This question is harder than I thought!

I think that a natural, but at this point in the argument rather useless, definition of an n'th row ladder escape template, is simply that it's a position which will escape an n'th row ladder however far away that ladder is starting (assuming that all rows from 1 to n are vacant).

It's easy to check that a position is a 2nd row ladder escape template; black (with the 2nd row ladder) just ladders towards the position and white is forced to play under. When black gets there the question is whether they can break through. So really a 2nd row ladder escape template is just a position such that if you add a black stone to the 2nd row just off to the left (say) of the template, and a vacant piece below it, black will win even if it's white to play. In short, a 2nd row ladder escape template (such as those seen at Dr King's template site [www.drking.org.uk] is simply a template such that if you remove those two dotted arrows and replace the top one with a black stone with an up-arrow in it, and the bottom with a vacant hex, then the resulting position is an edge template.

But what is a 3rd row ladder escape template? This is a bit more subtle, because white has more than one way to defend. For example white could drop black down to the second row at the last minute. So basically a 3rd row ladder escape template should be a picture with three arrows going into it (like on Dr King's site) and such that if you replace the top arrow with a black up-arrow stone and then put an empty triangle of three hexes under it then it's an edge template, *and* if instead you change the top and bottom arrow into vacant hexes and the middle one into a black up-arrow stone, then this is also an edge template. I am too lazy to draw pictures at this point but I might get round to it later. I believe one can check that this is a finite algorithm for testing whether something is a 3rd row ladder escape template. It is also a way of formulating and proving a precise statement of the form “a 3rd row ladder escape template is a 2nd row ladder escape template”. As it stands this is not quite true, but there's a simple way of making it true.

For 4th row things get worse because white has multiple ways to defend by playing pieces on rows other than the third row, and he can play these lines whenever they like (e.g. pretty much right at the end when black has no time to respond other than by following their nose) so a 4th row template has to be able to deal with all sorts of possibilities. I am not entirely clear about what the complete list is though! I will think more about this, but has anyone else written down or understood anything about this situation? The main issue is that a 4th row ladder escape template is supposed to be able to connect a 4th row ladder but if this ladder is 10 columns away then a lot can happen before the ladder gets to the template. I'd like to quantify exactly what can happen via a finite list.

I will probably write up what I have understood about 2nd and 3rd row ladders in the near future on hexwiki. I'm not sure that too many people will be interested but there's no reason not to get these thoughts down in a precise form so that other people don't have to re-invent them later.

Here you have at least one person who is interested! :-)

I need to be more precise about what I mean; I worked some more precise stuff out over the weekend (e.g. an algorithm to check that something is a third row ladder escape) but I think the natural place to put this is hexwiki and my understanding is that the only boards you can put up there are parallelograms, which is a real pain when writing about templates. I'm currently investigating various ways of displaying board fragments. Once I've written this then my question will be much clearer.

Thank you for your contributions to our game.

I think I've today worked out a sufficient condition for a position to be a 4th row ladder escape. But it was a messy calculation and the situation will only get worse for higher row ladder escapes. On the other hand I think I've not run into 5th row ladders which seriously need to be escaped in my practical hex experience thus far.

I wrote up a formal definition of what it means to be a 4th row ladder escape (it must escape a 4th row ladder however far away this ladder is) plus an algorithm which can prove in a finite time that a certain pattern is a 4th row ladder escape.

Proving that a pattern is a ladder escape [hexwiki.amecy.com]

It was more subtle than I had thought. It would be some work, but perhaps not impossible, to push the theory up to 5th and 6th rows. But because of issues explained here

Open problems about edge templates [hexwiki.amecy.com]

I am not at all sure about how to push it to the 7th row.

I still need to write up my thoughts on whether a 3rd row ladder escape is a 2nd row ladder escape etc. I can only make sense of this currently for 3 to 2, 4 to 3 and 4 to 2, but for each of these I can give it a go later on today.

Nice work wccanard

Wccanard, awesome work!  I fully admit to having a hard time following along with the proofs, but your write up is excellent, and must have taken a lot of time and effort, especially with the number of diagrams that you put in.  Thanks for this enormous contribution!

Yes, very good work!

Can anyone check the ABCD=>4th row ladder escape is a theorem?

Also what about the converse theorem that a 4th row ladder escape needs ABCD, in other words, do all 4th row ladder escapes follow P+A,P+B,P+C and P+D?

My guess is that you can just build random 4th row ladder escapes that escape A+n, B+n, C+n and D+n for all n>=m (some fixed m) and also escape A,A+1,..,A+m, but not, say, D+m. This probably isn't too hard – i did it for the 3rd row escape.

All this means is that these comments like “any 3rd row escape is a 2nd row escape” – you have to be really careful what you mean by this.