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Revision as of 02:43, 8 April 2026

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:* ''Answers:''

:* ''Answers:''

:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.

:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.

−

:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).

+

:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.4-stone mistake, and Blue's best reply is cutting through at the adjacent (a).

−

:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.

+

:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.35-stone mistake.

:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />

:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />

−

:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.

+

:** (d): Blue can respond at either (a) and create a pattern congruent to the one where Red plays (c), so this is at least a 0.35-stone mistake. KataHex says Blue can do even better responding with tenuki or (e), so it thinks (d) is actually a 0.4-stone mistake.

−

:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model &mdash; our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.

+

:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex says it's a 0.2-stone mistake.

−

:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.

+

:** (f): This is a retroactive near allow-cut. Blue can play at (a) or (b) to reach a position congruent to the one where Red plays (e), so it's at least a 0.2-stone mistake. Blue tenuki may be slightly stronger; KataHex says (f) is a 0.25-stone mistake because of that.

:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here &mdash; taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing &mdash; is a powerful tool when thinking about real games.)

:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here &mdash; taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing &mdash; is a powerful tool when thinking about real games.)

+

:* Deviations between KataHex and our mental model are unsurprising and don't invalidate the mental model &mdash; our reasoning was a bit handwavy, Blue often has stronger alternatives, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.

:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.

:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.

* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)&ndash;(g) and tenuki.

* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)&ndash;(g) and tenuki.

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:* ''Answers:''

:* ''Answers:''

:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.

:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.

−

:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.

+

:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.2-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.25-stone mistake.

−

:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.

+

:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.35-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.4-stone mistake.

−

:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.

+

:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.4-stone mistake.

−

:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake &mdash; 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.

+

:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake &mdash; 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex nearly agrees and says it's a 0.35-stone mistake.

−

:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

+

:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.5-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.

That was a pretty dense section, so let's take a breather and summarize.

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