Difference between revisions of "User:Hexanna"

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19×19 is the most popular of the "large" board sizes. This board size offers a lot of room for strategic freedom (unlike 11×11 or 13×13), but tactics and local play remain highly important.
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==Insights and tidbits from KataHex (hzy's bot)==
  
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An average well-played game lasts about 72-90 moves before one side resigns, or 20-25% of the board, though it varies considerably from game to game.
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* Two very fair openings using two-move equalization, on [https://hexworld.org/board/#11n,a9d3 11×11] and [https://hexworld.org/board/#13n,c2e9 13×13]. Fairer than any opening with one-move equalization; KataHex thinks win probability is very close to 50% even if you let it think for a long time.
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* A self-play game on [https://hexworld.org/board/#15n,f3l4d12g8d8f5c7e4n3k3l3e10c11c12d11f11m5n2m3d9i8n4m4i9g11g10h10h9j9i10k10k12l12k13m13l6m6l7m7l14m14l9j7k2m1k5k8m12j12l8h7h4b6c3b3c2b2b4b10c9g6i3j4k4d4c8a9b7c6b8e7d7e6d5:rb 15×15], where KataHex thinks long enough to have around 50k visits on the top move, and more if it's unsure between two moves. f3 is among the fairest openings on 15×15.
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The advice in this guide is heavily influenced by hzy's KataHex bot, the strongest known (and easily superhuman) bot as of March 2023.
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* Two 19×19 self-play games, with [https://hexworld.org/board/#19n,a14j5o9d16n11l4f7e4b12e17p15g16c7h4i7i15d4d5b6k14f17f16d17d15i13c3c15b17b5e2e3f2f3g2h6h7e16d18i6h13c18f14i12p16o16i4g3h2h3i2i3j2j3k2l3k3k4j4k5n13l14m15k16k15l15l13m13n6r4q2q3p3j7h11f12j10k9n8m11n2m1n12m12l7g13h12k10l11f15m10g14f13g12f11d13e12e13e10b9b11c11d12h9h10c13d9c10c8b8c9b10e6f5g6f6g7g10g11f9:rw a14] and [https://hexworld.org/board/#19n,a19j5n8i15d15k14n13n15o15n16p16p17q16q17o17p15r14r15q15o5q6m18n19o3f7c17b17f5c14c16k7i11m10m8e5f4d4d3b3c2e3c15p3q5e16d18f17e19g18d14a16b15a15b14g12h10f11n7a14b13l8m4n6g13o6o7h12e6p5m7l9h14h13j12e12g7p6p4q10e14f14m5d8c7h4k8j8l7i10h11j10i9j9i7r3p8i8k5g6e10c10d11l4l5f8g9b12h5g5a13i6j4j6n12o12n10j11i12r6l12q8q9p9p10:rb a19] openings. Only 1k visits on the top move for these games. I think it's interesting how different the opening strategies are in these two games.
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* The b4 opening appears to be weaker than all 6 of its neighbors. On a large enough board, maybe even 27×27, b4 could be a losing opening, and the swap map could contain a hole:
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==Differences from smaller boards==
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<hexboard size="5x4"
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* While [[corner move|corner moves]] are still good moves, playing near the middle of your opponent's 5th row is often just as good. This starts to become true for boards 18&times;18 and larger.
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* Ladders and ladder escapes are less important. Human games often have long ladders across a side of the board, but it's usually a mistake for the defending side to keep pushing the ladder. Often, it's best for the defender to jump, allowing their opponent to connect in exchange for territory. Here is a [https://hexworld.org/board/#19n,c2d16p15o5r4q2q3r2p3p2o3m3o2 common example].
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* There is a lot more room to ignore your opponent's threats and [[tenuki|play elsewhere]] in the early opening. Moves are less forcing, and there's a much larger variety of different strategies you can try.
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* On smaller boards, the game becomes quite tactical after the opening, and playing well often means playing stones that "work well" with existing stones near the corner. On 19&times;19, there is room to start a local fight near the middle of the board, relatively far away from existing stones.
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==Common human mistakes==
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* Playing too close to your own edge is by far the most common mistake in the opening. There are exceptions where it can be a good idea, like when you're playing a corner move or joseki, or your opponent has intruded heavily into one of your edges, or you're responding to a local tactical situation. However, if your opponent hasn't played near one of your edges, it's almost always a bad idea to play a move closer to that edge than one of your opponent's edges.
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==General principles==
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* Corner and edge moves: In the absence of other stones nearby, Red would do well to play in one of the following spots:
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<hexboard size="19x19"
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   coords="show"
 
   coords="show"
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   contents="R d5 e6 *:f7 g8 f9 *:e10 *:e11 *:d16 p15 o14 *:n13 m12 n11 *:o10 *:o9 *:p4"
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  edges="top left"
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   contents="S red:all blue:(a1--d1 a2--d2 a3 b4)"
 
   />
 
   />
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* A 13&times;13 swap map, with KataHex's self-play Elo estimate of the swap advantage for each opening. Generated using around 30k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Hexes without numbers are unfair openings that confer Blue more than a 300 Elo advantage. For example, the fairest opening is g3 (or g11), which KataHex thinks Blue should swap, leaving Blue with a 51.5% win rate, or 10 Elo.
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This is far from an exhaustive list; many other moves near the middle of Blue's 4th to 6th rows are often just as good. Of course, the presence of other stones even moderately nearby can influence things. KataHex prefers the spots marked (*) especially often.
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** Key takeaways: The "common" human openings c2, k2, a10, a13 are all reasonably fair. g3 has become more popular recently, for good reason. b4 is rarely played, but it seems fair enough to be suitable for even high-level human play.
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<hexboard size="13x13"
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* If Blue plays too closely to her edge, Red usually has some good local responses. In particular, if Blue plays near the middle of her 4th row, Red can choose one of the following blocks:
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   coords="show"
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<hexboard size="7x7"
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   contents="S red:all
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   coords="hide"
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              blue:(a1--l1 a2--k2 a3 a11)
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  edges="left"
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              blue:(b13--m13 c12--m12 m11 m3)
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   contents="B d4 E A:f3 B:e4 C:e5"
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            E 239:(d3 j11)
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              187:(e3 i11)
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              48:(f3 h11)
 +
              10:(g3 g11)
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              68:(h3 f11)
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              185:(i3 e11)
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              158:(j3 d11)
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              107:(a13 m1)
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              161:(k2 c12)
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              258:(d2 j12)
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              110:(c2 k12)
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              184:(b2 l12)
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              189:(a2 m12)
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              207:(a3 m11)
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              143:(b4 l10)
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              226:(b11 l3)
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              247:(a4 m10)
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              211:(a6 m8)
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              219:(a7 m7)
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              197:(a8 m6)
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              171:(a9 m5)
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              158:(a10 m4)
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              131:(a11 m3)"
 
   />
 
   />
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KataHex prefers A the most often on a relatively empty board.
 
  
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* If Blue plays near the middle of her 5th row:
 
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<hexboard size="7x8"
 
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  coords="hide"
 
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  edges="left"
 
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  contents="B e4 E A:g3 B:f4 C:f5 D:d6 E:c5 F:d3 *:f2"
 
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  />
 
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KataHex usually prefers A or E, though B/C/D/F are also common. The move marked (*) is usually less good, because Blue can respond at A.
 
  
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* If Blue plays near the middle of her 6th row:
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==Random unsolved questions==
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<hexboard size="7x9"
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−
  coords="hide"
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  edges="left"
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  contents="B f4 E A:h3 B:e6 C:d5 D:e3 *:(g2 g5) +:g4"
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−
  />
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Here, all of A/B/C/D are often good choices. The moves marked (*) are usually worse because Blue can respond at A. The move marked (+) is also worse, and Blue usually does well to tenuki.
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* If Blue plays close to the center, Red would do well to block at a distance, rather than using an adjacent or near block.
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Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:
  
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* A well-played game between equally matched players should "use" almost the whole board. In particular, large templates like [[edge template VI1a]] rarely matter on 19&times;19. Many players are tempted to play a stone in the middle of their 6th row, because such a stone is connected. However, the opponent has good responses intruding into the template (see above).
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* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n&times;n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
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** A. P(n) is always true. If so, can we prove this?
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** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
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** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
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** D. P(n) is true for finitely many n. If so, what's the largest such n?
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* Kriegspiel Hex (Dark Hex), a variant with incomplete information
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** Under optimal mixed strategies, what is Red's win probability on 4&times;4?
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** For larger boards (say, 19&times;19), is Red's win probability close to 50%?
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*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
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*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?
  
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* Suppose Red has played the 5-4 opening. It turns out that a [https://hexworld.org/board/#19nc1,d5g3:pd4 decent response by Blue] is playing at 3-7 (from Red's perspective), partially due to the threat of Blue 4-4 as a followup. This would imply that, had Blue ''first'' played at 3-7 before Red played in the corner, Red should not respond with 5-4, because that would make Blue's 3-7 (which was placed first) unnecessarily effective. Red should instead play a move that works well against Blue's stone. It turns out that the [https://hexworld.org/board/#19nc1,:pg3d4 4-4 corner] is such a move. This is an important concept &mdash; you don't want to play a move close to your opponent's, if that would make your opponent's stone efficiently placed relative to yours.
 
  
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* Here's another [https://hexworld.org/board/#19nc1,e4d4d5 example]. Red accidentally played the 4-5 corner move instead of 5-4. Blue should not play 4-4, because then Red could play 5-4, and he would be in the same position that he would've been, had he played the first move correctly (via the Red 5-4, Blue 4-4, Red 4-5 joseki). Blue essentially let Red out of his mistake. A better move for Blue here is simply to tenuki.
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replies by [[User:Demer|Demer]]:
  
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==Acute corner theory==
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* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
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**​ It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
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** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
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* As far as I'm aware, even 3&times;4 Dark Hex has not been solved. ​ (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)
  
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Corner joseki on 19&times;19 can be quite involved. Here's a sampler for inspiration.
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hexanna:
  
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('''TODO''' elaborate, add diagrams)
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* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13&times;13 and smaller, using transfer learning to train larger nets on top of the 13&times;13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
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** The results for up to 15&times;15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13&times;13]:
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*** a1&ndash;c1 are stronger than d1; a2&ndash;c2 &ge; d2 &ge; e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
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*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
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*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
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*** a10 is the weakest of a4&ndash;a10, while a5 is the strongest.
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*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
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** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13&times;13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
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** On the other hand, and the author seems to agree, the 37&times;37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
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** The 27&times;27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.
  
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===5-4 acute corner===
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==Article ideas==
  
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* High intrusion is by far the most common: [https://hexworld.org/board/#19nc1,p15p16o16p14 here]
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* '''Motifs''' &mdash; very loosely related to joseki; small local patterns that occur in the middle of the board, usually representing optimal play from at least one side but not necessarily both sides
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** Motifs have some notion of '''"local efficiency"''' (not to be confused with [[efficiency]]) &mdash; some motifs are, on average, good or bad for a particular player. Strong players anecdotally try to play locally efficient moves on large boards where calculating everything is impractical. It would be useful to have some of these rules of thumb written down. Can be thought of as a generalization of dead/captured cells, where LE(dead cell) = 0, and LE(X) &le; LE(Y) if Y capture-dominates X.
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Blue typically doesn't play 4 if she already occupies the obtuse corner on that side, but in other cases it's often the best move. Other bots like leela_bot also play this joseki often, so even if the benefit of Blue 4 isn't immediately obvious to humans, the move still deserves serious consideration.
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** Here are some examples. In the first motif, Red 1 is often a weak move. Blue's best response is usually at a, or sometimes at b or c as part of a minimaxing play. But d is rarely (possibly never) the best move, because Red can respond with a, and Blue's central stone is now a dead stone. So, for any reasonable working definition of "local efficiency" LE, we have LE(d) < LE(a), and LE(b) = LE(c) due to symmetry. KataHex suggests that LE(a) > LE(b).
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* An extended version: [https://hexworld.org/board/#19nc1,p15p16o16p14q14o17n17 here]
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===6-5 acute corner===
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* Low intrusion by Blue, high intrusion by Red: [https://hexworld.org/board/#19nc1,o14o16p16p15n16n17m17 here]
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* A much longer variation: [https://hexworld.org/board/#19nc1,o14o16p16o17q17p18q18p15q13p17n16n15o15m18l17m16m17 here]
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Blue has a couple ways to gain territory from Red 15, either playing at j18 or k19, but it seems better to defer the [[question]] and wait until one option is clearly preferable.
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* High intrusion by Blue: [https://hexworld.org/board/#19nc1,o14o15p14p15n15m17l16m15m16k18q13p16j17k16k17 here]
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Red 11 is a good minimaxing move, but he can only play it after Blue 10, since otherwise Blue has a [https://hexworld.org/board/#19nc1,o14o15p14p15n15m17q13l18 strong minimaxing reply].
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===7-6 acute corner===
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−
 
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* Here's a standard one that KataHex prefers: [https://hexworld.org/board/#19nc1,n13n15o15n16p16p17r17q18o17 here]
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==Obtuse corner theory==
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===4-4 obtuse corner===
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It's highly instructive to go through the many possible Blue responses to Red 4-4 in the obtuse corner.
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−
 
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<hexboard size="7x7"
+
−
  coords="hide"
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−
  edges="bottom left"
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−
  contents="R d4 E A:c5 B:d3 C:e3 D:e2 E:f2 F:d6 G:c6 H:b6 I:e5 J:d5 K:e4"
+
−
  />
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'''A:''' KataHex's favorite response on 19&times;19 by far. Blue's move 3 gives her a 3rd row ladder escape in the form of [[edge template III2a]].
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<hexboard size="7x7"
+
−
  coords="hide"
+
−
  edges="bottom left"
+
−
  contents="R d4 B 1:c5 R 2:d5 B 3:c6 R 4:d6"
+
−
  />
+
−
 
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Unless Red's acute corner is free, Red usually connects directly to the bottom with move 4. This may be counterintuitive since it goes against the principle of minimaxing, but most Red attempts to minimax allow Blue to gain territory. For instance, if Red plays at 4 below, Blue gets move 7 for free, and the result is favorable to Blue.
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<hexboard size="7x7"
+
−
  coords="hide"
+
−
  edges="bottom left"
+
−
  contents="R d4 B 1:c5 R 2:d5 B 3:c6 R 4:e5 B 5:d7 R 6:f6 B 7:f5 R 8:e6"
+
−
  />
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If the acute corner is free, Red can play an alternative joseki on move 4:
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<hexboard size="7x19"
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−
  coords="hide"
+
−
  edges="bottom left right"
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−
  contents="R d4 B 1:c5 R 2:d5 B 3:c6 R 4:p3 B 5:m5 R 6:q4"
+
−
  />
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This joseki is quite tactical. After Blue's move 3, Red has a third row ladder from the obtuse corner, even if he plays elsewhere, but no ladder escape. Instead of connecting outright, Red plays 4 to give himself a ladder escape at a distance. Blue can defend the ladder by pushing for a few turns, but it's a mistake to push all the way to the acute corner where Red can escape the ladder. So, Blue jumps at a distance on move 5. Note that Blue deliberately chooses the 3-7 point, which works well against Red's 4.
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−
 
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After Red responds at 6, Blue has several reasonable options. Blue can push the ladder defensively, which Red can't escape outright because of Blue 5, but eventually Red can climb or carry out a complex switchback with the help of 4 and 6 (neither of which are overly strong for Red). Alternatively, Blue can start a fight in the acute corner for territory or ladder escapes. Since this is a joseki, it represents excellent play by both sides without big mistakes, but the exact best continuation will depend on the surrounding board situation.
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'''B:''' Interestingly, this move is relatively common on 11&times;11 but not 13&times;13. The usual purpose of this move is to block Red from playing at (+) below. It appears slightly worse than move '''A''', but it's still very playable. Red has many reasonable responses marked (*):
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<hexboard size="7x7"
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−
  coords="hide"
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−
  edges="bottom left"
+
−
  contents="R d4 B 1:d3 E +:e2 *:(c4 e3 f2)"
+
−
  />
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−
 
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'''C:''' This move is often effective on smaller boards when Blue has a ladder escape at her acute corner. However, the acute corner is much farther away on 19&times;19, and Blue's 5th row ladder is much less threatening, so Blue gains less from playing this move. Red, who is defending the ladder, usually pushes the ladder by playing at (*) below, or he jumps a couple hexes forward on the 3rd or 5th row (either immediately or after pushing a few times), indicated by (+):
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+
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<hexboard size="8x7"
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−
  coords="hide"
+
−
  edges="bottom left"
+
−
  contents="R d5 B 1:e4 E *:d4 +:(c3 e2)"
+
−
  />
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−
 
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'''D:''' This blocking move is common on 13&times;13 but less so on 19&times;19. Blue's idea, if Red ignores the threat, is to follow up with this move 2, which is quite strong since it neutralizes Red's 4-4 stone significantly:
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<hexboard size="7x7"
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−
  coords="hide"
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−
  edges="bottom left"
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  contents="R d4 B 1:e2 2:f3"
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−
  />
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Indeed, Red usually responds to the threat, and the following sequence is a common joseki on 13&times;13:
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<hexboard size="8x7"
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−
  coords="hide"
+
−
  edges="bottom left"
+
−
  contents="R d5 B 1:e3 R 2:c4 B 3:d2 R 4:e2 B 5:d3 E *:b3 +:b4 -:f3"
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−
  />
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Red 6 is often at one of (*), (+), or (-). The move (*) allows Red to gain territory, while (+) creates a [[Flank#Capped_flank|capped flank]] that blocks Blue 3rd row ladders under Red's 4-4 stone. It's not obvious to me why, but KataHex tends to think Red is slightly better after this sequence on 19&times;19, so Blue usually doesn't play '''D''' in the first place.
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'''E:''' Usually not the best move for Blue. Depending on local tactics, Red should either tenuki, or play one of (*):
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<hexboard size="7x7"
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−
  coords="hide"
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−
  edges="bottom left"
+
−
  contents="R d4 B 1:f2 E *:(e2 c2)"
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−
  />
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'''F:''' This 4-2 obtuse corner block is strong on small boards like 11&times;11, but it's rarely a good move on 19&times;19, whether as the first stone in the obtuse corner, or in response to 4-4. There are exceptions &mdash; the 4-2 move works well in combination with a "middle of third row" opening stone, for example. Red would do well to connect directly with 2:
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<hexboard size="7x7"
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  coords="hide"
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−
  edges="bottom left"
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  contents="R d4 B 1:d6 R 2:c6"
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−
  />
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'''G:''' This block is a "surprise weapon" of sorts &mdash; it's a weak move on an empty board, but for local tactical reasons it can be very strong. The standard example is with the q2 opening, where an unsuspecting Blue who plays 4-4 in response is faced with an unpleasant surprise (more on that later).
+
  
 
<hexboard size="5x5"
 
<hexboard size="5x5"
−
   coords="hide"
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   coords="none"
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   edges="top right"
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   edges="none"
−
   contents="R c2 B 1:b4 R 2:d3"
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   contents="R b3 B c3 R 1:d2 E a:c2 b:b4 c:d3 d:c4"
 
   />
 
   />
  
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Move '''G''' is also a threat if Blue already has a stone in either of (*) below.
+
The motif below seems quite common on large boards, and in my experience it is ''usually'' good for Red, who allows Blue to connect 2 and 4 in exchange for territory.
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 E *:(e5 f6) G:c6"
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  />
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If '''G''' is played, Red should consider blocking the 3rd row ladder at a, or minimaxing at b.
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 B 1:c6 E a:b6 b:c3"
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  />
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'''H:''' Another "surprise weapon," arguably even more so. Anecdotally, when KataHex thinks '''H''' is the best move in a position, it rarely assigns a high policy to the move, only liking the move after some search. In other words, KataHex's policy "intuition" rarely considers the move a top choice, or even top 10, until it realizes that the move works tactically in the particular situation.
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 B 1:b6 E *:d5"
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−
  />
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This 2-2 obtuse corner move typically works as an unusual minimaxing move, providing ladder escapes for Blue while simultaneously blocking Red and threatening a move like (*).
+
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+
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'''I:''' This move is sometimes played on 13&times;13, but it rarely works on 19&times;19. The standard joseki is favorable to Red, probably because Blue 1 and 5 function mainly as a ladder escape blocker, and ladders/ladder escapes are themselves less important on 19&times;19.
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 B 1:e5 R 2:c4 B 3:c5 R 4:d5 B 5:d6 R 6:b5"
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−
  />
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'''J:''' Like many other Blue responses, this is a bad move in isolation. Red's 4-4 is already connected to the bottom via [[edge template IV1d]], so Blue attempts to block are futile unless she gets useful territory in exchange (like with '''A'''), but the territory gained by '''J''' is not nearly as good. However, this move can become useful if there are other blue stones present.
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 B 1:d5 R 2:c5"
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  />
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'''K:''' Also a weak response. Can you see why?
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<hexboard size="7x7"
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  coords="hide"
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  edges="bottom left"
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  contents="R d4 B 1:e4 R 2:e2"
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  />
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Red 2 is strong, but that's not the only reason why. It turns out that had Blue played 1 first (before Red played the initial 4-4 stone), then a good Red response would be playing at 4-4. Going back to our general principles, it's a bad idea to play a move that would make your opponent's existing stone unnecessarily well-placed relative to yours, and that's exactly what '''K''' does.
+
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+
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===5-5 obtuse corner===
+
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+
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'''TODO'''
+
−
 
+
−
==The first move==
+
−
 
+
−
See [[Swap_rule#Size_19]] for a swap map.
+
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+
−
We'll now go through the general strategy of specific first moves. For simplicity, everything will be from Red's point of view, assuming Blue doesn't swap. Unlike the guides for smaller board sizes, we won't think too hard about ladder escapes or switchbacks, and instead we will just mention some brief notes for some selected openings.
+
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===Acute corner openings===
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<hexboard size="5x5"
 
<hexboard size="5x5"
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   coords="hide"
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   coords="none"
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   edges="top left"
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   edges="none"
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   contents="S red:all blue:(a1--e1 a2--e2 a3)
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   contents="R 1:b2 B 2:b4 R 3:d3 B 4:c2 R 5:b3 B 6:c3"
−
            E *:(c2 d3 e3 b4)"
+
 
   />
 
   />
  
−
The stone in the acute corner affects which moves are locally efficient for Red and Blue.
+
The following motif is quite clearly good for Blue, who captures the two hexes marked (*):
  
−
====c2====
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<hexboard size="3x4"
−
 
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   coords="none"
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On 13&times;13, b5 or c6, marked with (*) below, are common Red moves that combine well with c2. On 19&times;19, these moves are a bit too close to the corner. Playing a bit further along the b5-c6 diagonal, such as A or B below, is often a better move:
+
   edges="none"
−
 
+
   contents="R 1:a2 B 2:c1 R 3:d2 B 4:b3 E *:b2 *:c2"
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<hexboard size="9x6"
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   coords="show"
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   edges="top left"
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   contents="R c2 E *:(b5 c6) A:d7 B:e8"
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   />
 
   />
  
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====b4====
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Sometimes, a player will attempt to minimax by placing two stones adjacent to each other, like the unmarked blue stones below. Red has several options, such as the adjacent block 1, though a far block is often possible too. It would be enlightening to know, absent other considerations, which block is the most "efficient" for Red, so that on a large board, Red could play this block without thinking too hard. Of course, in general the best move depends on the other stones on the board, and there's no move that strictly dominates another. The best move may even plausibly be to "[[tenuki|play elsewhere]]."
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+
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Under the right circumstances, Blue c2 (followed by Red tenuki) can be a good local response, though this happens less in the early opening.
+
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+
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====e3====
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e3 is notable because KataHex thinks it's the fairest opening with the swap rule, with KataHex assigning a 49.2% win percentage for Blue, assuming no swap, after 100k visits.
+
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===First column openings===
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If Red starts with a move near the middle of his first column, like a10, a good followup for Red is to play one of the hexes marked A or B, or sometimes C (or both). These moves combine very efficiently with the opening stone to split up Blue's edge. KataHex nearly always plays one of these in the early opening.
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<hexboard size="7x5"
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  coords="hide"
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  edges="left"
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  contents="R a4 E A:b5 B:c6 C:b2"
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  />
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If Red plays at A or C, Blue often peeps in Red's bridge, as follows. Red typically responds at one of the hexes marked (*) or elsewhere, instead of defending the bridge.
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<hexboard size="7x5"
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  coords="hide"
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  edges="left"
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  contents="R a4 1:b5 B 2:b4 E *:(b2 c6)"
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  />
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====a10&ndash;a15====
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Some of the fairer openings in this category are a10, a14, a15. Blue's best response to a10&ndash;a15 in the obtuse corner is usually 4-4, but there's no rush to play it:
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<hexboard size="7x6"
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  coords="hide"
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  edges="bottom left"
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  contents="R a2 B 1:d4"
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  />
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On 19&times;19, a15 is weaker than it looks, because the 4-2 obtuse corner, marked (*) below, is less potent for Red than on smaller boards:
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<hexboard size="6x5"
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  coords="hide"
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  edges="bottom left"
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  contents="R a2 E *:b3"
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  />
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====a16====
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a16 is also a relatively fair opening. Blue can play 2-2 obtuse corner like on smaller boards, but it's less clearly the best option. The 4-4 obtuse corner also works, and if Blue instead waits for Red to play 1 as follows, then Blue 2 is a strong response.
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<hexboard size="7x5"
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  coords="hide"
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  edges="bottom left"
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  contents="R a4 1:b2 B 2:b5"
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−
  />
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===Obtuse corner openings===
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There are several openings that affect play in the obtuse corner, but they are quite different from each other so we'll consider them separately.
+
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====a19====
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A common joseki for Red is to play at 1, which is basically the 4-4 opening shifted up one row. Blue often responds at 2, and Red has a couple good responses marked (*):
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<hexboard size="6x6"
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  coords="hide"
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  edges="bottom left"
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  contents="R a6 1:d2 B 2:c4 E *:(b4 c1)"
+
−
  />
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My subjective opinion is that this is the most beginner-friendly opening:
+
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* A beginner who opens with Red c2 could accidentally play b3 instead, or alternatively his opponent who wishes to swap Red c2 could implement swap-pieces incorrectly and replace it with Blue c2 instead of b3. Though a19 should technically be swapped to s1 under the swap-pieces convention, it doesn't really matter.
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* Aesthetically, a19 retains the "most" symmetry of any fair opening. Beginners who don't want to think about the swap rule could play Hex without swap, where Red must open in an obtuse corner, and such a ruleset would be quite elegant and still balanced, even on large boards.
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* For beginners who don't want to learn too much opening theory, "obtuse corner" is easy to remember and a good [https://en.wikipedia.org/wiki/Focal_point_(game_theory) Schelling point]. It's relatively likely that other beginners who look at the swap map and just want to try a random opening will pick a19 or s1.
+
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+
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====q2====
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+
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If Red opens q2, the most important advice for Blue is to refrain from playing 4-4 in the nearby obtuse corner, because of Red's strong response:
+
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<hexboard size="7x6"
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−
  coords="hide"
+
−
  edges="top right"
+
−
  contents="R d2 B 1:c4 R 2:e3 B 3:d5 R 4:f4 B 5:e6 R 6:d6 B 7:e5 R 8:c5 B 9:d4"
+
−
  />
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====b17====
+
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+
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For the adventurous, while b17 should be swapped, it is weaker than it looks and quite playable. It's not overly strong, because Blue can play b18, either immediately or later. I consider it the obtuse-corner analog of b4, which is surprisingly weak because of the threat of Blue c2.
+
  
 
<hexboard size="5x5"
 
<hexboard size="5x5"
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   coords="hide"
+
   coords="none"
−
   edges="bottom left"
+
   edges="none"
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   contents="R b3 B 1:b4"
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   contents="B c2 d2 R 1:d3 B 2:b4 R 3:b3 B 4:c3"
 
   />
 
   />
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−
===Third and fourth row openings===
 
−
 
−
According to KataHex, the fairest openings in this category are e3 (mentioned above), n3, and p3.
 
−
 
−
Openings in the middle of Red's 4th row are surprisingly playable, but most people prefer not to have their opening stone swapped, and playing against a 4th row opening stone can seem daunting, so a 3rd row opening is often preferable. If you strongly prefer having the first stone, or you think your opponent is overly eager to swap, you can play a weaker opening like g3 or h3.
 
−
 
−
Third row openings, especially those near an obtuse corner (except p3), tend to combine well with the 4-2 obtuse corner move:
 
−
 
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<hexboard size="5x7"
 
−
  coords="hide"
 
−
  edges="top right"
 
−
  contents="R b3 1:f4"
 
−
  />
 
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[[category: Opening]]
 
−
[[category: Advanced Strategy]]
 

Revision as of 01:12, 30 March 2023

Insights and tidbits from KataHex (hzy's bot)

  • Two very fair openings using two-move equalization, on 11×11 and 13×13. Fairer than any opening with one-move equalization; KataHex thinks win probability is very close to 50% even if you let it think for a long time.
  • A self-play game on 15×15, where KataHex thinks long enough to have around 50k visits on the top move, and more if it's unsure between two moves. f3 is among the fairest openings on 15×15.
  • Two 19×19 self-play games, with a14 and a19 openings. Only 1k visits on the top move for these games. I think it's interesting how different the opening strategies are in these two games.
  • The b4 opening appears to be weaker than all 6 of its neighbors. On a large enough board, maybe even 27×27, b4 could be a losing opening, and the swap map could contain a hole:
abcd12345
  • A 13×13 swap map, with KataHex's self-play Elo estimate of the swap advantage for each opening. Generated using around 30k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Hexes without numbers are unfair openings that confer Blue more than a 300 Elo advantage. For example, the fairest opening is g3 (or g11), which KataHex thinks Blue should swap, leaving Blue with a 51.5% win rate, or 10 Elo.
    • Key takeaways: The "common" human openings c2, k2, a10, a13 are all reasonably fair. g3 has become more popular recently, for good reason. b4 is rarely played, but it seems fair enough to be suitable for even high-level human play.
abcdefghijklm12345678910111213107189184110258161207239187481068185158226131247143158171211197219219197211171158143247131226158185681048187239207161258110184189107


Random unsolved questions

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

  • Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
    • A. P(n) is always true. If so, can we prove this?
    • B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
    • C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
    • D. P(n) is true for finitely many n. If so, what's the largest such n?
  • Kriegspiel Hex (Dark Hex), a variant with incomplete information
    • Under optimal mixed strategies, what is Red's win probability on 4×4?
    • For larger boards (say, 19×19), is Red's win probability close to 50%?
      • If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
      • If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?


replies by Demer:

  • https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
    • ​ It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
    • On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
  • As far as I'm aware, even 3×4 Dark Hex has not been solved. ​ (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

  • Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the swap rule article later with some insights.
    • The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on 13×13:
      • a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See Openings on 11 x 11#d2.
      • b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
      • j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
      • a10 is the weakest of a4–a10, while a5 is the strongest.
      • b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
    • That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
    • On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
    • The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

Article ideas

  • Motifs — very loosely related to joseki; small local patterns that occur in the middle of the board, usually representing optimal play from at least one side but not necessarily both sides
    • Motifs have some notion of "local efficiency" (not to be confused with efficiency) — some motifs are, on average, good or bad for a particular player. Strong players anecdotally try to play locally efficient moves on large boards where calculating everything is impractical. It would be useful to have some of these rules of thumb written down. Can be thought of as a generalization of dead/captured cells, where LE(dead cell) = 0, and LE(X) ≤ LE(Y) if Y capture-dominates X.
    • Here are some examples. In the first motif, Red 1 is often a weak move. Blue's best response is usually at a, or sometimes at b or c as part of a minimaxing play. But d is rarely (possibly never) the best move, because Red can respond with a, and Blue's central stone is now a dead stone. So, for any reasonable working definition of "local efficiency" LE, we have LE(d) < LE(a), and LE(b) = LE(c) due to symmetry. KataHex suggests that LE(a) > LE(b).
a1cbd

The motif below seems quite common on large boards, and in my experience it is usually good for Red, who allows Blue to connect 2 and 4 in exchange for territory.

145632

The following motif is quite clearly good for Blue, who captures the two hexes marked (*):

2134

Sometimes, a player will attempt to minimax by placing two stones adjacent to each other, like the unmarked blue stones below. Red has several options, such as the adjacent block 1, though a far block is often possible too. It would be enlightening to know, absent other considerations, which block is the most "efficient" for Red, so that on a large board, Red could play this block without thinking too hard. Of course, in general the best move depends on the other stones on the board, and there's no move that strictly dominates another. The best move may even plausibly be to "play elsewhere."

3412
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