On David King's hex template page, www.drking.plus.com/hexagons/hex/templates.html, he says

“Guaranteed Ladder Escape templates:

These will complete a ladder with a successful connection.

All 3rd row templates are also 2nd row templates. All 4th row templates are also 3rd and 2nd row templates….. and so on.”

Has anyone actually proved that an nth-row ladder escape template is always also a ladder escape template for rows 2 through n-1, with the same vigor with which Nash proved that a hex game played to completion always has exactly one winner?

The only way to do that is to first define exactly what an nth row ladder escape template is. David King doesn't do that. He just provides examples of them. I don't believe he was attempting to state any general theorem. He was merely saying that for any of the templates he shows, if you shift the stones closer to their border row, you still have a win. After all, the only way to connect to the border is to pass through all the rows of cells above the border.

Oh, is that what he means. I've wondered about that for years. I thought he meant that if you shift the ladder sontes, but not the escape stones closer to the edge, you'd still have a valid escape. That seems like a likely suppostion to me, but I could never prove it to myself with the amount of effort I was willing to spend.

Well, I really think what the site is trying to say is that if you move the approaching, laddering, stones closer to the edge that the unmoved template still acts as a ladder escape. If that's really true, then the defender can never prevent the escape just by moving his line of defense closer to the opponent's edge.

I think we said the same thing in different ways, and you gave a good reason to believe that it's true. Still not a proof, though.

There must be some mathematically minded Hex players who could come up with a proof. Having said that, I must admit that I have a BS in math, although my career has been 95% computer programming.

1. Would having such a proof be worthwhile?

Knowing that just laddering along, defending on a row as far as possible from the edge, is a tactic that cannot be improved upon would simplify analysis of a situation.

In any event, finding a proof, or understanding a proof someone else has found, often leads to insights, which may be more valuable than a laddering tactic.

2. Would such a proof be difficult, time-consuming to find?

I don't know.

I think this is a very interesting question. It seems to me there are several interpretations of the question, at least one of them trivial and others hard. For the purposes of diagram-drawing let me just stick to the following sort of question: “If template X is a 3rd row ladder escape template, is it a second row ladder escape template?“. Of course if you allow 4th row templates etc then you can ask more questions of this form and more variants of these question.

Here's the an example of the trivial variant of the question (in all positions it's black to play, by the way). If black wins this:

then does black win this?

Of course you can replace the bottom right hand area with any template you like. This question is trivial (unless I made a mistake) because black can just play from the bottom position to get into the top one.

But here's an example of a question I don't know how to answer for all templates (for any one specific template one could attempt to answer it, but I'm talking about a general question which would deal with all templates at once). Here's an example of the question.

If black wins this:

then does black win this:

One could ask a similar question about all the third row ladder escape templates at Dr King templates [drking.org.uk] and in each case you would have to decide where to stop the 3rd row and 2nd row templates in order to make sense of the question; my initial example was supposed to indicate that if you're not too careful about this then the question might admit an easy answer, but my second question is just one example of a class of questions of the form “if you win with a 3rd row ladder starting here, do you with with a second row ladder starting here?“. Where the second row ladder starts seems to me to depend on the template, so this question needs to be made a little more precise before one can attempt to answer it I guess. This post is I think a summary of what others have said here. I should say now that my understanding of Dr King's templates is that the third position I posted above is a win for black, and I am less clear about the 4th position. This is my attempt to explain what I know about Dr King's statement. Does anyone know anything more?

Can I upload pictures? I just used gimp on trmph to get this.

OK so it worked. Great. I have a relatively trivial thing to say. Is this a third row ladder escape for black? Well I would say that it depends on what you mean by a third row ladder escape. A rather naive definition would simply be that if if we just add three rows to the left of this picture, and give black a stone connected to the top on the third row:

, and then we leave black and white to just ladder with black along the third row and on the second row, then we could ask what happens. And of course black will win with this strategy:

However, keeping black on the third row is clearly a terrible strategy for white – they can let black drop to the second row because this is further from safety, and easily win like this:

So this is kind-of interesting. I am pretty sure that a good (i.e. useful) definition of a *second-row* ladder escape is simply that a second row ladder escape is a template where if you give black a stone connected to the top on the second row just next to the escape, and then let white play under it to stop the immediate connection, then black wins anyway. However a _bad_ definition of a third row ladder escape is a template where if you give black a stone connected to the top on the third row just to the left of the template, and let white play directly under it, then black wins. For the board fragment at the top of this post has this property, and we've just seen that black does not win if they have a third row ladder heading for it.

In general, the picture above shows that for black coming in on the third row, which can drop them to the second row at any point they please and in particular they can do it right at the end. My conclusion is that if we want a third row ladder escape to mean a template that black is guaranteed a win if they're laddering towards it on the third row, then a third row ladder escape template should *by definition* be a template such that black can connect a stone on the third row to the left of the template, to the bottom edge, *and* that black can connect a stone on the 2nd row to the left of the template to the bottom edge. The problem with the template at the top is that it does not have this latter property.

So my conclusion, finally, is that all third row templates are second row templates *essentially by definition*, because if you do not make this part of the definition of a third row template then your definition is not of any practical use. In particular I think that the fragment I posted above resolves the question Dvd and MarleysGhost were thinking about above in 2007: you can have a template which connects to a third row stone but not to a second row stone. I know that this is basically a trivial observation but somehow it's a crucial one when trying to formulate a sensible notion of a third row ladder escape.

Curse this uneditable forum. White not which in penultimate para.

Wccanard, thank you for all of your comments and ideas.

With regard to the question if an escape work from the third row, will it work from the second row, I have looked at this question over the years and I have come with a general conjecture:

If an escape works from row N then it will also work from any row M where M < N.

I am 99% confident of this, I have never found a counterexample.

I have a way of formalising this statement in such a way that it's false. I have another way of formalising it in such a way that it's true. These things are far more subtle than I had realised. Even the *definition* of an N'th row ladder escape has some subtleties.

I will try and write something on hexwiki to explain more clearly what I want to say, but let me just make the following comment (which I think I might have already made in another thread): if by a 2nd row ladder escape we mean something which will escape any 2nd row ladder even if the 3rd row is full of white pieces, then there are 3rd row ladder escapes which aren't 2nd row ladder escapes because the row of white pieces that you can insert on the 3rd row blocks the 2nd row escape. So you have to be a bit more careful about what you mean. But something of this nature is definitely true.