think it is likely that the variety available in 50x50 Hex would exceed the variety of all the smaller grids combined. Whatever that means.

David, the second player doesn't have the option to add stones, he draws an entirely different position, with at least one stone less.

For a sufficiently large board, the first player will choose N=1 and it'll be equivalent to usual swap rule. For smaller boards, first player can choose N=2 or N=3 or whatever is necessary.  If the other player thinks a too larger number has been chosen, then he can choose a smaller numbers and we start over again. So basically we find the optimal N with this procedure.

And like I've said in another post, I don't think opening books are inherently bad. They add a different element of strategy.

> In general, the more stones are on the board, the simpler the position is. I cannot conceive of any position created by the first player in which it would be desirable, from the second player’s point of view, to add more stones to the board and hand the swap choice over to the opponent, instead of making the swap choice their self.

The position may be objectively simpler but it'll be unfamiliar to you and familiar to your opponent. This is the point. You may want to choose to start over instead of choosing which side to play. In this case, we can say that the N picked was too large and the player drawing the position had a too large advantage (also taking into account the time control).

So you choose a SMALLER N and the game starts over with swapped roles. Eventually, if we keep going, we'll reach N=1. But we can easily stop at N=2 or N=3. Hence, my swap rule solve all your concerns. I think it's really the optimal way to generalize swaps. It's basically the FULL generalization of it, with dynamic computation of N. ;)

Playing a bunch of stones at once changes the game substantially.  Different strategic skills are required to play multiple stones at once.   It's a matter of personal taste how good the game is with the changed swap rules.  Me personally: I like the elegant simplicity of Hex as is.  And it's not like 19x19 has been analyzed to death already.

Tom, I absolutely agree, I think double moves like connect6 are a very bad idea. In my swap rule you play multiple stones only to initialize the board. And if your opponent doesn't like your initialization, he can choose another. So yes there are multiple stones but only once and with opponent approval.

If I start with 6 stones, you can counter with 5, I can counter with 4, … .  Even though only one position continues through to the rest of the game, there are potentially several instances of choosing multiple stone plays.  It feels like bolting a different game onto the beginning of Hex.

realistically it would wind down to n = 1 and just be a normal swap rule, no one is going to want to risk falling into a trap from an opening that their opponent has deeply studied

Thank you for clarifying, lazy. BTW I am not against opening preparation either. But I don't necessarily agree that a future-proof ruleset is preferable to a simpler one, especially for Hex. I have enough difficulty explaining to newbies the difference between swapping sides and swapping the first stone. I mentioned 3MEQ as an alternative way of avoiding AI published analysis which would benefit players with better memory than skill. But such analysis is not here just yet, and may not be so much a problem when it does.

I left off a word at the end. I should have said, … when it does arrive.

My and lazyplayer we've discussed this opening rule outside of the forum. We basicaly agreed that this is not elegant enough in practice. Despite that, I admit that this produces the most fair game possible and it is coherent with current Hex gameplay.

I cannot think of any rule which would be elegant enough for the beautiful game of Hex. My best idea is to keep the swap button active for 2 moves instead of just 1 so that you can play a weaker stone and swap next, but even that I do not like so much.

Luckily so far it's enough to adjust the board size. Recently the bots have proven some opening moves unplayable (we already knew that but now having the proof there is a different psychological situation)

…so we restricted ourselves to basically just 2 choices: b3 and a13. Moreover, a13 between the strongest players is very imbalanced and, studying it for around last 3 years, I've not found too many interesting positions…

Now, with b3 we have a decent rivalry and the all time battle between long and short diagonal, but how long before we solve this?

https://littlegolem.net/jsp/game/game.jsp?gid=2124659&nmove=7

https://littlegolem.net/jsp/game/game.jsp?gid=2124650&nmove=7

https://littlegolem.net/jsp/game/game.jsp?gid=2124667&nmove=8

I say we sincerely need to request for a bigger board here on LG. 15x15 will solve our opening boredom.

Arek, do you mean c2?

I'm all in with 15x15. Keep the bots off the lawn!

Another possible idea could be this:

1) Start with empty board

2) The system randomly adds a stone of a random color on the board, black to play next

3) Both players, independently, they choose which side they would like to play. If they choose different sides, then the game can begin, otherwise, we go back to step (2).

This also leads to basically same game, an arbitrary position can happen, the difference is that this time we don't have the issue of prepared openings.

Btw, the side to play next can also be decided randomly at each iteration of step (2).

Iguser, yes realistically people would end up with low values of N, but not necessarily N=1. It depends on board size and time control

Tom, yes if I start with 6, you can counter with 5, but you can also counter with 2 or 1, you can skip numbers if you want. ;)

Anyway, I don't like randomized opening either. I think the decreasing N opening rule is the best, although it's very unfamiliar to us right now and it seems a kludge :P

with your system of randomized opening, it would seem like good players would either start early or keep choosing the same side until too much of the board is filled.

That there are only one or two fair openings on the 13x13 doesn't seem to be a problem to me, because it is just the first move, and with the swap rule it is meant that player's settle on the fairest move, and as we study more it is reasonable to see the fairest placement get narrowed down like this. But if later moves are also solved, then it's time to move on to a bigger size, which the hex playerbase can keep doing up until some very large size between 50 and 100, but the bigger the board, the more time it takes for it to become overstudied.

There are c2 and m1 but there are also g3, a4, a10 and a8. It just happens that the bot don't like the moves in the A column.

It's not that bots don't like them. They are too strong and we've already known it before bots but we had hope…

After the bots we lost hope. Maybe a4 and a10 are still playable for risk-lovers, I actually played a10 in the current championship ;)

Arek, the bots even play stuff like 1 J3, I think they may be clueless actually, they're based on statistics rather than perfect play. I used to think they're strong at openings but maybe they're not. It's possible they just see statistically that stuff like J3 or g3 is balanced in practice and stuff like a8 or a10 is not. But being balanced in practice doesn't mean much in theory.

You have a point here, but here is what I think.

Bots are bad at openings, relatively to their play in later stages. But they are very very consistent.

From consistency come highly precise statistics i.e. evaluations for the opening map that we got from leela.

To add to that, from my 15 years experience, the statistically strong position is usually also a winning (forcibly) position.

Among the top players still it's possible that a4, a10 and g3 produce interesting games, but like I said, it takes one player to willingly take a high risk.