Difference between revisions of "Openings on 11 x 11"

Revision as of 00:45, 10 October 2022

11×11 is a tactical board size and games can be over very quickly. Nevertheless, 11×11 is good for learning Hex, and many of the concepts here apply to larger board sizes as well.

Some of the most popular opening moves among strong players on Little Golem are c2, i2, a3, and a11 (and their mirror-image equivalents). Here is a swap map based on the opinions of strong players and bots, with asterisks on some recommended moves that are likely to be pretty fair:

As a general point, the opening sequences suggested in this article are certainly not the only ways to respond to each move, and even on 11×11 there is some room for creativity!

Specific openings

a2/b2/c2

These moves all serve as 2nd and 3rd row ladder escapes. They are all quite fair openings, with c2 the most popular nowadays. It can be shown that c2 ≥ b2 ≥ a2 in strength, because c2 capture-dominates b2, which in turn capture-dominates a2. However, many strong players seem to think even c2 is (very) slightly on the losing side and choose not to swap it.

A common opening sequence has Red taking the acute corners and Blue the obtuse corners. On 11×11, Blue usually plays the 4-4 opening at both obtuse corners, or the 4-4 in one corner and 5-5 in the other:

Of course, this is not the only way to play this opening. Red could play blocking moves to Blue's obtuse corner moves:

Or Blue could play in the acute corners:

One of Red's threats, especially on 11×11, is the combination of b5 and b8 cutting up Blue's edge. For example, if Blue plays as follows, the position is completely winning for Red:

This is because the red stone marked "3" acts as a 2nd- and 3rd-row ladder escape, so Blue cannot yield the 4th row ladder starting from 7. But 3 can also switchback 4th row ladders to the 6th row via (*), so either 7 connects to bottom or 3 connects to top.

a3

This is a curious move in that it is "between" a2 and a4, but while a4 is strong, a3 is surprisingly weak and only about as strong as a2, so most strong players prefer not to swap it. It escapes 2nd but not 3rd row ladders, and it can complete a 3rd-to-5th row switchback. One explanation for why a3 is weak is that edge template II fits in the acute corner. Even though b1 is not a strong immediate response to a3, the threat of b1 can be strong, which explains why there are many templates that have a "hole" and don't require the equivalent of a3 in their carriers, for instance template III, template IV2e, or template V2d.

d2

d2 is less commonly played, and in fact weaker than c2. Why? It escapes 2nd and 3rd row ladders like c2, but it doesn't do as good of a job blocking Blue's edge. In fact, with Red c2, if Blue wants to play in the same acute corner, she has to move over two hexes:

With Red d2, Blue can instead play closer to Red's edge:

Because many edge templates have an empty corner of height 3, but none with height 2, Red d2 allows for enough space for Blue's move to fit the "corner", whereas c2 doesn't leave enough space for Blue to do that. In fact, a move like f2 is even weaker than d2, because Blue can fit edge template V1a perfectly in the corner:

As an aside, the same reasoning about template shapes can be used to deduce that while a1–c1 are equally weak, d1 is even weaker, because Blue doesn't have to move over by one hex in the acute corner. The swap maps of leela_bot and neurobenzene confirm this, with leela_bot estimating a 24% win probability for each of a1—c1 and only 15% for d1 on 13×13. This reasoning works on 3rd row openings too — leela_bot thinks d3 and e3 (64% win probability) are weaker than c3 (76%) on 13×13, while f3 (55%) is even weaker because it allows Blue to play e4. I would expect a similar relation to hold on other board sizes like 19x19.

a4

This is a strong move that is probably on the winning side, and it's recommended that you swap if your opponent opens with a4. It serves as 2nd-to-4th row and 3rd-to-5th row switchbacks and cuts significantly into Blue's edge. When combined with b5, it can escape ladders as high as the 6th row.

b4

An interesting move that is probably winning, but weaker than it looks, for a similar reason as the a3 opening. Neurobenzene actually thinks it's one of the fairest moves on 11×11. It can escape 2nd and 3rd row ladders, but it leaves room for the ziggurat in the acute corner. Blue could do well to respond with c2 immediately, which blocks Red's ladder escapes:

Furthermore, if Red connects b4 to the edge, Blue can play an undermining move and gain territory:

This move is rarely played, probably because it looks too strong, but it seems under-explored and could be a decent opening move.

a5

Likely even stronger than a4 and not recommended. Like a4, it can switchback 2nd and 3rd row ladders. Can combine well with b3 and/or b6.

a6

Fairly strong move that cuts up Blue's edge in half; can be combined with b7. It can switchback 2nd row ladders, but the large amount of space required to carry out the switchback makes this rare on 11×11. Probably weaker than a5, because a6 locally combines well with b4, but b4 itself is usually a weak move for Red in the acute corner. Example opening:

a7

Probably stronger than a6 because it combines well with b8, which leaves no room for the ziggurat in the obtuse corner. Blue can attempt to counter this by playing the 3-3 obtuse corner and playing a minimaxing move to achieve edge template III2a:

a8

Fairly strong move, but certainly playable. leela_bot thinks the equivalent move a10 on 13×13 is winning but pretty close to equal. Blue can respond with b10 or d3:

a9

Somewhat fair but on the losing side, only because Blue can play b10, which almost totally neutralizes the a9 stone! But I find this move effective against players who don't know about the b10 trick, because if they swap then you can play b10, and if they don't swap but play something other than b10, Red can play b10 which combines efficiently with a9, and Red's position actually seems quite strong. For example:

Red's a9 and b10 cut significantly into Blue's edge, they can escape 2nd and 3rd row ladders, and importantly they can also deflect 4th and 5th row ladders effectively.

a10

Similar to a12 on 13×13, a10 on 11×11 is too strong and not recommended. Can combine well with b8 to split up Blue's edge, or more simply h7 will escape the 2nd row ladder threatened by a10.

a11

Somewhat strong on 11×11 but playable. Strong players typically swap this move. It escapes 2nd row ladders and strengthens Red's obtuse corner, so Red can play moves in that corner over by one hex. Here is a standard sequence:

i2

i2 feels pretty fair on 11×11, though top players consider it likely on the losing side and rarely swap it. It escapes 2nd and 3rd row ladders, and even though it sits below the obtuse corner, it can still affect play in that corner. For example, Blue usually cannot play the 4-4 opening in the same corner, because of the undermining move j3:

One reason i2 is stronger than a move farther from Blue's edge, like h2, is that with h2, Blue can in fact play the 4-4 opening, because of the minimaxing move h3 (which i2 prevents):

In some cases, i2 combines well with a move like h4.

f3

On 13×13, some moves in the middle of the third row (f3, g3) are quite fair. The equivalent move f3 on 11×11 is probably quite strong but can still be interesting. The trade-off with a move like c2 is that f3 escapes 4th row ladders, but it doesn't intrude into Blue's edges at all (and it demonstrates the priniciple that moves in the middle of your own edge are usually weak). Blue often responds in the far obtuse corner, and Red has a couple strong options marked with (*):

h3

Some circumstantial evidence from leela_bot's swap map suggests j3 is quite fair on 13×13, so the equivalent h3 on 11×11, while strong, could be playable. The idea is that if Blue sets her position up well, she can play the minimaxing move j2, and suddenly h3 no longer escapes 3rd or 4th row ladders:

Without swap

If you are playing a game without the swap rule against a stronger player, the center opening is a good choice. But a corner move like c4 or d8 could give you an easier game, for the same reason that playing in the center is arguably not the best response to a move like a2. While f6 helps you connect your two sides, it doesn't do much to block your opponent. A move like d5 helps you connect and attacks your opponent's edge. Of course, on 11×11 any of these central moves should be easily winning against an equal opponent.

External links

leela_bot's swap map

neurobenzene's swap map for 11×11 and 13×13