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

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:

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:

Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by edge template V2m. The latter template is itself based on Tom's move at "x". It works, for example, like this:

Now Red is connected by the (ordinary) Tom's move.

Proof that Red is connected

Red has two main threats. Via a ziggurat:

And via edge template IV2b:

Blue must play in the overlap:

Intrusion at p, q, r

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 IV-2-b. ​ Otherwise, Red connects via Tom's move.

Intrusion at s

Red is connected by edge template IV1d.

Intrusion at t

Now Red is connected by edge template V2m. If Blue plays 4 on the first row instead, Red connects by Tom's move:

Intrusion at u

Red is connected by Tom's move.

Intrusion at w

Red responds with

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

and

Blue must play in the overlap:

If Blue plays at a, then Red plays 5, after which Red connects via

or

If Blue plays at b, then Red plays 5, after which Red connects via

or

If Blue plays at c, then Red responds with

This forces Blue to defend towards the top, after which

connects directly or via red 1.

If Blue plays at d or e, then Red plays like this:

If Blue plays at f, then Red plays like this:

Now Red threatens to play in one of the cells marked "*", and the only overlap (apart from the two captured cells below 5) is at "+", so Blue must play there. Then Red responds like this:

Now Red is connected left by double threats "*" and right by double threats "+".

Intrusion at x

Red can respond here:

Red's main threat is as follows,

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

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

then Red still connects via Tom's move,

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 remaining tries for Blue:

If Blue plays the right of those two,

then Red connects via double threat.

Thus Blue instead plays the left of those two. ​ Red responds as follows.

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.

So this is equivalent to Red having a clipped version of Edge template IV1d.

In particular, Red is connected down.

Intrusion at v or y or z

If Blue plays one of the three highlighted cells, then Red responds as shown below.

This captures the two cells marked "*", forcing Blue 4, and then Red 5

connects by double threat or edge template IV2e or edge template IV2p.