### Marius' first theorem on openings Hex, Havannah

21 replies. Last post: 2003-10-25

Marius' first theorem on openings
• Marius Halsor at 2003-10-23

This might as well have been posted in the “hex wiki” hroup, and when the wiki is up and running, that will definitely be the place for such topics. If someone else has mentioned this before, I appologize. And obvious as it may seem, I’ve never read it before. Anywy, here it is:

For any given number n, there exists a number m, such that for board sizes larger or equal to m, all opening moves that are n or less hexes away from a “pointy” corner (e.g. A1) is a losing move.

Does this seem obvious? Does anyone object? Has anyone an idea as to how we might prove it – if it has not already been proven? I believe that it is NOT true if you replace “pointy corner” with “non-pointy corner”, as I believe that even that cornere itself will be a winning move on every board size – although I guess I don’t believe it strong enough to call it a “theorem” :-)

Marius

• Tasmanian Devil at 2003-10-23

I object to calling it a “theorem” before there is a proof!

• michael at 2003-10-23

conjecture is what we call that

• Marius Halsor at 2003-10-23

Sorry; yes, ofcourse it should only be called a conjecture until it’s been proved. So what are you waiting for, then! Come on, prove it! :-)

Marius

• Herb Doughty at 2003-10-23

Marius,
I will be surprised if your guess turns out to be true.
I had held the same opinion 40 years ago because of feelings from potential theory which I could not turn into a proof.
Some much stronger players changed my mind. While playing each other with no swap rule on boards 19x19 and larger, they often choose moves like E4. Vaguely: Choosing an edge connectable move that is “out of the way” seems to more clearly invoke the theorem that in nx(n+1) hex the narrow side has the advantage no matter who goes first. These moves may not be better than other edge connectable moves, but may be easier to see how to use.
Edge connectable opening moves in hex seem like 3rd line moves in go, and some moves near them seem like 4th line moves in go. On small boards strong go players play center moves, on large boards they don’t. I think Go Sei Gen said that this is not because they are know to be inferior, but because they are known to be overwhelmingly harder to use.
I expect the same in hex. I would consider it a major achievement if you could prove your claim. --Herb

• Marius Halsor at 2003-10-23

Yes, it would be a major achievement. I doubt I’ll be able to do it, though. At least not for a while. So for now, I’ll just state another conjecture – I think I’ve heard others feel the same way about this:

Any opening move on the short diagonal is a winning move.

I’m not really sure which one of these will be the harder to prove...

Marius

• Bill LeBoeuf ★ at 2003-10-24

Marius,
I too have been coming up with conjectures for opening theorems.
One of my first is that all opening moves on the short diagonal seem to win (with the non-pointy corner being the weakest but still a winning move on the short diagonal).
I also held your first conjecture. Believe it or not, it has already been disproved! The losing moves on the 7x7 board are as follows: A1 through F1, A2, B2, D2, A3, A5. Now given that A5 and D2 lose, then A4 and C2 should lose, but they are both winning! This result has surprised at least a few of us hex players.
Cheers, Bill

• Tasmanian Devil at 2003-10-24

This does not disprove the conjecture...

• Dvd Avins at 2003-10-24

Taking Jan’s conjectures together, we can start to build a framework. For any size board, if the conjectures are right, the board may be divided into three regions, based on the distance from the short diagonal.

Within sime minimum distance(claimed to be at least 0 for all size boards) an initial move wins.

Beyond some maximum distance, all opening moves lose. The claim is that as the board-size n gets arbitrarily large, that distance also goes infinite, though it may be of order less than n.

In between those limits, there is a region where the value of an opening move is not determined by the conjectures.

• Tasmanian Devil at 2003-10-24

I am getting increasingly confused. I (Jan) have a conjecture now? I thought about posting a counter-conjecture to Marius' one (e.g., C3 wins on all boards with a few exceptions on the smallest ones) but I don’t recall posting it before. ;-)

I think you mean: All initial moves within a certain distance from A1 are losing and beyond a certain distance they are winning. But you need a further restriction, so that you don’t get to the opposite corner where a first move is also losing.

• Tasmanian Devil at 2003-10-24

Nevermind the second paragraph; I didn’t see that the distance was from the short diagonal.

• David J Bush ★ at 2003-10-24

Marius’s first conjecture says nothing about what first moves are winning. It merely proposes that for large enough boards, a triangular region in an acute corner of the board will contain only losing initial moves, and this region will grow larger without bound as boards get larger. I have serious doubts about that; I see no reason why 1.E5 would ever be unswappable regardless of the board size. Corner battles are crucial on any size board. An initial move which “wins the corner,” so to speak, seems like a good move to me. Anyway, the example of the 7x7 board does not disprove this conjecture; the triangular region is of size 2 in that instance, but the conjecture is concerned with larger boards.

Marius’s second conjecture says nothing about what initial moves lose. Results from analysis of smaller boards suggests that an initial move on your border row other than the obtuse corner is losing. So maybe your border cell adjacent to the obtuse corner always loses regardless of board size.

I see no reason to think the locus of all initial losing moves on a Hex board would have any easily definable shape for larger size boards.

• Bill LeBoeuf ★ at 2003-10-24

I agree with David.

I have some conjectures for winning and losing moves (for any board size):

1. Any move on the short diagonal wins.

2. Any move on the first row loses except for the square on the short diagonal (at the obtuse angle).

3. Any move wins with the exception of moves on the first row and possible exceptions of moves on the second row or of moves on the A column.

These rules do hold without exception for all currently resolved boards (1x1 through 7x7).

• Rex Moore ★ at 2003-10-24

If I might drag down the discussion just a little...

Would you please define “winning move” and “losing move”?

Thanks. ;-)

• Bill LeBoeuf ★ at 2003-10-24

Rex,
In any given position on the hex board there can never be a draw (this has been proved), so with best play on both sides it is always the case that one player has lost and one player has won.

So after the very first move the game is (theoretically) over, one side has lost and one side has won (with best play). Of course, the trick is to know what the best play is.

Since the second player has the choice to swap or not, the second player has a theoretical win and the first player a loss in every game.

See David Bushes games for ways of turning theoretical losses into practical wins.

• Dvd Avins at 2003-10-25

Sorry, I did mean Marius’s conjecture’s, not Jan’s.

And at least for now, I stand by what I said about them. Consider, for any hex, the distance to the short diagonal and the distance to the acute corner add up to a constant. So to say that a hex is within a certain distance of the actue corener is the same as saying that it is more than certain distance from the short diagonal.

Using that eqality, I was putting the conjectures in the same terms as each other, categorizing each hex by its distacne from the same referant.

All of which is rather academic if, as the strong players suggest, the conjectures are false.

• Bill LeBoeuf ★ at 2003-10-25

Not only do I think the conjecture is false, I would propose the exact opposite conjecture, namely that a move as close to the acute edge as B3 wins on a board size of any m.

• Marius Halsor at 2003-10-25

Well, then your conjeture has a great advantage over mine – it is easy to disprove if it’s wrong :-)

However, we probably need to solve boards of size over 17x17 before b3 is a losing opening, I think. But I believe there ARE boards on which it is a losing move.

Marius

• Bill LeBoeuf ★ at 2003-10-25

Marius,
I think you are making a good point, namely that patterns I observe on small boards may or may not prove true on larger boards. Still that all we have to go by at this point.

Of course, David wants us to start playing 19x.

• Marius Halsor at 2003-10-25

Playing on 19x19 might eventually start to give us a few pointers as to which stones should be swapped and not. However, I think it will take more than a year of playing before some of us are good enough at 19x19 to make a valid statement about openings on such boards. But it’s a start. Where do we play? :-)

Marius

• David J Bush ★ at 2003-10-25

Back in the glory days when Paul (Batchis?) had his Newton Games server running, all size Hex grids from 10x10 up to 19x19 were available, with very nice graphics and a resizeable board. Now the URL doesn’t even work any more. All that remains is an image from a 19x19 game played there, which is posted on the BoardGameGeek website. Maybe if someone else with a broadband connection would be willing to host a server, Paul might be willing to share his code, I don’t know.

I halfway finished a “Connection Collection” package for Zillions, with Hex and some similar games such as Y. Hex and
Y are already available for Zillions, but my version will include 19x19 Hex and official Y with the curved board. I already finished the Hex part actually, but I want to do all of it before I post it. But Zillions is not an ideal way to play anyway, since it is not a server, and it costs money to buy the software.

PBMserv has a -size option which would certainly allow a 19x19 game. They have the swap rule there also. You can view all current Hex games on that server here.

Playsite has 18x18 which is pretty close, but it’s difficult to find an opponent there lately, and the client window does not resize, so the board cells can be difficult to view. The Beginner’s lobby there has been choked with “ghosts” for weeks now. The admins don’t care.