Difference between revisions of "Edge template V1b"

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(Intrusion at x: slightly shortened analysis for one case)
(Added "see also", and some copy-editing.)
 
(3 intermediate revisions by the same user not shown)
Line 106: Line 106:
 
   />
 
   />
  
Now Red is connected by [[Tom's_move#Tom.27s_move_for_3rd-and-5th_row_parallel_ladders|Tom's move for 3rd and 5th row parallel ladders]].
+
Now Red is connected by [[Tom's move for 3rd and 5th row parallel ladders]].
 
+
Continuation:
+
 
+
<div class="toccolours mw-collapsible  mw-collapsed">
+
 
+
Now Red has two main threats. Via a [[ziggurat]]:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 E *:f4 S f2 g2 f3 e4 area(d6,f4,g4,g6)"
+
  />
+
And via [[edge template IV2b]]:
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 E *:f4 S f2 g2 f3 e4 area(e6,i6,i4,h3,f4)"
+
  />
+
 
+
Blue must play in the overlap:
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 S f2 g2 f3 e4 area(e6,f4,g4,g6)
+
            E p:f2 q:g2 r:f3 s:e4 t:f4 u:g4 v:f5 w:g5 x:e6 y:f6 z:g6"
+
  />
+
 
+
 
+
==== Intrusion at p, q, r ====
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:(f3 f2 g2) R 3:e4 S d6 e5 e6"
+
  />
+
 
+
Now Blue must play in one of the 3 shaded cells. ​If Blue plays in the left 2 of those 3, then Red connects via [[Edge_template_IV2b|IV-2-b]]. ​ Otherwise, Red connects via [[Tom's move]].
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:(f3 f2 g2) R 3:e4 5:e5 7:f5 9:i4 B 2:(f3 f2 g2) 4:e6 6:d6 8:f6"
+
  />
+
 
+
==== Intrusion at s ====
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:e4 R 3:f2"
+
  />
+
 
+
Red is connected by [[edge template IV1d]].
+
 
+
==== Intrusion at t ====
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:f4 R 3:e4 B 4:e5 R 5:h2"
+
  />
+
 
+
Now Red is connected by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. If Blue plays 4 on the first row instead, Red connects by [[Tom's move]]:
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:f4 R 3:e4 B 4:e6 R 5:e5 B 6:d6 R 7:f5 B 8:f6 R 9:i4"
+
  />
+
 
+
==== Intrusion at u ====
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g4 R 3:f4 B 4:e6 R 5:f5 B 6:f6 R 7:i4"
+
  />
+
Red is connected by [[Tom's move]].
+
 
+
==== Intrusion at w ====
+
 
+
Red responds with
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
Continuation:
+
<div class="toccolours mw-collapsible  mw-collapsed">
+
 
+
We will assume that Red simply defends the pink [[ziggurat]], and therefore we will not need to consider any Blue intrusions there.
+
 
+
Red's main threats are
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 B 2:g5 S d6 e6 e5 e4 f4 f3 h3 h4 i3 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
and
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:f2 B 2:g5 S f2 g2 e6 f6 f5 f4 g4 f3 h3 h4 i3 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
Blue [[mustplay region|must play]] in the overlap:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 B 2:g5 S e6 f4 f3 h3 h4 i3 E c:e6 b:f4 a:f3 d:h3 f:h4 e:i3 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
If Blue plays at a, then Red plays 5, after which Red connects via
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:f4 B 2:g5 4:f3 6:e5 6:d6 E *:f5 *:h3 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
or
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:e5 9:f5 11:h3 B 2:g5 4:f3 6:e6 8:d6 10:f6 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
If Blue plays at b, then Red plays 5, after which Red connects via
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:f4 R 5:e4 B 6:(e5 d6) R 7:h2 B 8:h3 R 9:g4 S red:area(i4,g6,j6,j4) E *:(g1 f3 f5 h4)"
+
  />
+
 
+
or
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:e5 9:f5 11:h3 B 2:g5 4:f4 6:e6 8:d6 10:f6 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
If Blue plays at c, then Red responds with
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 B 2:g5 4:e6 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
This forces Blue to defend towards the top, after which
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 7:e4 B 2:g5 4:e6 6:(f2 f3 g2) S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
connects directly or via red 1.
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 7:e4 9:e5 11:f5 B 2:g5 4:e6 6:(f2 f3 g2) 8:f4 10:d6 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
If Blue plays at d or e, then Red plays like this:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:(h3 i3) R 5:e5 B 6:e4 R 7:f2 B 8:f4 R 9:g4 E *:(f5 h4) S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
If Blue plays at f, then Red plays like this:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:h4 R 5:f5 E *:(f2 e4) +:f4 S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
Now Red threatens to play in one of the cells marked "*", and the only overlap (apart from the two [[captured cell]]s below 5) is at "+", so Blue must play there. Then Red responds like this:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:h4 R 5:f5 B 6:f4 R 7:h2 E *:(g1 e4) +:(g4 i3) S red:area(i4,g6,j6,j4)"
+
  />
+
 
+
Now Red is connected left by double threats "*" and right by double threats "+".
+
 
+
</div>
+
 
+
==== Intrusion at x ====
+
 
+
Red can respond here:
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6"
+
  />
+
 
+
Continuation:
+
<div class="toccolours mw-collapsible  mw-collapsed">
+
 
+
Red's main threat is as follows,
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  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"
+
  />
+
 
+
connecting back via one of the cells marked "*".
+
 
+
If Blue plays either of the two of those cells adjacent to the edge, then Red plays the
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  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)"
+
  />
+
 
+
same move anyway, connecting down via the other of those two cells or [[Tom's move]].
+
 
+
If Blue plays one of the five upper-right of those cells
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S blue:(f3 f2 g2 g4 f4)"
+
  />
+
 
+
, ​ then Red still connects via [[Tom's move]],
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  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.
+
 
+
This leaves only two
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S e4 f5"
+
  />
+
 
+
remaining tries for Blue.
+
 
+
If Blue plays the right of those two,
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 1:g3 3:e5 5:e4 B c3 d5 e3 c5 2:e6 4:f5 E *:d6 *:h4"
+
  />
+
 
+
then Red connects via [[double threat]].
+
 
+
Thus Blue instead plays the left of those two. ​ Red responds as follows.
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  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"
+
  />
+
 
+
Now, it does not matter which of the two shaded cells Blue 2 is at, since
+
 
+
* if Blue plays the other too then the order in which they were played doesn't matter
+
* and
+
* if instead Red plays the other, then whichever one Blue played will be [[Dead cell|dead]]
+
 
+
, ​ so this is equivalent to Red having a [[Theorems_about_templates#Corner_clipping|clipped]] version of [[Edge template IV1d]].
+
 
+
In particular, Red is connected down.
+
 
+
</div>
+
 
+
==== Intrusion at v or y or z ====
+
 
+
If Blue plays one of the three highlighted cells, then Red responds as shown below.
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:e5 E *:d6 *:e6 S blue:(f5 f6 g6)"
+
  />
+
 
+
This [[captured cell]]s the two cells marked "*", forcing Blue 4, and then Red 5
+
 
+
<hexboard size="6x14"
+
  coords="hide"
+
  edges="bottom"
+
  visible="area(f1,a6,n6,n4,l2,h1)"
+
  contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:e5 5:f2 B 4:e4 E *:d6 *:e6 S blue:(f5 f6 g6)"
+
  />
+
 
+
connects by [[double threat]] or [[edge template IV2e]] or [[Fourth_row_edge_templates#IV-2-q|edge template IV2p]].
+
 
+
</div>
+
  
 
=== Continuation after 2nd row ladder ===
 
=== Continuation after 2nd row ladder ===
Line 467: Line 121:
 
Now Red connects in essentially the same way as [[Tom's move]].
 
Now Red connects in essentially the same way as [[Tom's move]].
  
Continuation:
+
Specifically, Red has these threats:
<div class="toccolours mw-collapsible  mw-collapsed">
+
 
+
Red has these threats:
+
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
Line 493: Line 144:
 
   />
 
   />
 
    
 
    
The overlap in which Blue must play is:
+
The overlap in which Blue [[mustplay region|must play]] is:
  
 
<hexboard size="6x14"
 
<hexboard size="6x14"
Line 524: Line 175:
 
Note that Red 4 connects to the bottom with [[Edge_template_IV2b|IV-2-b]].
 
Note that Red 4 connects to the bottom with [[Edge_template_IV2b|IV-2-b]].
  
</div>
+
== See also ==
  
 +
* [[Tom's move]]
 +
* [[Tom's move for 3rd and 5th row parallel ladders]]
 +
* [[Fourth row edge templates]]
  
 
[[category:edge templates]]
 
[[category:edge templates]]

Latest revision as of 20:44, 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 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.

Specifically, Red has these threats:

13524
132
1

The overlap in which Blue must play is:

Four of these five possible moves can be analysed together. In the following diagram, assume Blue has played 1 in any one of the cells marked with +:

64532

After Red 2, that group is safely connected to them bottom, now matter which of the pluses Blue chose before.

That leaves only one Blue move to deal with:

76108495213

Note that Red 4 connects to the bottom with IV-2-b.

See also