Difference between revisions of "User:StillYetAnother11"
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== log2(1.5) ratio == | == log2(1.5) ratio == | ||
| − | I was thinking about why katahex evaluates a7 as the most balanced opening on 12x12 which is very anomalous because openings in the middle of the 1st column tend to be quite strong, although in Hayward's 2003 paper, a4 on 7x7 has the highest number of nodes of all solved openings. The smallest board size after 12x12 where such an opening seems quite playable is 17x17 since katahex evaluates a10 as more balanced than | + | I was thinking about why katahex evaluates a7 as the most balanced opening on 12x12 which is very anomalous because openings in the middle of the 1st column tend to be quite strong, although in Hayward's 2003 paper, a4 on 7x7 has the highest number of nodes of all solved openings. The smallest board size after 12x12 where such an opening seems quite playable is 17x17 since katahex evaluates a10 as more balanced than its neighbors and is still one of the most balanced openings on the first column. A pattern starts to emerge where if you encode the acute corner as 0 and the obtuse corner as 1 you'll find that these openings match ratios that well approximate a particular ratio namely log2(1.5). The first semiconvergents of log2(1.5) are 1/1, 1/2, 2/3, 3/5, 4/7, 7/12, 10/17, 17/29, 24/41, 31/53 and 55/94. The next fraction after 10/17 is 17/29 so is a17 on 29x29 more balanced than its neighbors? I've run 200 engine games for each opening: a16, a17 and a18 and the results were 110 wins for a16, 102 wins for a17 and 112 wins for a18. I also ran 100 games on 41x41 for each of the following openings. The results were 57 wins for a23, 49 wins for a24 and 56 wins for a25 so the pattern seems to be consistent. This has a benefit when it comes to opening variety because most balanced openings in hex are predominantly on your own edge and not on your opponent's edge. I believe that having the denominators of semiconvergents of log2(1.5) as board sizes is the most logical and consistent way of deriving board sizes in hex. |
Latest revision as of 18:58, 20 March 2026
log2(1.5) ratio
I was thinking about why katahex evaluates a7 as the most balanced opening on 12x12 which is very anomalous because openings in the middle of the 1st column tend to be quite strong, although in Hayward's 2003 paper, a4 on 7x7 has the highest number of nodes of all solved openings. The smallest board size after 12x12 where such an opening seems quite playable is 17x17 since katahex evaluates a10 as more balanced than its neighbors and is still one of the most balanced openings on the first column. A pattern starts to emerge where if you encode the acute corner as 0 and the obtuse corner as 1 you'll find that these openings match ratios that well approximate a particular ratio namely log2(1.5). The first semiconvergents of log2(1.5) are 1/1, 1/2, 2/3, 3/5, 4/7, 7/12, 10/17, 17/29, 24/41, 31/53 and 55/94. The next fraction after 10/17 is 17/29 so is a17 on 29x29 more balanced than its neighbors? I've run 200 engine games for each opening: a16, a17 and a18 and the results were 110 wins for a16, 102 wins for a17 and 112 wins for a18. I also ran 100 games on 41x41 for each of the following openings. The results were 57 wins for a23, 49 wins for a24 and 56 wins for a25 so the pattern seems to be consistent. This has a benefit when it comes to opening variety because most balanced openings in hex are predominantly on your own edge and not on your opponent's edge. I believe that having the denominators of semiconvergents of log2(1.5) as board sizes is the most logical and consistent way of deriving board sizes in hex.
The best way for red to fight in c2 opening on 11x11
I prefer to play 5.c9 here even though it's not the best move according to katahex. Moves marked with * are the only moves winning for blue.
7.j4 is again not the top engine move but here is why I prefer it. Moves marked with * are the only winning moves for blue but there are still ways for blue to lose if they don't play this very concrete position correctly.
At this point red can try a bunch of different moves. Blue needs to know that the cell marked with + is the only winning move to a lot of red's responses. You should continue to analyze this position yourself with katahex.
g3 opening on 13x13
Additional tests with katahex showed that after d10 i7 is the only winning move for red.
While j9 is losing for red. If red plays * then + is the only winning response for blue.
