View source for Tom's move for 3rd and 5th row parallel ladders

There is a version of [[Tom's move]] that works for [[parallel ladder]]s on the 3rd and 5th rows. It requires a large amount of space:
<hexboard size="6x12"
   coords="hide"
   edges="bottom"
   visible="area(d1,b3,b6,l6,l4,j2,f1)"
   contents="R arrow(12):c2 b3 b4 B b5 c3 R 1:e3"
   />

== Basic idea ==

By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:
<hexboard size="6x12"
   coords="hide"
   edges="bottom"
   visible="area(d1,b3,b6,l6,l4,j2,f1)"
   contents="R arrow(12):c2 b3 b4 B b5 c3 E x:i4
             R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1"
   />
Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. The latter template is itself based on Tom's move at "x". It works, for example, like this:
<hexboard size="6x12"
   coords="hide"
   edges="bottom"
   visible="area(d1,b3,b6,l6,l4,j2,f1)"
   contents="R arrow(12):c2 b3 b4 B b5 c3
             R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1
             B 4:f4 R 5:g3 B 6:g4 R 7:e4 B 8:d6 R 9:e5 B 10:e6 R 11:f5 B 12:f6 R 13:i4"
   />
Now Red is connected by the (ordinary) Tom's move.

== Proof that Red is connected ==

Red has two main threats. Via a [[ziggurat]]:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,n6,n4,l2,h1)"
   contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 R 3: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,d3,d6,n6,n4,l2,h1)"
   contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 R 3: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,d3,d6,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, or r ====

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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)"
   />

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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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 "+". 

==== Intrusion at x ====

Red can respond here:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,n6,n4,l2,h1)"
   contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6"
   />

Red's main threat is as follows,

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,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,d3,d6,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,d3,d6,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,d3,d6,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 for Red 1 to connect to Red's left group.

This leaves only two remaining tries for Blue:

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,n6,n4,l2,h1)"
   contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S e4 f5"
   />

If Blue plays the right of those two,

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,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,d3,d6,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.

==== Intrusion at v, 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,d3,d6,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|captures]] the two cells marked "*", forcing Blue 4, and then Red 5

<hexboard size="6x14"
   coords="hide"
   edges="bottom"
   visible="area(f1,d3,d6,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]].

== See also ==

* [[Parallel ladder]]
* [[Tom's move]] (for 2nd and 4th row parallel ladders)
* [[Edge template V1b]]

[[category:ladder]]
[[category:Advanced Strategy]]