Difference between revisions of "Edge template VI1a"

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This template is the first one stone 6th row [[edge template|template]] for which a proof has been written out.
+
Template VI1-a is a 6th row [[edge template]] with one stone.
  
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R j2"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
 
Sa6
+
This template is the first one stone 6th row [[edge template|template]] for which a proof of validity has been written out. The template has been verified by computer, and also verified to be minimal.
</hex>
+
  
 
== Elimination of irrelevant Blue moves ==
 
== Elimination of irrelevant Blue moves ==
  
Red has a couple of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.
+
Red has a number of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.
  
=== [[edge template IV1a]] ===
+
=== [[Edge template IV1a]] ===
  
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R i4 j2 S i4 d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h6 h7 i3 i5 i6 i7 j3 j5 j6 j7"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
Sa6
+
  
Pi3 Pj3
+
<hexboard size="7x14"
Ph4 Ri4
+
  coords="none"
Pf5 Pg5 Ph5 Pi5 Pj5
+
  edges="bottom"
Pe6 Pf6 Pg6 Ph6 Pi6 Pj6
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Pd7 Pe7 Pf7 Pg7 Ph7 Pi7 Pj7
+
  contents="R i4 j2 S i4 e7 f6 f7 g5 g6 g7 h5 h6 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
</hex>
+
/>
  
<hex>
+
=== [[Edge template IV1b]] ===
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Pi3 Pj3
+
<hexboard size="7x14"
Ri4 Pj4
+
  coords="none"
Pg5 Ph5 Pi5 Pj5 Pk5
+
  edges="bottom"
Pf6 Pg6 Ph6 Pi6 Pj6 Pk6
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Pe7 Pf7 Pg7 Ph7 Pi7 Pj7 Pk7
+
  contents="R i4 j2 S i4 d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
</hex>
+
/>
  
=== [[edge template IV1b]] ===
 
  
<hex>
+
=== Using [[Tom's move]] ===
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Pi3 Pj3
+
6 intrusions can furthermore be discarded thanks to [[Tom's move]], also known as the [[parallel ladder]] trick. Of course, symmetry will cut our work in half!
Ph4 Ri4 Pj4
+
Pf5 Pg5 Ph5 Pi5 Pj5 Pk5
+
Pe6 Pf6 Pg6 Pi6 Pj6 Pk6
+
Pd7 Pe7 Pf7 Pg7 Ph7 Pi7 Pj7 Pk7
+
</hex>
+
=== Using the [[parallel ladder]] trick ===
+
  
6 moves can furthermore be discarded thanks to the [[Parallel ladder]] trick.  Of course, symmetry will cut our work in half!
+
If Blue moves in any of the cells marked "1" on the left (and, using mirror symmetry, in the corresponding 3 cells on the right), Red can respond as follows:
  
We can dispose of 3 moves on the left (and, using mirror symmetry, the corresponding 3 moves on the right), as follows:
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 4:h5 6:h6 2:i4 j2 B 5:g7 6:h7 3:i5 B 1:(e7 f6 g5)"
 +
/>
  
<hex>
+
At this point, Red can use [[Tom's move]] to connect:
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Pg5
+
<hexboard size="7x14"
Pf6
+
  coords="none"
Pe7
+
  edges="bottom"
N:on Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
+
  visible="area(a7,n7,n5,k2,i2,c5)"
</hex>
+
  contents="R h5 h6 i4 4:i6 j2 8:k3 2:k5 6:l4 B g7 h7 i5 5:i7 7:j5 3:j6 B 1:(e7 f6 g5)"
 +
/>
  
At this point, we can use the [[Parallel ladder]] trick as follows:
+
=== Remaining intrusions ===
  
<hex>
+
The only possible remaining intrusions for Blue are the following:
R7 C14 Q0
+
<hexboard size="7x14"
1:BBBBBBBBBRBBBBB
+
  coords="none"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  edges="bottom"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa4 Sb4 Sc4 Sd4 Sn4
+
  contents="R j2
Sa5 Sb5
+
            S f7 g6 g7 h5 h7 i3 i4 i5 i6 i7 j3
Sa6
+
            E a:f7 b:g7 c:g6 d:h5 e:i4 f:i3"
 +
/>
 +
By symmetry, if is sufficient to consider the six possible intrusions at a &ndash; f.
  
Pg5
+
== Specific defense ==
Pf6
+
Pe7
+
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
+
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
+
</hex>
+
  
=== [[Overlapping connections|Remaining possibilities]] for Blue ===
 
Blue's first move must be one of the following:
 
<hex>
 
R7 C14 Q0
 
1:BBBBBBBBBRBBBBB
 
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
 
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
 
Sa4 Sb4 Sc4 Sd4 Sn4
 
Sa5 Sb5
 
Sa6
 
 
[[#The_remaining_intrusion_on_the_fifth_row|Pi3]] [[#The_remaining_intrusion_on_the_fifth_row|Pj3]]
 
[[Template_VI1/Intrusion_on_the_4th_row|Pi4]]
 
[[Template_VI1/Intrusion_on_the_3rd_row|Ph5]]
 
[[Template_VI1/Intrusion_on_the_3rd_row|Pi5]]
 
[[#The_remaining_intrusion_on_the_second_row_.28stub.29|Pg6]] [[#The_remaining_intrusion_on_the_second_row_.28stub.29|Pi6]]
 
[[#One_remaining_intrusion_on_the_first_row_.28stub.29|Pf7]]
 
[[#The_other_remaining_intrusion_on_the_first_row_.28stub.29|Pg7]]
 
[[#The_other_remaining_intrusion_on_the_first_row_.28stub.29|Ph7]] [[#One_remaining_intrusion_on_the_first_row_.28stub.29|Pi7]]
 
</hex>
 
 
== Specific defense ==
 
 
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!
 
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!
  
===One remaining intrusion on the first row (stub) ===
+
=== Intrusion at a ===
<hex>
+
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bf7
+
If Blue intrudes at a, Red has several winning responses. For example, Red can play at 2:
</hex>
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:f7 R 2:i4 E x:i5 y:h7"
 +
/>
 +
Apart from intrusion into the bridge, which Red defends, Blue's only possible moves are at x and y. If Blue plays at x,  Red can set up a [[parallel ladder]] and connect using [[Tom's move]].
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:f7 R 2:i4 B 3:i5 R 4:g5 B 5:e6 R 6:g6 B 7:g7 R 8:h6 B 9:h7 R 10:k5"
 +
/>
 +
If Blue plays at y, Red has the following simple win, using the [[trapezoid]] template:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:f7 R 2:i4 B 3:h7 R 4:i5 B 5:j6 R 6:g5 B 7:e6 R 8:g7"
 +
/>
  
Details to follow
+
=== Intrusion at b ===
  
===The other remaining intrusion on the first row (stub)===
+
If Blue intrudes at b, Red can respond at 2:
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R j2 B 1:g7 R 2:i4 E x:i5 y:i6 z:i7 w:h7"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
Apart from intrusions into the bridge, which Red defends, Blue has only four possible moves x,y,z,w, because if Blue moves anywhere else, Red connects with either the [[ziggurat]] or [[edge template III1b]].
Sa6
+
  
Bg7
+
If Blue intrudes at x, Red can set up a [[parallel ladder]] and connect using [[Tom's move]]:
</hex>
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 B 3:i5 R 4:h5 B 5:f6 R 6:h6 B 7:h7 R 8:k5"
 +
/>
 +
If Blue intrudes at y or z (both shown simultaneously in the following diagram), Red can set up Tom's move on the opposite side:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 B 3:(i6 i7) R 4:h6 B 5:h7 R 8:f5"
 +
/>
 +
Finally, if Blue intrudes at w, Red can connect by the following variant of Tom's move:
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:g7 R 2:i4 B 3:h7 R 4:h5 B 5:f6 R 6:k5"
 +
/>
  
Red should go here:
+
=== Intrusion at c ===
  
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R j2 B 1:g6"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
Sa6
+
  
Bg7  MR Mh5
+
Red may play here:
</hex>
+
  
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:i5 B 1:g6
 +
            S blue:area(g7 m7 m5 l5 l3 k3)
 +
            E a:k2 b:j3 c:k3 d:j4"
 +
/>
  
===The remaining intrusion on the second row (stub)===
+
Note that if Red plays at c, then in the blue area there is a strategy such that both Red 2 and c connect down without choice, unless Blue first plays at d. Also, with this strategy the paths for 2 connecting down would not pass c or d in any case.
<hex>
+
Therefore, Blue must spend one move at either a,b,c or d in order to block Red on the right side, while Red 2 is always guaranteed to connect down. Thus we have this forcing sequence:
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bg6
+
<hexboard size="7x14"
</hex>
+
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:i5 B 1:g6 R 4:i3 B 5:i4 R 6:h4 B 7:h5  R 8:f5
 +
            S blue:area(g7 m7 m5 l5 l3 k3)
 +
            E a:k2 b:j3 c:k3 d:j4"
 +
/>
  
===The remaining intrusion on the third row (stub)===
+
(Assume that Blue 3 is played at either a,b,c or d, and there were no extra turns in the blue area.)
<hex>
+
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bh5
+
=== Intrusion at d ===
</hex>
+
  
Red should go here:
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:h5"
 +
/>
  
<hex>
+
Red may go here:
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bh5  MR Mi5
+
<hexboard size="7x14"
</hex>
+
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:h3 B 1:h5"
 +
/>
  
See more details [[Template VI1/Intrusion on the 3rd row| here]].
+
Details to follow. See more details [[Template_VI1/Intrusion_on_the_3rd_row|here]].
  
===The remaining intrusion on the fourth row===
+
=== Intrusion at e ===
<hex>
+
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bi4
+
<hexboard size="7x14"
</hex>
+
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:i4"
 +
/>
  
 
Red should move here (or the equivalent mirror-image move at "+"):
 
Red should move here (or the equivalent mirror-image move at "+"):
  
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R 2:h3 j2 B 1:i4 E +:k3"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
 
Sa6
+
Now the shaded area is a [[ladder creation template]], giving Red at least a 3rd row ladder as indicated.
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) E arrow(3):(h5 h6 h7)"
 +
/>
 +
 
 +
Red can escape both 2nd and 3rd row ladders using a [[ladder escape fork]] via "+". Specifically, Red escapes a third row ladder like this, and is connected by a [[ziggurat]] and double threat at "+":
 +
 
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) R 1:h5 B 2:h6 R 3:j5 E +:i5"
 +
/>
  
Bi4 Rh3 Pk3
+
If Blue [[ladder handling|yields]], or Red starts out with a 2nd row ladder, the escape fork works anyway:
</hex>
+
  
For more details, see [[Template VI1/Intrusion on the 4th row|this page]].
+
<hexboard size="7x14"
===The remaining intrusion on the fifth row===
+
  coords="none"
<hex>
+
  edges="bottom"
R7 C14 Q0
+
  visible="area(a7,n7,n5,k2,i2,c5)"
1:BBBBBBBBBRBBBBB
+
  contents="R h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) R 1:h5 B 2:h7 R 3:h6 B 4:g7 R 5:j6 B 6:i6 R 7:j5 E +:i5"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
/>
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bi3
+
=== Intrusion at f ===
</hex>
+
  
First establish a [[double ladder]] on the right.
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:i3"
 +
/>
  
<hex>
+
First establish a [[parallel ladder]] on the right.
R7 C14 Q0
+
1:BBBBBBBBBRBBBBB
+
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
Sa4 Sb4 Sc4 Sd4 Sn4
+
Sa5 Sb5
+
Sa6
+
  
Bi3
+
<hexboard size="7x14"
N:on Rj3 Bi5 Rk4 Bk5 Rj5 Bi7 Ri4 Bh5
+
  coords="none"
</hex>
+
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 8:i4 j2 2:j3 6:j5 4:k4 B 9:h5 1:i3 3:i5 7:i7 5:k5"
 +
/>
  
 
Then use [[Tom's move]]:
 
Then use [[Tom's move]]:
  
<hex>
+
<hexboard size="7x14"
R7 C14 Q0
+
  coords="none"
1:BBBBBBBBBRBBBBB
+
  edges="bottom"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
+
  contents="R 12:f4 10:f5 14:h3 i4 j2 j3 j5 k4 B 11:f6 13:g5 h5 i3 i5 i7 k5"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
Sa6
+
  
Bi3
+
There are two marginal cases. If Blue 3 blocks at a or b, then Red 4 can reduce the situation to "Intrusion at a" or "Intrusion at b" as before, since intruding the vertical bridge is irrelevant in these two cases.
Rj3 Bi5 Rk4 Bk5 Rj5 Bi7 Ri4 Bh5 Rf5 Bf6
+
N:on Rf5 Bf6 Rf4 Bg5 Rh3
+
</hex>
+
  
 +
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:j3 4:i4 B 1:i3 E a:i7 b:g7"
 +
/>
  
 
[[category:edge templates]]
 
[[category:edge templates]]
 
[[category:theory]]
 
[[category:theory]]

Latest revision as of 05:15, 10 May 2024

Template VI1-a is a 6th row edge template with one stone.

This template is the first one stone 6th row template for which a proof of validity has been written out. The template has been verified by computer, and also verified to be minimal.

Elimination of irrelevant Blue moves

Red has a number of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.

Edge template IV1a

Edge template IV1b


Using Tom's move

6 intrusions can furthermore be discarded thanks to Tom's move, also known as the parallel ladder trick. Of course, symmetry will cut our work in half!

If Blue moves in any of the cells marked "1" on the left (and, using mirror symmetry, in the corresponding 3 cells on the right), Red can respond as follows:

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At this point, Red can use Tom's move to connect:

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Remaining intrusions

The only possible remaining intrusions for Blue are the following:

fedcab

By symmetry, if is sufficient to consider the six possible intrusions at a – f.

Specific defense

For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!

Intrusion at a

If Blue intrudes at a, Red has several winning responses. For example, Red can play at 2:

2x1y

Apart from intrusion into the bridge, which Red defends, Blue's only possible moves are at x and y. If Blue plays at x, Red can set up a parallel ladder and connect using Tom's move.

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If Blue plays at y, Red has the following simple win, using the trapezoid template:

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Intrusion at b

If Blue intrudes at b, Red can respond at 2:

2xy1wz

Apart from intrusions into the bridge, which Red defends, Blue has only four possible moves x,y,z,w, because if Blue moves anywhere else, Red connects with either the ziggurat or edge template III1b.

If Blue intrudes at x, Red can set up a parallel ladder and connect using Tom's move:

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If Blue intrudes at y or z (both shown simultaneously in the following diagram), Red can set up Tom's move on the opposite side:

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Finally, if Blue intrudes at w, Red can connect by the following variant of Tom's move:

246513

Intrusion at c

1

Red may play here:

abcd21

Note that if Red plays at c, then in the blue area there is a strategy such that both Red 2 and c connect down without choice, unless Blue first plays at d. Also, with this strategy the paths for 2 connecting down would not pass c or d in any case. Therefore, Blue must spend one move at either a,b,c or d in order to block Red on the right side, while Red 2 is always guaranteed to connect down. Thus we have this forcing sequence:

a4bc65d8721

(Assume that Blue 3 is played at either a,b,c or d, and there were no extra turns in the blue area.)

Intrusion at d

1

Red may go here:

21

Details to follow. See more details here.

Intrusion at e

1

Red should move here (or the equivalent mirror-image move at "+"):

21

Now the shaded area is a ladder creation template, giving Red at least a 3rd row ladder as indicated.

Red can escape both 2nd and 3rd row ladders using a ladder escape fork via "+". Specifically, Red escapes a third row ladder like this, and is connected by a ziggurat and double threat at "+":

132

If Blue yields, or Red starts out with a 2nd row ladder, the escape fork works anyway:

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Intrusion at f

1

First establish a parallel ladder on the right.

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Then use Tom's move:

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There are two marginal cases. If Blue 3 blocks at a or b, then Red 4 can reduce the situation to "Intrusion at a" or "Intrusion at b" as before, since intruding the vertical bridge is irrelevant in these two cases.

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