View source for Edge template V1b

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

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2"
   />
   
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  [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=669 Little Golem thread], but likely predates that post.



== Defense against intrusions ==

=== Reduction ===

Red has 3 main threats. Using the [[ziggurat]]:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 E *:d4 S d4 d3 e3 c4 S b5 c5 d5 a6 b6 c6 d6"
   />
   
Using [[edge template III1b]]:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 E *:d4 S d3 e3 c4 d4 e4 b5 c5 d5 e5 a6 b6 d6 e6"
   />
   
And using [[edge_template_IV1d]]:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 E *:f3 S area(e3,c6,k6,k4,i2,f2)"
   />

For a blocking attempt, 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 E a:e3 E b:d5 E c:d6 S e3 d5 d6"
   />

=== Intrusion at a ===

If Blue intrudes at a, Red can start by forcing a 3rd or 2nd row ladder like this
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B 1:e3 R 2:d3 B 3:c5 R 4:d4 B 5:d5"
   />
or like this:
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B 1:e3 R 2:d3 B 3:d4 R 4:c4 B 5:b6 R 6:c5 B 7:c6 R 8:d5 B 9:d6"
   />

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:
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B 1:d5 R 2:d4 B 3:e3 R 4:d3 B 5:c5"
   />

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:
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B 1:d6 R 2:d4 B 3:e3 R 4:d3 B 5:c5 R 6:d5 B 7:c6"
   />

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.

<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"
   />

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]].

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 ===

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

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R 1:e5 B 2:e6 R 3:h4"
   />

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

Continuation:
<div class="toccolours mw-collapsible  mw-collapsed">

Red has these threats:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 R 1:f5 B 2:f6 R 3:g5 B 4:g6 R 5:i5 S area(f5,f6,i6,i4,h4)"
   />
   
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 R 1:f5 B 2:f6 R 3:h5 S area(f5,f6,h6,h4,g4)"
   />
   
<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 R 1:g3 S area(f2,f3,g4,f6,i6,i4,g2)"
   />
   
The overlap in which Blue must play is:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 S area(f6,h6,h4)"
   />

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 +:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 R 2:g5 B 3:f5 R 4:g3 B 5:f4 R 6:f2 E +:h5 E +:f6 E +:g6 E +:h6"
   />

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:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,a6,n6,n4,l2,h1)"
   contents="R e2 B e3 R d3 B d4 R c4 B b6 R c5 B c6 R d5 B d6 R e5 B e6 R h4 R 10:f3 R 8:g3 R 4:i3 B 9:f4 B 5:g4 R 2:f5 B 1:g5 B 3:f6 R 6:h2 B 7:g2"
   />

Note that Red 4 connects to the bottom with [[Edge_template_IV2b|IV-2-b]].

</div>


[[category:edge templates]]