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 handwritten.
+
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
 
 
Pi3 Pj3
 
Pi4
 
Ph5 Pi5
 
Pg6 Pi6
 
Pf7 Pg7 Ph7 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
+
</hex>
+
 
+
===The other remaining intrusion on the first row (stub)===
+
<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
+
 
+
Bg7
+
</hex>
+
 
+
===The remaining intrusion on the second row (stub)===
+
<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
+
 
+
Bg6
+
</hex>
+
 
+
===The remaining intrusion on the third row (stub)===
+
<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
+
</hex>
+
 
+
===The remaining intrusion on the fourth row===
+
<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
+
</hex>
+
 
+
Red should move here:
+
 
+
<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 Rh3
+
</hex>
+
 
+
==== Elimination of irrelevant Blue moves ====
+
This gives Red several immediate threats:
+
From III1a:
+
<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 Rh3
+
Rg5
+
Pg4 Ph4
+
Ph5
+
Pf6 Pg6 Ph6
+
Pe7 Pf7 Pg7 Ph7
+
</hex>
+
 
+
From III1a again:
+
<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 Rh3
+
Rg5
+
Pg4 Ph4
+
Pf5
+
Pe6 Pf6 Pg6
+
Pd7 Pe7 Pf7 Pg7
+
</hex>
+
 
+
From III1b :
+
<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 Rh3
+
Rg5
+
Pg4 Ph4
+
Pf5 Ph5
+
Pe6 Pf6 Pg6 Ph6
+
Pd7 Pe7 Pg7 Ph7
+
</hex>
+
 
+
From IV1a:
+
<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 Rh3
+
Rg4
+
Pf4
+
Pd5 Pe5 Pf5 Pg5 Ph5
+
Pc6 Pd6 Pe6 Pf6 Pg6 Ph6
+
Pb7 Pc7 Pd7 Pe7 Pf7 Pg7 Ph7
+
</hex>
+
 
+
From IV1b:
+
<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 Rh3
+
Rg4
+
Pf4 Ph4
+
Pd5 Pe5 Pf5 Pg5 Ph5 Pi5
+
Pc6 Pd6 Pe6    Pg6 Ph6 Pi6
+
Pb7 Pc7 Pd7 Pe7 Pf7 Pg7 Ph7 Pi7
+
</hex>
+
 
+
The intersection of all of these leaves:
+
<hex>
+
R7 C14 Q1
+
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 Rh3
+
Pg4
+
  Pg5
+
  Pg6
+
Pe7 Pg7
+
</hex>
+
 
+
==== Specific defense ====
+
So we must deal with each of these responses.  (Which will not be too hard!)
+
 
+
===== Bg4 =====
+
<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 Rh3
+
 
+
N:on Bg4 Rh4 Bg6 Rh5
+
</hex>
+
And now either
+
<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 Rh3
+
 
+
Bg4 Rh4 Bg6 Rh5
+
N:on Bh6 Rj5
+
Pk3 Pi5
+
</hex>
+
 
+
or
+
 
+
<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 Rh3
+
 
+
Bg4 Rh4 Bg6 Rh5
+
N:on Bh7 Rh6 Bg7 Rj6 Bi6 Rj5
+
Pk3 Pi5
+
</hex>
+
===== Bg5 =====
+
<hex>
+
R7 C14 Q1
+
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 Rh3
+
 
+
N:on Bg5 Rf4
+
</hex>
+
Threatening:
+
<hex>
+
R7 C14 Q1
+
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 Rh3
+
If Blue intrudes at a, Red has several winning responses. For example, Red can play at 2:
Bg5 Rf4
+
<hexboard size="7x14"
            Pe4
+
  coords="none"
      Pc5 R4d5 Pe5
+
  edges="bottom"
  Pb6 Pc6 Pd6
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Pa7 Pb7 Pc7 Pd7
+
  contents="R j2 B 1:f7 R 2:i4 E x:i5 y:h7"
</hex>
+
/>
<hex>
+
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]].
R7 C14 Q1
+
<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 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"
Sa5 Sb5
+
/>
Sa6
+
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"
 +
/>
  
Bi4 Rh3
+
=== Intrusion at b ===
Bg5 Rf4
+
      Pe5 Pf5
+
      R4e6
+
    Pd7 Pe7
+
</hex>
+
  
<hex>
+
If Blue intrudes at b, Red can respond at 2:
R7 C14 Q1
+
<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 B 1:g7 R 2:i4 E x:i5 y:i6 z:i7 w:h7"
Sa5 Sb5
+
/>
Sa6
+
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]].
  
Bi4 Rh3
+
If Blue intrudes at x, Red can set up a [[parallel ladder]] and connect using [[Tom's move]]:
Bg5 Rf4
+
<hexboard size="7x14"
      Pd5 R4e5 Pf5
+
  coords="none"
  Pc6 Pd6 Pe6 Pf6
+
  edges="bottom"
Pb7 Pc7    Pe7 Pf7
+
  visible="area(a7,n7,n5,k2,i2,c5)"
</hex>
+
  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"
So the only hope for Blue lies in the intersection of the threats, Be5, but it is unsufficient:
+
/>
 +
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"
 +
/>
  
<hex>
+
=== Intrusion at c ===
R7 C14 Q1
+
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 Rh3
+
<hexboard size="7x14"
Bg5 Rf4
+
  coords="none"
N:on Be5 Rf5 Be7 Rf6 Bf7 Rg6 Bg7 Rj5
+
  edges="bottom"
Pk3 Pi5
+
  visible="area(a7,n7,n5,k2,i2,c5)"
</hex>
+
  contents="R j2 B 1:g6"
===== Bg6 =====
+
/>
<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 Rh3
+
Red may play here:
  
N:on Bg6 Rg5 Bf6 Rh5
+
<hexboard size="7x14"
Pe7
+
  coords="none"
</hex>
+
  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"
 +
/>
  
3 could be played at + with the same effect; in any case
+
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.
now either
+
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:
  
<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 2:i5 B 1:g6 R 4:i3 B 5:i4 R 6:h4 B 7:h5  R 8:f5
Sa4 Sb4 Sc4 Sd4 Sn4
+
            S blue:area(g7 m7 m5 l5 l3 k3)
Sa5 Sb5
+
            E a:k2 b:j3 c:k3 d:j4"
Sa6
+
/>
  
Bi4 Rh3
+
(Assume that Blue 3 is played at either a,b,c or d, and there were no extra turns in the blue area.)
  
Bg6 Rg5 Bf6 Rh5
+
=== Intrusion at d ===
N:on Bh6 Rj5
+
Pi5 Pk3
+
</hex>
+
  
or
+
<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
+
  
Bi4 Rh3
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 2:h3 B 1:h5"
 +
/>
  
Bg6 Rg5 Bf6 Rh5
+
Details to follow. See more details [[Template_VI1/Intrusion_on_the_3rd_row|here]].
N:on Bh7 Rh6 Bg7 Rj6 Bi6 Rj5
+
Pk3 Pi5
+
</hex>
+
  
===== Be7 =====
+
=== 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 Rh3
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R j2 B 1:i4"
 +
/>
  
N:on Be7 Rg5 Bg6 Rh5 Bh6 Rj5
+
Red should move here (or the equivalent mirror-image move at "+"):
Pi5 Pk3
+
  
</hex>
+
<hexboard size="7x14"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 2:h3 j2 B 1:i4 E +:k3"
 +
/>
  
===== Bg7 =====
+
Now the shaded area is a [[ladder creation template]], giving Red at least a 3rd row ladder as indicated.
  
<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 h3 j2 B i4 E +:k3 S area(h3,g3,e4,c5,a7,h7) E arrow(3):(h5 h6 h7)"
Sa4 Sb4 Sc4 Sd4 Sn4
+
/>
Sa5 Sb5
+
Sa6
+
  
Bi4 Rh3
+
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 "+":
  
N:on Bg7 Rg5 Bf6 Rh6 Bh7 Rj6 Bi6 Rj5
+
<hexboard size="7x14"
Pi5 Pk3
+
  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"
 +
/>
  
</hex>
+
If Blue [[ladder handling|yields]], or Red starts out with a 2nd row ladder, the escape fork works anyway:
  
===The remaining intrusion on the fifth row===
+
<hexboard size="7x14"
<hex>
+
  coords="none"
R7 C14 Q0
+
  edges="bottom"
1:BBBBBBBBBRBBBBB
+
  visible="area(a7,n7,n5,k2,i2,c5)"
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
+
  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"
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:

214316156

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

8617214315

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.

24310568179

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

26475183

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:

24385617

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:

2843153

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:

1736542

Intrusion at f

1

First establish a parallel ladder on the right.

128493657

Then use Tom's move:

1412101311

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.

124ba