Difference between revisions of "Edge template VI1a"

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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.
+
Template VI1-a is a 6th row [[edge template]] with one stone.
  
 
<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 B i4"
+
   contents="R j2"
 
/>
 
/>
  
−
Red should move here (or the equivalent mirror-image move at "+"):
+
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.
 +
 
 +
== 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 14: Line 20:
 
   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 i4 j2 S d7 e6 e7 f5 f6 f7 g5 g6 g7 h4 h5 h6 h7 i3 i5 i6 i7 j3 j5 j6 j7"
 
/>
 
/>
−
 
−
== Elimination of irrelevant Blue moves ==
 
−
This gives Red several immediate threats:
 
−
From III1a:
 
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 25: Line 27:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R g5 h3 j2 B i4 E +:e7 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"
+
   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"
 
/>
 
/>
  
−
From III1a again:
+
=== [[edge template IV1b]] ===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 34: Line 36:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:f7 +:g4 +:g6 +:g7 +:h4"
+
   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"
 
/>
 
/>
  
−
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 43: Line 50:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R g5 h3 j2 B i4 E +:d7 +:e6 +:e7 +:f5 +:f6 +:g4 +:g6 +:g7 +:h4 +:h5 +:h6 +:h7"
+
   contents="R 3:h5 5:h6 1:i4 j2 B 4:g7 6:h7 2:i5 S e7 f6 g5"
 
/>
 
/>
  
−
From IV1a:
+
At this point, we can use [[Tom's move]] as follows:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 52: Line 59:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   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"
+
   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"
 
/>
 
/>
  
−
From IV1b:
+
=== [[Overlapping connections|Remaining possibilities]] for Blue ===
−
 
+
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 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"
+
   contents="R j2 S f7 g6 g7 h5 h7 i3 i4 i5 i6 i7 j3"
 
/>
 
/>
  
−
The intersection of all of these leaves:
+
See
−
 
+
[[Template_VI1/Intrusion_on_the_3rd_row]],
−
<hexboard size="7x14"
+
[[Template_VI1/Intrusion_on_the_4th_row]],
−
  coords="full bottom right"
+
[[Template_VI1/The_remaining_intrusion_on_the_fifth_row]].
−
  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 ==
−
So we must deal with each of these responses.  (Which will not be too hard!)
+
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!
−
 
+
−
=== 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 h3 2:h4 4:h5 j2 B 1:g4 3:g6 i4"
+
   contents="R j2 B f7"
 
/>
 
/>
−
And now either
+
 
 +
Details to follow
 +
 
 +
===The other remaining intrusion on the first row===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 90: Line 95:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R h3 h4 h5 j2 2:j5 B g4 g6 1:h6 i4 E +:i5 +:k3"
+
   contents="R j2 B g7"
 
/>
 
/>
  
−
or
+
Red should go here:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 99: Line 104:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   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"
+
   contents="R 1:h5 j2 B g7"
 
/>
 
/>
  
−
=== Bg5 ===
+
See more details [[Template VI1/Other Intrusion on the 1st row| here]].
  
−
<hexboard size="7x14"
+
===The remaining intrusion on the second row (stub)===
−
  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="full bottom right"
+
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 4:d5 f4 h3 j2 B g5 i4 E +:a7 +:b6 +:b7 +:c5 +:c6 +:c7 +:d6 +:d7 +:e4 +:e5"
+
   contents="R j2 B g6"
 
/>
 
/>
  
−
<hexboard size="7x14"
+
===The remaining intrusion on the third row (stub)===
−
  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="full bottom right"
+
   coords="none"
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 4:e5 f4 h3 j2 B g5 i4 E +:b7 +:c6 +:c7 +:d5 +:d6 +:e6 +:e7 +:f5 +:f6 +:f7"
+
   contents="R j2 B h5"
 
/>
 
/>
−
So the only hope for Blue lies in the intersection of the threats, Be5, but it is unsufficient:
 
  
−
<hexboard size="7x14"
+
Red should go here:
−
  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 146: Line 133:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 2:g5 h3 4:h5 j2 B 3:f6 1:g6 i4 E +:e7"
+
   contents="R j2 1:k3 B h5"
 
/>
 
/>
  
−
3 could be played at + with the same effect; in any case
+
Details to follow.
−
now either
+
 
 +
===The remaining intrusion on the fourth row===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 156: Line 144:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R g5 h3 h5 j2 2:j5 B f6 g6 1:h6 i4 E +:i5 +:k3"
+
   contents="R j2 B i4"
 
/>
 
/>
  
−
or
+
Red should move here (or the equivalent mirror-image move at "+"):
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 165: Line 153:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   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"
+
   contents="R h3 j2 B i4 E +:k3"
 
/>
 
/>
  
−
=== Be7 ===
+
For more details, see [[Template VI1/Intrusion on the 4th row|this page]].
−
Either this
+
===The remaining intrusion on the fifth row===
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 175: Line 163:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   contents="R 2:g5 h3 4:h5 j2 6:j5 B 1:e7 3:g6 5:h6 i4 E +:i5 +:k3"
+
   contents="R j2 B i3"
 
/>
 
/>
  
−
or a minor variation
+
First establish a [[double ladder]] on the right.
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 184: Line 172:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   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"
+
   contents="R 7:i4 j2 1:j3 5:j5 3:k4 B 8:h5 i3 2:i5 6:i7 4:k5"
 
/>
 
/>
  
−
=== Bg7 ===
+
Then use [[Tom's move]]:
  
 
<hexboard size="7x14"
 
<hexboard size="7x14"
Line 193: Line 181:
 
   edges="bottom"
 
   edges="bottom"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
 
   visible="area(a7,n7,n5,k2,i2,c5)"
−
   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"
+
   contents="R 3:f4 1:f5 5:h3 i4 j2 j3 j5 k4 B 2:f6 4:g5 h5 i3 i5 i7 k5"
 
/>
 
/>
 +
  
 
[[category:edge templates]]
 
[[category:edge templates]]
 +
[[category:theory]]

Revision as of 02:32, 9 March 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 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

edge template IV1b


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:

132546

At this point, we can use Tom's move as follows:

7561324

Remaining possibilities for Blue

Blue's first move must be one of the following:

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

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)

Details to follow

The other remaining intrusion on the first row

Red should go here:

1

See more details here.

The remaining intrusion on the second row (stub)

The remaining intrusion on the third row (stub)

Red should go here:

1

Details to follow.

The remaining intrusion on the fourth row

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

For more details, see this page.

The remaining intrusion on the fifth row

First establish a double ladder on the right.

17382546

Then use Tom's move:

53142
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