Difference between revisions of "Edge template VI1a"

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Template VI1-a is a 6th row [[edge template]] with one stone.
+
This page is devoted to details on how to [[Defending against intrusions in template VI1|defend against intrusions in template VI]]. This page explores what are the possibilities for Red to defend the template when Blue intrude on the 4th row.
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 5: Line 5:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R j2"
+
   contents="R j2 B i4"
 
/>
 
/>
  
−
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.
+
Red should move here (or the equivalent mirror-image move at "+"):
−
 
+
−
== 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.
+
−
 
+
−
=== [[edge template IV1a]] ===
+
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 20: Line 14:
 
   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 h3 j2 B i4 E +:k3"
 
/>
 
/>
 +
 +
== Elimination of irrelevant Blue moves ==
 +
This gives Red several immediate threats:
 +
From III1a:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 27: Line 25:
 
   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 g5 h3 j2 B i4 E +:e7 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"
 
/>
 
/>
  
−
=== [[edge template IV1b]] ===
+
From III1a again:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 36: Line 34:
 
   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 g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4"
 
/>
 
/>
  
−
 
+
From III1b :
−
=== 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!
+
−
 
+
−
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"
 
<hexboard size="7x14"
Line 50: Line 43:
 
   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 g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"
 
/>
 
/>
  
−
At this point, we can use [[Tom's move]] as follows:
+
From IV1a:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 59: Line 52:
 
   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 g4 h3 j2 B i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:d7 +:e5 +:e6 +:e7 +:f4 +:f5 +:f6 +:f7 +:g5 +:g6 +:g7 +:h5 +:h6 +:h7"
 
/>
 
/>
  
−
=== [[Overlapping connections|Remaining possibilities]] for Blue ===
+
From IV1b:
−
Blue's first move must be one of 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 g4 h3 j2 B i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:d7 +:e5 +:e6 +:e7 +:f4 +:f5 +:f7 +:g5 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7 +:i5 +:i6 +:i7"
 
/>
 
/>
  
−
See
+
The intersection of all of these leaves:
−
[[Template_VI1/Intrusion_on_the_3rd_row]],
+
 
−
[[Template_VI1/Intrusion_on_the_4th_row]],
+
<hexboard size="7x14"
−
[[Template_VI1/The_remaining_intrusion_on_the_fifth_row]].
+
  coords="full bottom right"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R h3 j2 B i4 E +:e7 +:g4 +:g5 +:g6 +:g7"
 +
/>
  
 
== 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!
+
So we must deal with each of these responses.  (Which will not be too hard!)
 +
 
 +
=== Bg4 ===
  
−
===One remaining intrusion on the first row (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 h3 2:h4 4:h5 j2 B 1:g4 3:g6 i4"
 
/>
 
/>
−
 
+
And now either
−
Details to follow
+
−
 
+
−
===The other remaining intrusion on the first row===
+
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 95: Line 90:
 
   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 h3 h4 h5 j2 2:j5 B g4 g6 1:h6 i4 E +:i5 +:k3"
 
/>
 
/>
  
−
Red should go here:
+
or
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 104: Line 99:
 
   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 h3 h4 h5 2:h6 j2 6:j5 4:j6 B g4 g6 3:g7 1:h7 i4 5:i6 E +:i5 +:k3"
 
/>
 
/>
  
−
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
+
=== Bg5 ===
  
−
===The remaining intrusion on the second row (stub)===
+
<hexboard size="7x14"
 +
  coords="full bottom right"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 2:f4 h3 j2 B 1:g5 i4"
 +
/>
 +
Threatening:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
−
   coords="none"
+
   coords="full bottom right"
 
   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 4:d5 f4 h3 j2 B g5 i4 E +:a7 +:b6 +:b7 +:c5 +:c6 +:c7 +:d6 +:d7 +:e4 +:e5"
 
/>
 
/>
  
−
===The remaining intrusion on the third row (stub)===
+
<hexboard size="7x14"
 +
  coords="full bottom right"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R 4:e6 f4 h3 j2 B g5 i4 E +:d7 +:e5 +:e7 +:f5"
 +
/>
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
−
   coords="none"
+
   coords="full bottom right"
 
   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 4:e5 f4 h3 j2 B g5 i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:e6 +:e7 +:f5 +:f6 +:f7"
 
/>
 
/>
 +
So the only hope for Blue lies in the intersection of the threats, Be5, but it is unsufficient:
  
−
Red should go here:
+
<hexboard size="7x14"
 +
  coords="full bottom right"
 +
  edges="bottom"
 +
  visible="area(a7,n7,n5,k2,i2,c5)"
 +
  contents="R f4 2:f5 4:f6 6:g6 h3 j2 8:j5 B 1:e5 3:e7 5:f7 g5 7:g7 i4 E +:i5 +:k3"
 +
/>
 +
=== Bg6 ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 133: Line 146:
 
   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 2:g5 h3 4:h5 j2 B 3:f6 1:g6 i4 E +:e7"
 
/>
 
/>
  
−
Details to follow.
+
3 could be played at + with the same effect; in any case
−
 
+
now either
−
===The remaining intrusion on the fourth row===
+
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 144: Line 156:
 
   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 g5 h3 h5 j2 2:j5 B f6 g6 1:h6 i4 E +:i5 +:k3"
 
/>
 
/>
  
−
Red should move here (or the equivalent mirror-image move at "+"):
+
or
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 153: Line 165:
 
   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 g5 h3 h5 2:h6 j2 6:j5 4:j6 B f6 g6 3:g7 1:h7 i4 5:i6 E +:i5 +:k3"
 
/>
 
/>
  
−
For more details, see [[Template VI1/Intrusion on the 4th row|this page]].
+
=== Be7 ===
−
===The remaining intrusion on the fifth row===
+
Either this
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 163: Line 175:
 
   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 2:g5 h3 4:h5 j2 6:j5 B 1:e7 3:g6 5:h6 i4 E +:i5 +:k3"
 
/>
 
/>
  
−
First establish a [[double ladder]] on the right.
+
or a minor variation
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 172: Line 184:
 
   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 2:g5 h3 4:h5 6:h6 j2 10:j5 8:j6 B 1:e7 3:g6 7:g7 5:h7 i4 9:i6 E +:i5 +:k3"
 
/>
 
/>
  
−
Then use [[Tom's move]]:
+
=== Bg7 ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 181: Line 193:
 
   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 2:g5 h3 4:h6 j2 8:j5 6:j6 B 3:f6 1:g7 5:h7 i4 7:i6 E +:i5 +:k3"
 
/>
 
/>
−
 
  
 
[[category:edge templates]]
 
[[category:edge templates]]
−
[[category:theory]]
 

Revision as of 02:31, 9 March 2021

This page is devoted to details on how to defend against intrusions in template VI. This page explores what are the possibilities for Red to defend the template when Blue intrude on the 4th row.

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

Elimination of irrelevant Blue moves

This gives Red several immediate threats: From III1a:

From III1a again:

From III1b :

From IV1a:

From IV1b:

The intersection of all of these leaves:

abcdefghijklmn234567

Specific defense

So we must deal with each of these responses. (Which will not be too hard!)

Bg4

1243

And now either

21

or

625431

Bg5

abcdefghijklmn23456721

Threatening:

abcdefghijklmn2345674
abcdefghijklmn2345674
abcdefghijklmn2345674

So the only hope for Blue lies in the intersection of the threats, Be5, but it is unsufficient:

abcdefghijklmn23456712846357

Bg6

2431

3 could be played at + with the same effect; in any case now either

21

or

625431

Be7

Either this

246351

or a minor variation

24103698175

Bg7

28347615
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