Theory of ladder escapes

The object of this page is to formalise precisely what it means for a pattern to be a ladder escape. To do this we first need to formalise what it means to be a ladder! After we have done this, we can formalise what it means to be a ladder escape, but now there is a catch. A ladder escape (say, to make things precise, a 4th row ladder escape) is supposed to give the attacker a guarantee that their 4th row ladder will be able to connect to the edge, however far away it is. So strictly speaking to check that a pattern is a 4th row ladder escape we need to check that red can connect to the edge from an *infinite set* of positions (the ladder can be as far away as you like). This raises the issue of how one can possibly check in a finite time whether a given pattern is a 4th row ladder escape.

This issue is resolved on this page, for 2nd, 3rd and 4th row ladders. It might be possible to resolve it for 5th and 6th row ladders but these have no practical use and it will take a lot of work, so I have not done this. For 7th row ladders we run into a new difficulty -- blue can simply ignore the ladder and play near the escape, because no appropriate 6th row edge template seems to be known which will connect an ignored 7th row ladder to the edge. This presents a theoretical obstruction which I do not know how to resolve; I think that in theory there could be no 7th row ladder escapes at all!

Ladders
There is no issue at all with defining a 2nd row ladder: it looks like this. 

There is a minor issue with defining 3rd and higher row ladders though. We want a definition which is useful in practice and not too restrictive. For example we surely want this to be a third row ladder:  even though there are a couple of blue pieces on the first row. It is intuitively clear (and also provably true) that these blue pieces cannot be of any help to blue (they can never play a crucial role in any blue connection). So although we want a 3rd row ladder (moving from left to right, say) to have no stones on the first three rows to the right of the ladder (until we reach the escape of course), we do not want to also guarantee that there are no stones on the first row to the left of the ladder.

Here is the definition I came up with. A third row ladder is a pattern like this:  The asterisk is not part of the pattern. The red piece can be thought of as being connected to the top. The key part of the definition is that we are guaranteeing the triangle of three empty hexes under the red piece. This is a minimal requirement, because for example if one of these pieces were filled:  then in reality the game could look like this:  and blue can block the ladder with this move.  So that is my working definition of a third row ladder, and it seems to work in practice.

The problems compound themselves as we move up the board. Here is my working definition of a 4th row ladder:  You see that I have replaced the blue piece with an asterisk (indicating "not part of the pattern"). This is OK because hexes which are not part of the pattern may as well be blue pieces. What I am suggesting here is that for a 4th row ladder to become established we should demand that the triangle with 6 hexes below the laddering piece are all vacant. Note that even filling in one of these can break the ladder: even if we fill in the bottom left corner of the triangle:  then blue has this move  which is easily seen to stop the ladder. To to make the ladder established red needs at a minimum those 6 vacant hexes under their red stone.

I have said above that I will be restricting my attention to 2nd, 3rd and 4th row ladders, and one of the reasons for this is that I do not know a satisfactory definition of a 5th row ladder. What I do know is that this  will not do, because it turns out that if it is blue to play in the diagram below:  then blue can block the ladder with this move:  The main line is complex; see http://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=669 for example. The conclusion from this is that red needs more space in order to ensure that the ladder is established, and because I am only an amateur hex player I do not really know the minimal amount required or even if one can expect a "natural" answer to this -- perhaps there is more than one kind of 5th row ladder.

A similar issue arises with 6th row ladders -- I do not know how much space to guarantee red under the ladder stone. And for 7th row ladders I have already explained in open problems about edge templates that the situation is even worse -- no amount of space under the ladder (even if we demand that the entire 5th row is clear) seems to guarantee a red connection if blue just ignores the ladder and plays elsewhere, from which I deduce that I am struggling to make sense of the notion of a 7th row ladder from a theoretical point of view whilst simultaneously being aware that there is probably essentially no use for it from a practical point of view either.

2nd row ladder escapes
I have only formulated the definition of an n'th row ladder for 2<=n<=4 above. Let me start this discussion of ladder escapes with a discussion of 2nd row ladder escapes, because here we can see the main points without any of the technicalities about how much space one is supposed to allow under a ladder -- there is no space under the ladder.

Let me start with an example. An example of a second row ladder escape is a pattern that looks something like this.  Of course directly underneath the pattern is the bottom (red) edge. The pluses indicate where the ladder connects. The reason this is a second row ladder escape is that however far away the ladder is:

 red can guarantee a connection from the ladder stone (marked 1) to the bottom edge. Note that I have replaced asterisks with blue stones -- any hex not in the pattern or the ladder can be thought of as a blue stone, as red is not allowed to move there.

Let me clarify what the plussed hexes mean: they indicate the last point where the 2nd row ladder is allowed to start. So for example saying that the pattern above is a second row ladder escape means that red must win the following position:

 Here we regard stone 1 as connected to the top, and the claim (easily verified) is that even with blue to play, red can connect to the bottom: 

Now we can guess the general definition. Before I start I should say that for simplicity I won't allow mirror images and all 2nd row ladders should be thought of as coming in from the left. A second row ladder escape is the following data. It is a pattern, plus two hexes with pluses in them, such that one of the plussed hexes is on the first row, the other is on the second row up and directly to the left of the first hex, and such that neither of the plussed hexes nor any hex directly to the left of either of the plussed hexes are in the pattern. This data is subject to the following axiom: any position comprising of a second row ladder <hexboard size="2x1" coords="hide" contents="R 1:a1" /> followed directly to the right by as many pairs of vacant hexes as you like on the first and second rows: <hexboard size="2x1" coords="hide" contents="R q:a1" /> followed by the second row ladder escape pattern (where the ladder slots into the escape by putting the ladder onto the plussed hexes) is an edge template, in the sense that even if it is blue to move, red can guarantee a connection from the ladder stone marked 1 to the edge.

Notation: let's say that our 2nd row ladder pattern followed by n (an integer >= 0) pairs of vacant hexes is called a "second row ladder at distance n". What we are demanding of our second row ladder escape template is that it becomes an edge template when you slot in a second row ladder at distance n, for all values of n>=0.

This is an interesting definition because it allows the ladder to be an *arbitrary* distance away from the escape, which is of course what we want in practice; there is no reason that the escape should be right next to the ladder. However this means that to check that something is a 2nd row ladder escape we need to check that *infinitely many* patterns are edge templates. How can we do this? Well of course as every hex player knows,

Lemma: a pattern <hexboard size="4x4" coords="hide" contents="E *:a1 E *:b1 E *:c1 E *:d1 E *:a2 E *:b2 E *:c2 E *:d2 E +:a3 E *:b3 E *:c3 E *:d3 E +:a4 E *:b4 E *:c4 E *:d4" /> (where here the asterisks can indicate any pieces at all) is a second row ladder escape if, and only if, replacing the plussed hexes with a second row ladder (at distance zero) <hexboard size="4x4" coords="hide" contents="E *:a1 E *:b1 E *:c1 E *:d1 E *:a2 E *:b2 E *:c2 E *:d2 R a3 E *:b3 E *:c3 E *:d3 E *:b4 E *:c4 E *:d4" /> is an edge template for red.

Proof: If the pattern is a second row ladder escape then *by definition* replacing the plusses with a second row distance zero ladder right next to the escape gives an edge template (indeed a second row ladder escape means that this position and infinitely many other positions are an edge template). So this finishes the implication in one direction.

To go the other way we actually have to play some hex, but it's pretty trivial. Say the pattern becomes an edge template if we insert a second row ladder at distance zero. We now have to prove that the pattern becomes an edge template if we insert a second row ladder at distance n for all n, and this is an easy induction on n, because blue must play directly below red's ladder piece and now red plays along the ladder and by induction this is an edge template.

QED

Definition: A 2nd row ladder escape template is minimal if the following two things are true. Firstly, removing any hex from the pattern gives a new pattern which is not a 2nd row ladder escape template any more. And secondly, if the two hexes directly to the right of the two plussed stones are both vacant hexes in the pattern, then moving the plussed hexes one hex to the right results in a new pattern which again is not a 2nd row ladder escape.

Corollary [cf http://www.drking.org.uk/hexagons/hex/templates.html]: the following positions are minimal second row ladder escapes: <hexboard size="2x2" coords="hide" contents="E +:a1 E +:a2 R b2" /> <hexboard size="2x2" coords="hide" contents="E +:a1 R b1 E +:a2" /> <hexboard size="3x3" coords="hide" contents="E *:a1 R b1 E +:a2 E +:a3" /> (note that the corresponding pattern on Dr King's site is not minimal; I have moved the plusses) <hexboard size="3x3" coords="hide" contents="E *:a1 R c1 E +:a2 E +:a3" /> (note that the corresponding pattern on Dr King's site is not minimal; I have moved the plusses) <hexboard size="2x3" coords="hide" contents="E +:a1 R 10:c1 E +:a2 E *:b2 E *:c2" /> (here and below the 10 indicates a stone connected to the bottom edge, but the connection is not shown) <hexboard size="3x4" coords="hide" contents="E *:a1 E +:a2 R 10:d2 E +:a3 E *:b3 E *:c3 E *:d3" /> <hexboard size="4x4" coords="hide" contents="E *:a1 E *:b1 E *:c1 R 10:d1 E *:a2 E *:d2 E +:a3 E *:c3 E *:d3 E +:a4 E *:b4 E *:c4 E *:d4" /> <hexboard size="4x6" coords="hide" contents="E *:a1 E *:b1 E *:c1 R d1 E *:f1 E *:a2 E +:a3 E +:a4" /> <hexboard size="4x6" coords="hide" contents="E *:a1 E *:b1 E *:c1 R e1 E *:f1 E *:a2 E +:a3 E +:a4" /> and so on and so on (these templates are all taken from Dr King's site, and there are several more there).

Third row ladder escapes
We have seen a lot of the formalism of ladder escapes in the above section on second row escapes. However there is a new twist with third row ladder escapes, because blue can defend against a third row ladder in more than one way: blue can at some stage decide to drop to the second row.

The following definition is unsurprising:

Now a third row ladder escape should be a pattern which looks something like this: <hexboard size="3x3" coords="hide" contents="E +:a1 R b1 E +:a2 E +:a3" />