Hex on a n x (n+2) board

Let black have the shorter distance between his sides, like !http://www.trmph.com/hex/board#11,a2b1b2c1c2d1d2e1e2f1f2g1g2h1h2i1i2j1j2k1k2a1! where the first line is ignored. In return, let white begin the game with two moves in a row. This seems to make a surprisingly fair game - at least averaged over small board sizes. On a 1x3 board, black wins. On a 2x4 board, white wins. On a 3x5 board, black wins, and on a 4x6 board white wins. It seems to me that black wins on a 5x7 board and white wins on a 6x8 board, but maybe that's just wishful thinking. I conjecture that black wins for odd n, and white wins for even n (when n is at least 2). Since this is hex, I also conjecture that my previous conjecture is wrong. What do you think?

One last try to post the redundant image link: wonderful 9x11 board

This is interesting. I see how white wins on 6x8.

In the position given, shouldn't the black stone all the way on the left be white and one of the white stones (other than the one all the way on the left) be black?

Like http://www.trmph.com/hex/board#1example1,b2c1c2d1d2e1e2f1f2g1g2h1h2i1i2j1j2k1k2a2b1a1, for example? Maybe that's slightly better. Of course it's a bit unfortunate if white has won the game before the game even begins. That's what I meant with "where the first line is ignored".

I see, in the original position trmph.com doesn't allow play to continue. I didn't actually try playing it on a 9x11 board, only on sizes up to 6x8.

http://www.trmph.com/hex/board#11,b1a2b2a1c2c1d2d1e2e1f2f1g2g1h2h1i2i1j2j1k2k1  allows further play (and will correctly recognize a black win).

To allow white to begin and play to moves in a row, it may be best to use a larger board:

http://www.trmph.com/hex/board#13,a4l13b4l12c4l11d4l10e4l9f4l8g4l7h4l6i4l5j4l4k4e9. (I'm afraid one could miss the purpose of the thread f1h8

But the 9x11 board was just an unnecessary demonstration of what "Let black have the shorter distance between his sides" meant. The focus of my post was supposed to be the conjecture that black wins on odd sizes and white wins on even sizes.

Ugh. That didn't come out right. Let me try again.

To allow white to begin and play to moves in a row, it may be best to use a larger board:

http://www.trmph.com/hex/board#13,a4l13b4l12c4l11d4l10e4l9f4l8g4l7h4l6i4l5j4l4k4e9f1h8

But the 9x11 board was just an unnecessary demonstration of what Let black have the shorter distance between his sides meant. (I'm afraid one could think this thread was about how to realize a n x (n+2) board on trmph.com)

To me it seems white wins both 8x10 and 9x11.

I haven't checked thoroughly, it's just what my gut tells me

5x7 and 6x8 are 100% white wins

And 7x9 looks white, butindeed on the verge of losing...

I agree this game may be around equal for both players :)

Arek, I think your 6x8 is broken... how is it supposed to work?

White definitely wins on 5x7, but I think 6x8 is still open

Here's my 6x8.

and my 5x7 also.

Actually, 6x8 is trickier than I thought. I like this game.

Thanks for discussing. I agree now that white wins on 5x7. So it really was wishful thinking. Maybe that means white wins on all boards larger than 3x5.

alihv, I think in your first 6x8 variation, j9 refutes i10, but white can win like this . This was also my variation. Instead, black can try to attack on both sides simultaneously, but I don't think it helps. In your second 6x8 variation, d9 can be refuted by f8. I don't dare to comment on Arek's 6x8 :-)

(The link between "white can win like this" and "d9 can be refuted by f8" was not supposed to be a link. I think this editor is buggy with firefox.)

I'm convinced by alihv's 6x8, though the actual lines had some mistakes, as urmaul pointed out. So it seems white wins 6x8, but I'd like to check a little more before calling it with 100% certainty.

Arek's 7x9 is complicated... I'm not sure white wins.

urmaul, thanks for the clarifications!

As a mostly Slither player, I wonder how would that work there.

By the way, on n by n+1 boards where white starts with only one stone, black always has a forced win by playing symmetrically. I assume many people here already know this, but others might not.

The question of how many rows are worth how many stones is an interesting one in general. We could test larger tradeoffs as well.

hey, what was wrong with this 6x8?

oh, sorry, I was seeing something that wasn't there :)

anyway alihv's proposition seems correct to me, so.. white!

Arek, in your 7x9, what's the intended response to g8? Is it h6? If so, how do you respond to i8?

I would go this way

no, previous was losing, g7 has to be delayed, this one is better

I've finally been able to use Benzene to solve several positions.

5x7: Arek's 5x7 indeed wins. There are two other winning first turns for white: 1 and 2.

6x8: alihv's 6x8 indeed wins, and it is the only winning first turn for white.

7x9: I let it analyze all mentioned positions. Shall I reveal the results? All of them?

8x10 is probably out of reach for my combination of hard- and software.

So far, it seems like 3 pieces are worth about as much as 4 columns (since 2 pieces head start means there are 1.5 more white pieces than black pieces on the board on average). I think on boards that are much wider than high (for example 100x100000), 2 pieces will be worth 3 columns, since that's the efficiency of bridges, and I can think of no horizontal connection that is more efficient.

I saw one of the alternative 5x7 wins. I'm really surprised there are only 3 winning white positions in 5x7 and only one (!) for 6x8. Wow!

For 7x9, I think Arek's line works against g8, but that just raises the question of other first moves. Maybe i8 or e9 should be first, with the plan of doing this or this. I'm not convinced it's a white win.

Also, I don't think the 1.5 stone to 2 column exchange rate is likely to generalize. For example, it would imply that 4.5 stones are worth 6 columns, which would suggest testing n by n+6 boards with 5 starting stones for white. But that's pretty easy for white. I think on boards like 100x100000 a piece would might be worth slightly more than urmaul's 1.5 column estimate, because on large boards, pieces can form edge patterns as well as bridges.

I'm surprised how small is the margin, I would think white has an easy game 2 stones ahead, but not really.

10x8 I doubt black can beat this. An instinctive game of me vs me is white win

So, here are the 7x9 results. I want to stress that all lines have been found by the Benzene DFPN solver, not by me. In an attempt to prevent spoiling any possibly existing fun, I'm using light gray font and background color in the answers (Unfortunately this doesn't work for links)

Whether Arek's second defense against shalev's first attack works: Yes, it does.

Whether Arek's first defense against shalev's first attack works: Yes, it does.

Whether shalev's i8 against his "straw man" defense works: No, it doesn't. What works instead: i7 or e8.

Whether shalev's "straw man" defense against his own attack works: No, it doesn't ( PV).

Whether shalev's second attack works: No. I didn't find on my own how white can deviate, but it's actually quite easy.

Whether Arek's first turn suggestion is a white win: No, it isn't ( PV). The only black moves that win against Arek's first turn suggestion are j6, j9 and e8.

Whether white can force a win on the 7x9 board: Yes, white can, like this ( PV). I don't know whether there are other winning first turns for white, but at least i7 is the only move that works in combination with f8.

... Now that I have  used gray background and font color,  setting the background color to "transparent" doesn't work anymore, so I'll make another post.

The second link was supposed to be Arek's first defense against shalev's first attack.

Since more than seven hours (but only on one thread), I'm letting the program solve Arek's new 8x10. Maybe it would take weeks to finish, or worse.

Maybe n x (n+6) with 5 stones gets better on larger boards. It's unclear to me whether this 6x12, though I guess it does.

Ok, anyway, this 6x12 is an easy win for white. I speculate 5x11 and 6x12 are easy for white because the boards are wide enough that each move is worth about 3/2 instead of 4/3.

Huh, I didn't do very well, did I...

Well, you're playing against a cheater. The first "PV" link actually belongs to the line above it.

Neither of us did very well. Wow, e8! Brilliant play by computer :D (black win that I didn't see)

Now that I value short diagonal more than before I think this is better for white 10x8. Looks as unbeatable as the previous, but is less prone to human errors ;)

Arek, to me your other 8x10 seems more convincing. That one feels like a crushing white win, whereas this one... not so much.

Another potential white win on 8x10

Very nice and interesting analysis by all.

I see that you focus mainly on the  "n x (n+2)" (and some analysis for i=6).

Has anyone tried the generalized "n x (n+i)" where i>=1 , i.e other values besides 2 and 6? Any interestign results there?

That wasn't a perfect example, but I agree my short diagonal approach is losing after all. Also I think your center bridge example is actually the strongest one.

ypercube, as I mentioned earlier, n x (n+1) boards are a win for black if white goes first, for all values of n. If White gets two stones, this is strictly better for white than the n x (n+2) boards we're looking at, so that's not interesting either.

To understand how black always wins on n x (n+1), look for a symmetry argument. Here is an example of the symmetry strategy in play.

Ypercube -- the symmetry argument is explained in Cameron Browne's book, if you have access to it.

I guess people did not read carefully my comment or I wasn't very clear. The symmetry argument is only about n x (n+1) and white starting first. It says nothing about white starting with 2 or more stones.

So, does anyone know who wins at 100 x 101 where White starts with 2 stones?

Plus, I said n x (n+ i) (where i >= 1, not just i=1) and I didn't mention it but I thought it was implied, with arbitrary k stones for white.

So, to be explicit, has anyone investigated n x (n+3) or n x (n+4), etc. (with white starting with 1, 2, ..., k stones) ?

ypercube, the symmetry strategy is also described on MathWorld. ;)

I looked a little bit into n x (n+i), but didn't find anything particularly interesting. I just created a hexwiki article about it.

I wish to deliver some more analysis on this interesting trade off topic. I tried to look at a playable size of 13x11.

Candidates:

I consider 3 logical options for opening.

a) long/long, countered by short diagonal

b) short/long, countered by long diagonal

c) center, countered by long diagonal

Further analysis is open for discussion...

So far I only had time to look at my favourite b) which looks to me like white win this way or this way.