Difference between revisions of "Edge template VI1a"

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(Undo revision 7352: oops, committed wrong page.)
(Copy-editing, inlined intrusion on the 4th row (it doesn't need its own article).)
Line 12: Line 12:
 
== 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]] ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 20: Line 20:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R i4 j2 S d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h6 h7 i3 i5 i6 i7 j3 j5 j6 j7"
+
   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"
 
/>
 
/>
  
Line 27: Line 27:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R i4 j2 S e7 f6 f7 g5 g6 g7 h5 h6 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
+
   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"
 
/>
 
/>
  
−
=== [[edge template IV1b]] ===
+
=== [[Edge template IV1b]] ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 36: Line 36:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R i4 j2 S d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h7 i3 i5 i6 i7 j3 j4 j5 j6 j7 k5 k6 k7"
+
   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"
 
/>
 
/>
  
  
−
=== Using the [[parallel ladder]] trick ===
+
=== Using [[Tom's move]] ===
  
−
6 moves can furthermore be discarded thanks to the [[Parallel ladder]] trick. Of course, symmetry will cut our work in half!
+
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!
  
−
We can dispose of 3 moves on the left (and, using mirror symmetry, the corresponding 3 moves on the right), as follows:
+
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:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 50: Line 50:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 3:h5 5:h6 1:i4 j2 B 4:g7 6:h7 2:i5 S e7 f6 g5"
+
   contents="R 4:h5 6:h6 2:i4 j2 B 5:g7 6:h7 3:i5 B 1:(e7 f6 g5)"
 
/>
 
/>
  
−
At this point, we can use [[Tom's move]] as follows:
+
At this point, Red can use [[Tom's move]] to connect:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 59: Line 59:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R h5 h6 i4 3:i6 j2 7:k3 1:k5 5:l4 B g7 h7 i5 4:i7 6:j5 2:j6 S e7 f6 g5"
+
   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)"
 
/>
 
/>
  
−
=== [[Overlapping connections|Remaining possibilities]] for Blue ===
+
=== Remaining intrusions ===
−
Blue's first move must be one of the following:
+
 
 +
The only possible remaining intrusions for Blue are the following:
 
<hexboard size="7x14"
 
<hexboard size="7x14"
 
   coords="none"
 
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 S f7 g6 g7 h5 h7 i3 i4 i5 i6 i7 j3"
+
   contents="R j2  
 +
            S f7 g6 g7 h5 h7 i3 i4 i5 i6 i7 j3
 +
            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.
−
See
+
−
[[Template_VI1/Intrusion_on_the_3rd_row]],
+
−
[[Template_VI1/Intrusion_on_the_4th_row]],
+
−
[[Template_VI1/The_remaining_intrusion_on_the_fifth_row]].
+
  
 
== Specific defense ==
 
== 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 (stub) ===
 
<hexboard size="7x14"
 
<hexboard size="7x14"
 
   coords="none"
 
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B f7"
+
   contents="R j2 B 1:f7"
 
/>
 
/>
  
 
Details to follow
 
Details to follow
  
−
===The other remaining intrusion on the first row===
+
=== Intrusion at b (stub) ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 95: Line 95:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B g7"
+
   contents="R j2 B 1:g7"
 
/>
 
/>
  
Line 104: Line 104:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 1:h5 j2 B g7"
+
   contents="R 2:h5 j2 B 1:g7"
 
/>
 
/>
  
 
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
 
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
  
−
===The remaining intrusion on the second row (stub)===
+
=== Intrusion at c (stub) ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 115: Line 115:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B g6"
+
   contents="R j2 B 1:g6"
 
/>
 
/>
  
−
===The remaining intrusion on the third row (stub)===
+
=== Intrusion at d (stub)===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 124: Line 124:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B h5"
+
   contents="R j2 B 1:h5"
 
/>
 
/>
  
Line 133: Line 133:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 1:k3 B h5"
+
   contents="R j2 2:k3 B 1:h5"
 
/>
 
/>
  
 
Details to follow.
 
Details to follow.
  
−
===The remaining intrusion on the fourth row===
+
=== Intrusion at e ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 144: Line 144:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B i4"
+
   contents="R j2 B 1:i4"
 
/>
 
/>
  
Line 153: Line 153:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R h3 j2 B i4 E +:k3"
+
   contents="R 2:h3 j2 B 1:i4 E +:k3"
 +
/>
 +
 
 +
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"
 +
/>
 +
 
 +
If Blue [[ladder handling|yields]], or Red starts out with a 2nd row ladder, the escape fork works anyway:
 +
 
 +
<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:h7 R 3:h6 B 4:g7 R 5:j6 B 6:i6 R 7:j5 E +:i5"
 
/>
 
/>
  
−
For more details, see [[Template VI1/Intrusion on the 4th row|this page]].
+
=== Intrusion at f ===
−
===The remaining intrusion on the fifth row===
+
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 163: Line 189:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2 B i3"
+
   contents="R j2 B 1:i3"
 
/>
 
/>
  
−
First establish a [[double ladder]] on the right.
+
First establish a [[parallel ladder]] on the right.
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 172: Line 198:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 7:i4 j2 1:j3 5:j5 3:k4 B 8:h5 i3 2:i5 6:i7 4:k5"
+
   contents="R 8:i4 j2 2:j3 6:j5 4:k4 B 9:h5 1:i3 3:i5 7:i7 5:k5"
 
/>
 
/>
  
Line 181: Line 207:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 3:f4 1:f5 5:h3 i4 j2 j3 j5 k4 B 2:f6 4:g5 h5 i3 i5 i7 k5"
+
   contents="R 12:f4 10:f5 14:h3 i4 j2 j3 j5 k4 B 11:f6 13:g5 h5 i3 i5 i7 k5"
 
/>
 
/>
  

Revision as of 01:37, 30 April 2021

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 (stub)

1

Details to follow

Intrusion at b (stub)

1

Red should go here:

21

See more details here.

Intrusion at c (stub)

1

Intrusion at d (stub)

1

Red should go here:

21

Details to follow.

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
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