Difference between revisions of "Edge template V1b"

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(Continuation after 3rd row ladder: combined three 1-diagram cases into one 2-diagram case)
(Intrusion at x: slightly shortened analysis for one case)
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 R 3:f5 B 2:e6"
+
   contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6"
 
   />
 
   />
  
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<div class="toccolours mw-collapsible  mw-collapsed">
 
<div class="toccolours mw-collapsible  mw-collapsed">
  
Red has several threats. Via [[double threat]]s:
+
Red's main threat is as follows,
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 E *:e5 R f5 B e6 e3 R d3 S area(f2,d6,f6,g4,g2)-e6"
+
   contents="R e2 d4 d3 1:g3 3:e5 5:f5 B c3 d5 e3 c5 2:e6 E *:e4 *:f2 S d6 e4 f3 f2 g2 g4 f4 f5 f6"
 
   />
 
   />
  
Via [[double threat]] and [[edge template III2b]]:
+
connecting back via one of the cells marked "*".
  
<hexboard size="6x14"
+
If Blue plays either of the two of those cells adjacent to the edge, then Red plays the
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 S f2 g2 f3 R g3 R d4 S e4 f4 E *:g4 S h4 B c5 B d5 S e5 R f5 S g5 S h5 B e6 S f6 g6 h6 B e3 R d3 S g3 g4 f5"
+
  />
+
 
+
Via [[bridge]]s and [[Tom's move]]:
+
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 E *:f2 S area(f2,f6,l6,l4,j2)"
+
   contents="R e2 d4 d3 1:g3 3:e5 5:f5 B c3 d5 e3 c5 2:e6 E *:(d6 f6 i4) S blue:(d6 f6)"
 
   />
 
   />
  
Via a [[wheel]] and [[Tom's move]]:
+
same move anyway, connecting down via the other of those two cells or [[Tom's move]].
  
<hexboard size="6x14"
+
If Blue plays one of the five upper-right of those cells
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 E *:e4 S area(f3,e4,e5,f6,l6,l4,j2,h2)"
+
  />
+
 
+
Via a [[double threat]] and [[Tom's move]]:
+
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 E *:f4 S area(f2,f6,l6,l4,j2)+e4-g4"
+
   contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S blue:(f3 f2 g2 g4 f4)"
 
   />
 
   />
  
Via a different [[double threat]] and [[Tom's move]]:
+
, ​ then Red still connects via [[Tom's move]],
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 E *:e5 S area(f2,f6,l6,l4,j2)+e4+e5+d6-f6"
+
   contents="R e2 d4 d3 1:g3 3:e5 5:e4 7:f5 9:i4 B c3 d5 e3 c5 2:e6 6:d6 8:f6 E *:(f3 f4 g4) S blue:(f3 f2 g2 g4 f4)"
  />
+
/>
  
 +
since at least two of the three cells marked "*" are still available to Red 1 to connect to Red's left group.
  
Blue [[mustplay region|must play]] in the overlap:
+
This leaves only two
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 S f3 f4 E r:f3 t:f4"
+
   contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S e4 f5"
 
   />
 
   />
  
If Blue plays at r:
+
remaining tries for Blue.
 +
 
 +
If Blue plays the right of those two,
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 B 1:f3 R 2:e4"
+
   contents="R e2 d4 d3 1:g3 3:e5 5:e4 B c3 d5 e3 c5 2:e6 4:f5 E *:d6 *:h4"
 
   />
 
   />
  
And then, depending on Blue's next move, either a [[crescent]]
+
then Red connects via [[double threat]].
  
<hexboard size="6x14"
+
Thus Blue instead plays the left of those two. ​ Red responds as follows.
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 B 1:f3 R 2:e4 B 3:e5 R 4:f4 B 5:f6 R 6:h5"
+
  />
+
 
+
or [[Tom's move]]:
+
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
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   edges="bottom"
 
   edges="bottom"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
 
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 B 1:f3 R 2:e4 B 3:f6 R 4:e5 B 5:d6 R 6:i4"
+
   contents="R e2 d4 d3 1:g3 3:e5 5:f2 7:f3 B c3 d5 e3 c5 2:e6 4:e4 6:g2 S d6 e6"
 
   />
 
   />
  
If Blue plays at t:
+
Now, it does not matter which of the two shaded cells Blue 2 is at, since
  
<hexboard size="6x14"
+
* if Blue plays the other too then the order in which they were played doesn't matter
  coords="hide"
+
* and
  edges="bottom"
+
* if instead Red plays the other, then whichever one Blue played will be [[Dead cell|dead]]
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 B 1:f4 R 2:e4 E a:e5 b:d6 c:f6"
+
  />
+
  
If Blue plays at b or c, Red plays at a and connects via [[Tom's move]]. If Blue plays at a:
+
, ​ so this is equivalent to Red having a [[Theorems_about_templates#Corner_clipping|clipped]] version of [[Edge template IV1d]].
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 R g3 R d4 B c5 B d5 R f5 B e6 B e3 R d3 B 1:f4 R 2:e4 B 3:e5 R 4:h2"
+
  />
+
  
Now Red is connected left by [[double threat]], and right by a [[Theorems_about_templates#Corner_clipping|clipped]] version of [[Fifth_row_edge_templates#V-2-m|edge template V2m]].
+
In particular, Red is connected down.
  
 
</div>
 
</div>

Revision as of 05:46, 26 July 2022

Edge template V1b is a 5th row edge template with 1 stone.

The validity and minimality of this template has been checked by computer. The template was mentioned on 2016-05-19 by the user shalev in this Little Golem thread, but likely predates that post.


Defense against intrusions

Reduction

Red has 3 main threats. Using the ziggurat:

Using edge template III1b:

And using edge_template_IV1d:

For a blocking attempt, Blue must play in the overlap:

abc

Intrusion at a

If Blue intrudes at a, Red can start by forcing a 3rd or 2nd row ladder like this

21435

or like this:

214368579

Red's continuation will be discussed below.

Intrusion at b

If Blue intrudes at b, Red can start by forcing a 3rd row ladder like this:

43251

Red's continuation will be discussed below.

Intrusion at c

If Blue intrudes at c, Red can start by forcing a 2nd row ladder like this:

4325671

Red's continuation will be discussed below.

Continuation after 3rd row ladder

If Red achieved a 3rd row ladder after intrusions a or b as shown above, Red continues as follows.

1

Now Red is connected by Tom's move for 3rd and 5th row parallel ladders.

Continuation:

 [Expand

Continuation after 2nd row ladder

If Red achieved a 2nd row ladder after intrusions a or c above, Red continues as follows.

312

Now Red connects in essentially the same way as Tom's move.

Continuation:

 [Expand