Difference between revisions of "Edge template VI2a"
HappyHippo (Talk | contribs) (After checking with benezene, the template was not minimal. I've replaced it with the minimal version, and updated one of the lines to use the reduced space.) |
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− | + | Edge template IV2a is a 6th row [[edge template]] with two stones. | |
− | <hexboard size="6x9" | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R ↑:g1 h1" | ||
+ | /> | ||
+ | |||
+ | == Defending the template == | ||
Let us first see what possibilities [[Red (player)|Red]] has if he moves first. | Let us first see what possibilities [[Red (player)|Red]] has if he moves first. | ||
Line 7: | Line 14: | ||
There are two obvious options: | There are two obvious options: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="none" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R f3 g1 h1 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4 +:e5 +:e6 +:f2 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6" | ||
+ | /> | ||
− | < | + | <hexboard size="6x9" |
+ | coords="none" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R g1 g3 h1 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6 +:h2 +:h3 +:h4 +:h5 +:h6 +:i4 +:i5 +:i6" | ||
+ | /> | ||
In both diagrams the possible [[Template intrusion|intrusion]] points are marked by (+). So we only have to consider the [[Overlapping templates|intersection of the intrusion points]]. They are: | In both diagrams the possible [[Template intrusion|intrusion]] points are marked by (+). So we only have to consider the [[Overlapping templates|intersection of the intrusion points]]. They are: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="none" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R g1 h1 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6" | ||
+ | /> | ||
− | == Intrusion at E5 and F5 == | + | === Intrusion at E5 and F5 === |
If Blue blocks at E5 then Red plays F3, reducing to [[Template IVb]] | If Blue blocks at E5 then Red plays F3, reducing to [[Template IVb]] | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f3 g1 h1 B 1:e5 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4 +:e6 +:f4 +:f5 +:f6 +:g3 +:g4 +:g5 +:g6 +:h4 +:h5 +:h6" | ||
+ | /> | ||
Likewise if blue blocks at F5: | Likewise if blue blocks at F5: | ||
− | |||
− | == | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R g1 2:g3 h1 B 1:f5 E +:b6 +:c5 +:c6 +:d4 +:d5 +:d6 +:e4 +:e5 +:e6 +:f3 +:f4 +:f6 +:g4 +:g5 +:g6 +:h3 +:h4 +:h5 +:h6 +:i4 +:i5 +:i6" | ||
+ | /> | ||
− | < | + | === Intrusion at E6 === |
+ | |||
+ | <hexboard size="6x9" | ||
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f3 g1 h1 B 1:e6 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4" | ||
+ | /> | ||
Red threatens to connect via D4. Blue must respond in one of the marked hexes. | Red threatens to connect via D4. Blue must respond in one of the marked hexes. | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f3 g1 h1 4:h4 B 3:e4 1:e6" | ||
+ | /> | ||
− | The H4 piece is connected to the bottom with [[ | + | The H4 piece is connected to the bottom with [[ziggurat|template III-1a]], and is connected to the top in two non-overlapping ways: |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f3 g1 h1 4:h4 i2 B 3:e4 1:e6 E +:h2 +:h3 +:i1 +:i3" | ||
+ | /> | ||
and | and | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f3 g1 g4 h1 4:h4 B 3:e4 1:e6 E +:f2 +:f4 +:g2 +:g3" | ||
+ | /> | ||
− | == Intrusion at F4 == | + | === Intrusion at F4 === |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:d4 g1 h1 B 1:f4" | ||
+ | /> | ||
The Red piece at D4 is connected to the bottom. Blue has two direct attempts to block: | The Red piece at D4 is connected to the bottom. Blue has two direct attempts to block: | ||
− | === Block at F2 === | + | ==== Block at F2 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R d4 g1 2:g3 h1 4:h4 B 1:f2 3:f3 f4" | ||
+ | /> | ||
− | Red is now connected to the bottom via [[ | + | Red is now connected to the bottom via [[ziggurat|template III-1a]]. Note that neither of Red's threats overlapped. |
− | === Block at E3 === | + | ==== Block at E3 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R d4 2:f3 g1 4:g4 h1 10:h4 6:i2 8:i3 B 1:e3 3:e4 f4 5:g3 7:h3 9:h5" | ||
+ | /> | ||
And now Red can escape the ladder: | And now Red can escape the ladder: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R d4 6:d5 4:e5 f3 2:f5 g1 g4 h1 h4 i2 i3 B 5:d6 e3 e4 3:e6 f4 1:f6 g3 h3 h5" | ||
+ | /> | ||
And now Red has connected. | And now Red has connected. | ||
Attempts by Blue to block the use of the D4 piece as a ladder escape can be shown to not work. | Attempts by Blue to block the use of the D4 piece as a ladder escape can be shown to not work. | ||
− | == Intrusion at G2 == | + | === Intrusion at G2 === |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f2 g1 h1 B 1:g2" | ||
+ | /> | ||
Blue has four options that don't immediately reduce to another edge template: | Blue has four options that don't immediately reduce to another edge template: | ||
− | === Block at E4 === | + | ==== Block at E4 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R f2 g1 2:g3 h1 B 1:e4 g2" | ||
+ | /> | ||
Red's G3 piece is connected to the top via F3 or H2. | Red's G3 piece is connected to the top via F3 or H2. | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 4:c5 f2 2:f4 g1 g3 h1 B e4 3:e6 g2 1:g4" | ||
+ | /> | ||
Here Red has created a [[Ladder escape fork]]. If Blue blocks the ladder Red plays at D3. | Here Red has created a [[Ladder escape fork]]. If Blue blocks the ladder Red plays at D3. | ||
− | === Block at D5 === | + | ==== Block at D5 ==== |
− | < | + | |
+ | <hexboard size="6x9" | ||
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:e4 f2 g1 4:g4 h1 6:h2 B 1:d5 3:e5 5:f4 g2" | ||
+ | /> | ||
And Red has connected. If blue choose to play at E6 instead of E5: | And Red has connected. If blue choose to play at E6 instead of E5: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R e4 2:e5 f2 g1 6:g4 4:g5 h1 8:h2 B d5 3:d6 1:e6 7:f4 5:f5 g2" | ||
+ | /> | ||
− | === Block at C6 === | + | ==== Block at C6 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:e4 f2 g1 4:g4 h1 6:h2 B 1:c6 3:e5 5:f4 g2" | ||
+ | /> | ||
− | === Block at E6 === | + | ==== Block at E6 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:e4 4:e5 f2 g1 6:g5 h1 B 3:d5 5:d6 1:e6 g2" | ||
+ | /> | ||
Play continues... | Play continues... | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R e4 e5 f2 g1 2:g4 g5 h1 4:h2 B d5 d6 e6 3:f4 1:f5 g2" | ||
+ | /> | ||
− | == Intrusion at D6 or F6 == | + | === Intrusion at D6 or F6 === |
The D6 case is shown here, but Red's responses work symmetrically for the F6 case. | The D6 case is shown here, but Red's responses work symmetrically for the F6 case. | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f5 g1 h1 B 1:d6 E +:f4 +:g2 +:g3 +:g4 +:h2" | ||
+ | /> | ||
Red's F5 piece is connected to the bottom. To prevent its connection to the top, Blue must move in one of the marked tiles. | Red's F5 piece is connected to the bottom. To prevent its connection to the top, Blue must move in one of the marked tiles. | ||
− | === Block at F4 === | + | ==== Block at F4 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 8:c5 6:e4 4:f3 f5 g1 2:g4 h1 B d6 7:e5 1:f4 5:g3 3:h2" | ||
+ | /> | ||
Or, if for move three Blue played G3: | Or, if for move three Blue played G3: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R f5 g1 g4 10:g5 h1 8:h5 2:i2 4:i3 6:i4 B d6 f4 1:g3 9:g6 3:h3 5:h4 7:h6" | ||
+ | /> | ||
And Red is connected. This method can be used by Red in the symmetrical case of Blue intruding at F6. | And Red is connected. This method can be used by Red in the symmetrical case of Blue intruding at F6. | ||
− | === Block at G2 === | + | ==== Block at G2 ==== |
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 8:c5 6:e4 4:f3 f5 g1 h1 2:h2 B d6 7:e5 5:f4 1:g2 3:g4 E +:f2 +:g3" | ||
+ | /> | ||
Note that Red's F3 piece is connected via the two marked tiles. If Blue had played G3 for move three: | Note that Red's F3 piece is connected via the two marked tiles. If Blue had played G3 for move three: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R f5 g1 h1 2:h2 4:h3 6:h4 B d6 1:g2 3:g3 5:g4" | ||
+ | /> | ||
− | And Red connects via [[ | + | And Red connects via [[ziggurat|template III-1a]]. |
− | === Block at H2, G3, or G4 === | + | ==== Block at H2, G3, or G4 ==== |
Red's responses are similar in all three cases: | Red's responses are similar in all three cases: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:h2" | ||
+ | /> | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:g3" | ||
+ | /> | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:g4" | ||
+ | /> | ||
− | == Intrusion at C6 or G6 == | + | === Intrusion at C6 or G6 === |
The G6 case is shown here, but Red's responses work symmetrically for the C6 case. | The G6 case is shown here, but Red's responses work symmetrically for the C6 case. | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 2:f4 g1 h1 B 1:g6 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f5 +:f6" | ||
+ | /> | ||
− | Red's F4 piece is connected to the bottom via [[ | + | Red's F4 piece is connected to the bottom via [[ziggurat|template III-1a]]. The only move preventing the F4 piece from connecting to the top is G2. Intrusions into the template are met by Red with parallel moves which maintain the connection to the bottom while guaranteeing a connection to the top. F5, F6 and E5 are met by E4 while E4, E6, D5, D6 and C6 are met by G4, connecting Red to both the top and the bottom. |
− | === Block at G2 === | + | ==== Block at G2 ==== |
− | < | + | |
+ | <hexboard size="6x9" | ||
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R f4 g1 h1 2:h2 4:h3 6:h4 B 1:g2 3:g3 5:g4 g6" | ||
+ | /> | ||
And Red cannot be stopped, the F4 piece being a valid ladder escape. If Blue had played E6 for move five: | And Red cannot be stopped, the F4 piece being a valid ladder escape. If Blue had played E6 for move five: | ||
− | < | + | <hexboard size="6x9" |
+ | coords="bottom right" | ||
+ | edges="bottom" | ||
+ | visible="area(a6,f1,i1,i6)" | ||
+ | contents="R 8:d5 f4 g1 6:g4 h1 2:h2 4:h3 B 5:e6 7:f6 1:g2 3:g3 g6" | ||
+ | /> | ||
And if Blue had played F5 for move five: | And if Blue had played F5 for move five: | ||
− | < | + | <hexboard size="6x9" |
− | + | coords="bottom right" | |
− | + | edges="bottom" | |
− | + | visible="area(a6,f1,i1,i6)" | |
− | + | contents="R 6:e4 f4 g1 h1 2:h2 4:h3 B 5:f5 1:g2 3:g3 g6 E +:e3 +:f2 +:f3 +:g4" | |
+ | /> | ||
+ | Red threatens to connect in two non-overlapping ways, while the E4 piece is connected with [[ziggurat|template III-1a]]. | ||
[[Category:Edge templates]] | [[Category:Edge templates]] | ||
− |
Latest revision as of 14:36, 11 May 2023
Edge template IV2a is a 6th row edge template with two stones.
Contents
[hide]Defending the template
Let us first see what possibilities Red has if he moves first.
There are two obvious options:
In both diagrams the possible intrusion points are marked by (+). So we only have to consider the intersection of the intrusion points. They are:
Intrusion at E5 and F5
If Blue blocks at E5 then Red plays F3, reducing to Template IVb
Likewise if blue blocks at F5:
Intrusion at E6
Red threatens to connect via D4. Blue must respond in one of the marked hexes.
The H4 piece is connected to the bottom with template III-1a, and is connected to the top in two non-overlapping ways:
and
Intrusion at F4
The Red piece at D4 is connected to the bottom. Blue has two direct attempts to block:
Block at F2
Red is now connected to the bottom via template III-1a. Note that neither of Red's threats overlapped.
Block at E3
And now Red can escape the ladder:
And now Red has connected. Attempts by Blue to block the use of the D4 piece as a ladder escape can be shown to not work.
Intrusion at G2
Blue has four options that don't immediately reduce to another edge template:
Block at E4
Red's G3 piece is connected to the top via F3 or H2.
Here Red has created a Ladder escape fork. If Blue blocks the ladder Red plays at D3.
Block at D5
And Red has connected. If blue choose to play at E6 instead of E5:
Block at C6
Block at E6
Play continues...
Intrusion at D6 or F6
The D6 case is shown here, but Red's responses work symmetrically for the F6 case.
Red's F5 piece is connected to the bottom. To prevent its connection to the top, Blue must move in one of the marked tiles.
Block at F4
Or, if for move three Blue played G3:
And Red is connected. This method can be used by Red in the symmetrical case of Blue intruding at F6.
Block at G2
Note that Red's F3 piece is connected via the two marked tiles. If Blue had played G3 for move three:
And Red connects via template III-1a.
Block at H2, G3, or G4
Red's responses are similar in all three cases:
Intrusion at C6 or G6
The G6 case is shown here, but Red's responses work symmetrically for the C6 case.
Red's F4 piece is connected to the bottom via template III-1a. The only move preventing the F4 piece from connecting to the top is G2. Intrusions into the template are met by Red with parallel moves which maintain the connection to the bottom while guaranteeing a connection to the top. F5, F6 and E5 are met by E4 while E4, E6, D5, D6 and C6 are met by G4, connecting Red to both the top and the bottom.
Block at G2
And Red cannot be stopped, the F4 piece being a valid ladder escape. If Blue had played E6 for move five:
And if Blue had played F5 for move five:
Red threatens to connect in two non-overlapping ways, while the E4 piece is connected with template III-1a.