Difference between revisions of "Theory of ladder escapes"

Revision as of 23:24, 8 July 2020

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 formalise what it means to be a ladder.

Informally, a ladder escape (say, 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, no matter how far away from the ladder escape the ladder starts. So strictly speaking, to check that a pattern is a 4th row ladder escape, we must check that the attacker can connect to the edge from an infinite set of positions. This raises the issue of how one can 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 this has not yet been done, partly because such ladders are of little practical use. 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 is currently unresolved. It may in theory be that there are no 7th row ladders at all.

For the purpose of our analysis, we assume that all ladders move from left to right along the red bottom edge, with Red being the attacker. Of course, the analogous analysis also applies to ladders moving in the opposite direction or along different edges.

The analysis of 2nd–4th row ladders on this page was originally contributed by the user Wccanard in 2016.

Second row ladders

Definition of ladder

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

Note that at each point in the ladder, Blue's move is forced. Red can choose to continue pushing the ladder as long as she wants to. We formally define a second row ladder as follows:

Definition.. A second row ladder is a pattern like this:

Here, the red stone is on the second row, and we call it the ladder stone. Red's goal is to connect the ladder stone to the bottom edge. The cell immediately below and to the right of the ladder stone is empty. We denote this pattern by L2.

Definition of ladder escape

Before we give the formal definition of a second row ladder escape, let us consider an example. The following pattern is an example of a second row ladder escape.

Of course directly underneath the pattern is the bottom (red) edge. The cells marked "+" 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 the cells marked "*" can be of any color; but since the ladder escape must be valid regardless of the color of these cells, we may as well assume the worst case, i.e., all cells marked "*" can be thought of as blue stones.

Let us clarify what the hexes marked "+" in the ladder escape pattern 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 (among other things) that Red must win the following position:

Here, Red's ladder stone is marked "1", and the claim (easily verified) is that even with Blue to play, Red can connect the ladder stone to the bottom:

The reason that the pattern is a second row ladder escape is that this escape sequence works even if the ladder is a long way away:

Even here, Red can force a connection to the edge, even if it is Blue's move, because Blue must keep defending on the first row and Red keeps attacking on the second row,

and now we are back at the previous position with the ladder right next to the escape, where we have already seen that red can break through to the edge. We can now give a more formal definition of a second row ladder escape.

Definition. A second row ladder escape template (or simply second row ladder escape) is given by the following data. It is a pattern, plus two hexes marked "+", such that one of the hexes marked "+" is on the first row, the other is on the second row up and directly to the left of the first one, and such that neither of the hexes marked "+", nor any hex to their left, are in the pattern. This data is subject to the following axiom: any position consisting of a second row ladder,

followed directly to the right by an arbitrary number (zero or more) of pairs of vacant hexes on the first and second rows,

followed by the second row ladder escape pattern (where the ladder slots into the escape by putting the ladder or rightmost column of vacant hexes onto the hexes marked "+") is an edge template, in the sense that even if it is Blue's turn to move, Red can guarantee a connection from the ladder stone marked 1 to the edge.

Terminology and notation: If we have a pattern with two cells marked "+" that is of the shape required for a second row ladder escape, but we are not sure whether it satisfies the axiom of a second row ladder escape, then we refer to it as a candidate for a second row ladder escape (or simply candidate if the rest is clear from the context). A candidate is valid if it is actually a ladder escape.

If P is a candidate for a second row ladder escape and n ≥ 0 is an integer, we write n+P for the result of shifting the cells marked "+" to the left by n positions, filling the gap with n columns of two empty hexes each. Similarly, we write L2+n for the result of appending n columns of two empty hexes to the right of a second row ladder L2. Finally, we write L2+n+P for a 2nd row ladder, followed by n columns of empty hexes, followed by an escape candidate P. We call this a "second row ladder at distance n". When n = 0, we sometimes just write L2+P. With this notation, we can rephrase the definition of a 2nd row ladder escape as follows: a candidate P is a 2nd row ladder escape if and only if L2+n+P is a template connecting the ladder stone to the edge, for all n ≥ 0.

We also define what it means for a ladder escape template to be minimal.

Definition. A 2nd row ladder escape template is minimal if the following two things are true. First, removing any hex from the pattern, or removing a red stone from the pattern (and replacing it with an empty hex) gives a new pattern which is not a 2nd row ladder escape template any more. And second, if the two hexes directly to the right of the two cells marked "+" are both vacant hexes in the pattern, then moving the cells marked "+" one hex to the right results in a new pattern which is not a 2nd row ladder escape.

Below, we will use analogous terminology and notations for ladders and ladder escapes on the 3rd and higher rows.

Characterization of ladder escapes

The definition of a 2nd row ladder escape 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 we cannot directly use the definition to check that something is a 2nd row ladder escape, because this would require checking that infinitely many patterns are edge templates. Can we find some finite criterion for checking 2nd row ladder escapes? Fortunately, as every Hex player know, the answer is yes. We have the following theorem:

Theorem 1. Consider a candidate P for a 2nd row ladder escape. Schematically:

(Here the asterisks can indicate any stones at all.) Then P is a valid 2nd row ladder escape if and only if L2+P is an edge template for Red.

Proof. If the ladder escape is valid, then by definition, L2+n+P is an edge template for all n ≥ 0, and in particular for n = 0. This proves the left-to-right implication.

To go the other way we actually have to play some Hex, but it's pretty trivial. We must show that L2+n+P is an edge template for all n. This is an easy indunction on n. For n = 0, the claim is true by assumption. If n > 0, then Blue must play directly below Red's ladder stone (or else Red will connect to the edge immediately), and now Red can play a ladder stone at distance n−1 on the second row, which is an edge template by induction hypothesis. □

Examples

We can use Theorem 1 to prove that all of the following patterns are minimal second row ladder escapes. All of these templates are taken from David King's website, and there are several more there. Cells marked "*" are not part of the ladder escape template. For several of the templates, the corresponding pattern on David King's site is not minimal by our definition; for these templates, we have moved the cells marked "+" to the right to make the template minimal.

In the below templates, the stone marked 10 indicates a stone connected to the bottom edge, but the connection is not shown.

Third row ladders

Definition of ladder

There is a minor issue with defining 3rd and higher row ladders. We want a definition that 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 few blue stones on the first row. It is intuitively clear (and also provably true) that these blue stones 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 to have no stones on the first three rows to the right of the ladder (until we reach the escape), we do not want to also guarantee that there are no stones on the first row to the left of the ladder. We formally define third row ladders as follows.

Definition. A third row ladder is a pattern like this:

The red stone is again called the ladder stone, and Red's goal is to connect the ladder stone to the bottom edge. The cells marked "*" are not part of the pattern, and may be of any color or even outside the board. These cells can be assumed to be blue, although this is not required. We denote this pattern by L3.

The key part of the definition is that we are guaranteeing the triangle of three empty hexes under the red ladder stone. This is a minimal requirement, because for example if one of these cells were filled,

then in reality the game could look like this,

and blue can block the ladder with this move.

Definition of ladder escape

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 yield to the second row. The following definition is unsurprising.

Definition. A third row ladder escape is given by the following data. It consists of a pattern, and three hexes marked "+" (none of these hexes marked "+" are part of the pattern) going from the first to the third row, up and left as in the below picture, which is sitting on the bottom edge:

Here, the hexes marked "*" can be anything, they are just part of a general pattern, which could certainly go above the third row, and to the left above the hexes marked "+", and to the right, if necessary. The pattern must have the property that no hex directly to the left of any of the hexes marked "+" can be part of the pattern. To be a third row ladder escape, the pattern must satisfy the property that for all n ≥ 0, if we insert a third row ladder at distance n into the pattern at the three hexes marked "+", the resulting position is an edge template connecting the Red ladder stone to the bottom edge, with Blue to move. Or equivalently in symbols, if L3+n+P is an edge template connecting the ladder stone to the bottom edge, for all n ≥ 0.

As for second row escapes, a pattern that has the required shape for a ladder escape, but it is not (yet) known to be a valid ladder escape, is called a candidate.

In pictures, our pattern above (still marked with stars), if it is a 3rd row ladder escape, must give rise to an edge template when we attach a 3rd row ladder at distance 0,

or at distance 1,

or at distance 6,

or at any other distance.

Characterization of ladder escapes

Just like for second row ladder escapes, we again find ourselves in the situation that trying to use the definition to check that something is a 3rd row ladder escape involves checking that infinitely many positions are edge templates. Once again, we have a theorem that allows us to replace this by a finite condition.

Definition. If P is a candidate for a kth row ladder escape, then P′ denotes the candidate for a (k−1)st row ladder escape obtained by removing the topmost "+" from P and replacing it by an empty cell.

Theorem 2. Consider a candidate P for a 3rd row ladder escape. Assume that (a) L2+P′ is an edge template and (b) L3+P is an edge template, each such template connecting the ladder stone to the bottom edge. Then P is a valid third row ladder escape.

Proof. First note that by Theorem 1, because L2+P′ is an edge template, P′ escapes all 2nd row ladders. Now under the assumptions of the theorem, we must show that L3+n+P is an edge template for all n ≥ 0. We prove this by induction on n. For n = 0, the claim is true by assumption (b). Now suppose the claim is true for some n. To show the claim for n+1, consider the position L3+(n+1)+P. The first three columns of this position look like this:

(This is followed by n more columns of three empty hexes and by the pattern P). Blue has three possible moves in a triangle under stone 1, and Blue needs to play one of these or he will lose instantly. We analyze all three moves in turn.

For the first, Red pushes the ladder and will connect to the edge because by induction hypothesis, L3+n+P connects to the edge, so stone 3 connects to the edge, and so stone 1 does too.

For the second, Red just wins outright, i.e., we do not need to use the induction hypothesis.

And for the third, Red responds like this. Since stone 5 is a 2nd row ladder stone, it is connected to the edge because, as we noted above, P′ is a 2nd row ladder escape. Therefore stone 1 is also connected.

Note that the moves 3 and 4 are necessary; had Red bridged directly to 5, the cell above 5 would be both part of a bridge and potentially part of the template P′ (in case n = 0; remember that we replaced the topmost "+" by an empty cell), which means we could not necessarily guarantee the connection from 1.

The induction is now complete, showing that P is a 3rd row ladder escape. □

Remark: in more concrete terms, Theorem 2 states that a pattern P is a 3rd row ladder escape if the pattern becomes an edge template (connecting the ladder stone to the edge) when we attach each of the following two patterns to it at the cells marked "+":

A:

B:

In contrast to the situation with 2nd row ladders, while Theorem 2 is sufficient to show that a position is a 3rd row ladder escape, it is not necessary. For example, here is a somewhat strange third row ladder escape:

All of the cells marked "*" (and especially those that appear to the left of the cells marked "+") are not part of the template. One can check directly that if P is this pattern, then L3+1+P and L2+(1+P)' are both edge templates, so (1+P)' is a 2nd row ladder escape (by Theorem 1) and 1+P is a 3rd row ladder escape (by Theorem 2). Since moreover, L3+P is an edge template, it follows that P is a valid 3rd row ladder escape. It is, however, not a valid 2nd row ladder escape because in the position L2+P′,

the 1 stone cannot connect to the edge or to the 10 stones.

Theorem 2 is therefore not sufficient to check that a given pattern is a 3rd row ladder escape. We need to work a little harder to get a necessary and sufficient condition for 3rd row ladder escapes.

Theorem 3. Any 3rd row ladder escape also escapes 2nd row ladders that start at distance 2 or greater. More precisely, if P is a 3rd row ladder escape, then 1+(1+P)' is a 2nd row ladder escape. This is perhaps easier understood in pictures: given any 3rd row ladder escape, replacing the three cells marked "+"

by the pattern

yields a 2nd row ladder escape.

Proof. By Theorem 1, to prove that 1+(1+P)' is a 2nd row ladder escape, it is sufficient to prove that the following, with P added to it, is a template:

But Blue must play 2, and Red can jump to 3. Then 3 is a 3rd row ladder stone, and is connected to the edge because P is a 3rd row ladder escape. Therefore, 1 is also connected.

□

Theorem 4. Consider a candidate P for a 3rd row ladder escape. Then P is a valid 3rd row ladder escape if and only if L3+P, L3+1+P, and L3+2+P are edge templates.

Proof. The left-to-right implication is trivial, since by definition, P is valid if and only if L3+n+P is an edge template for all n, including n = 0, 1, 2. For the opposite implication, assume that L3+P, L3+1+P, and L3+2+P are edge templates. By Theorem 3, L2+1+(1+P)' is an edge template, and therefore also L2+(2+P)' (because (2+P)' differs from 1+(1+P)' only in that it has one more empty cell, which can only help Red). By Theorem 1, (2+P)' is a 2nd row ladder escape, so by Theorem 2 and the assumption about L3+(2+P), 2+P is a 3rd row ladder escape. It follows that L3+n+P is an edge template for all n ≥ 2. Since we additionally assumed this to be the case for n = 0 and n = 1, P is a valid third row ladder escape. □

Examples

Here are some examples of third row ladder escapes. Again these are all taken from David King's website. For several of the ladder escape templates, the version shown on David King's website is not minimal by our definition; in these cases, we have moved the cells marked "+" to the right to make the template minimal.

Here and below, the 10 indicates a stone connected to the bottom row.

This latter pattern above – a single stone on the second row – is Wccanard's favourite, because the Theorem 4 can be used to show that it is a 3rd row ladder escape even though the template is so small.

The version of this last pattern on David King's website has the cells marked "+" (he uses arrows) sloping in the other direction; the location that is shown here makes the template minimal.

All of the templates above have been proved to be third row ladder escapes using Theorem 4. All of them are minimal.

2nd row escapes from 3rd row escapes

One may ask whether all 3rd row ladder escapes are also 2nd row ladder escapes.

To some extent, this is true: we have already seen in Theorem 3 that every 3rd row ladder escape is also a 2nd row ladder escape, provided that the 2nd row ladder starts at distance 2 or greater, and provided that there is enough room on the 3rd row – specifically, we require the topmost "+" of the 3rd row ladder escape, and the cell immediately to its left, to be empty.

On the other hand, we must be careful. In the example just before Theorem 3, we saw that a 3rd row ladder escape may not be a 2nd row ladder escape if the 2nd row ladder starts at a very short distance, or if there are less than two empty hexes on the 3rd row near where the 2nd row ladder connects.

Fourth row ladders

Definition of ladder

Definition. A fourth row ladder is a pattern like this:

Again, the red stone is called the ladder stone and Red wants to connect the ladder stone to the bottom edge. The cells marked "*" are not part of the pattern and may be assumed to be blue. We denote this pattern by L4.

The key part of the definition is that the 6 hexes forming a triangle below the ladder stone 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 establish the ladder, Red needs at a minimum those 6 vacant hexes under her ladder stone.

Definition of ladder escape

The definition of a 4th row ladder escape is entirely analogous to that of 2nd and 3rd row ladder escapes.

Definition. A fourth row ladder escape is a pattern, plus four hexes marked "+" (not part of the pattern) going up and left from the first to the fourth row, such that there is no hex in the pattern to the left of any hex marked "+". This data must satisfy the following axiom: adding a 4th row ladder at distance n, for any n ≥ 0, connects the red ladder stone to the bottom edge, with Blue to move.

Again, a candidate is a pattern that has the correct shape, but is not known to be a valid escape.

Characterization of ladder escapes

We have already encountered all of the relevant ideas. If you have worked through the ideas in the second and third row escapes then this will be relatively easy, other than the actual Hex, which this time is quite fun!

Theorem 5. Consider a candidate P for a 4th row ladder escape. If L2+P′′, L3+P′, L4+P, and L4+1+P are valid edge templates, then P is a 4th row ladder escape.

Proof. The idea of the proof is the same as for 3rd row ladders. First observe that by Theorem 2, since L2+P′′ and L3+P′ are edge templates, P′ escapes all 3rd row ladders and P′′ escapes all 2nd row ladders. We must prove that L4+n+P is an edge template for all n ≥ 0. We proceed by induction on n. The base cases n = 0 and n = 1 hold by assumption (c). For the induction step, assume the claim is true for n ≥ 1. We need to prove the claim for n+1.

Consider the position L4+(n+1)+P. It looks like this, with n−1 additional columns of four vacant hexes and the pattern P attached on the right:

We need to prove that the ladder stone 1 is connected to the edge.

The five moves marked 2 below all lose instantly to Red 3:

The two moves marked 2 below also lose instantly:

The move marked 2 below can be answered by Red 3, moving us to position L4+n+P, which is an edge template by the induction hypothesis.

Move 2 below leads us to a 2nd row ladder, which P′′ escapes, so 5 (and therefore 1) is connected to the edge:

Both moves marked 2 below lead us to a 3rd row ladder, which P′ escapes, so 3 (and therefore 1) is connected to the edge:

Move 2 below also leads to a 3rd row ladder (note Blue 4 must be in the triangle left and below from Red 3; Blue can also play out the bridge between 1 and 3 but this doesn't help):

Move 2 below leads us to a 2nd row ladder:

The final choice for move 2 below also gives a 2nd row ladder.

This completes the induction, so P is a 4th row ladder escape. □

Remark: Theorem 5 is analogous to Theorem 2. It gives a sufficient, but not a necessary condition for a candidate to be a 4th row ladder escape. Once again, the criterion in Theorem 5 can be checked in a finite amount of time. Unlike the case for 3rd row ladder escapes, we do not currently have a theorem with a necessary and sufficient condition. However, we have the following, which gives a weaker sufficient condition (it is likely to also be necessary, but this has not been proved):

Theorem 6. Given a candiate P for a 4th row ladder escape. If there is some n ≥ 0 such that L4+P, L4+1+P, ..., L4+(n+1)+P, L3+(n+P)', and L2+(n+P) are edge templates, then P is a valid 4th row ladder escape.

Proof. This follows directly from Theorem 5 applied to n+P. □

Examples

Here are some examples of third row ladder escapes, taken from David King's website. In each case we have moved the column of "+"s as far as possible to the right to yield a minimal template. The validity of all of these escapes has been proved using Theorem 5.

Fifth row ladders

Definition of ladder

Definition. A fifth row ladder is a pattern like this:

As usual, the red stone is called the ladder stone and Red's goal is to connect it to the bottom edge. The cells marked "*" are not part of the pattern. We denote this pattern by L5.

Unlike in the case of 2nd, 3rd, and 4th row ladders, this time it is not sufficient for a triangle of cells below and to the right of the ladder stone to be empty. We also need three additional empty cells to the left of this triangle. This is a minimal requirement; if even one of these cells is occupied by Blue, for example like this:

then Blue can block the ladder with this move:

The main line is complex; see for example this Little Golem discussion thread. Therefore, to establish the ladder, Red needs at minimum the specified 13 vacant hexes under the ladder stone.

Definition of ladder escape

The definition of a 5th row ladder escape is as expected.

Definition. A fifth row ladder escape is a pattern, plus five hexes marked "+" (not part of the pattern) going up and left from the first to the fifth row, such that there is no hex in the pattern to the left of any hex marked "+". This data must satisfy the following axiom: adding a 5th row ladder at any distance n ≥ 0 connects the red ladder stone to the bottom edge, with Blue to move. As usual, a candiate is such a pattern that satisfies everything except perhaps the axiom.

Characterization of ladder escapes

Conjecture. Consider a candiate P for a fifth row ladder escape. Assume L5+P, L5+1+P, L5+2+P, L4+P′, L4+1+P′, L3+P′′, and L2+P′′′ are all edge templates. Then P is a 5th row ladder escape.

Proof. The proof idea is the same as for 3rd and 4th row ladders, but there are a lot more cases to consider. First, note that by previous theorems, P′ escapes all 4th row ladders, P′′ escapes all 3rd row ladders, and P′′′ escapes all 2nd row ladders. We prove by induction on n that L5+n+P is an edge template for all n ≥ 0. The base cases n = 0, 1, 2 are true by assumption. For the induction step, assume the claim is true for n ≥ 2. We need to prove the claim for n+1.

Consider the position L5+(n+1)+P, which looks like this (followed by an additional n−2 columns of five empty hexes and the pattern P):

The eight moves marked 2 below all lose instantly to Red 3 by edge template IV-1-a:

The three moves marked 2 below also lose instantly by edge template IV-1-a:

The six moves marked 2 below give a 4th row ladder, which P′ escapes.

This leaves us with 11 more moves to consider. If Blue pushes the ladder by making the move marked 2 below, Red can answer 3, moving us to position L5+n+P, which is an edge template by the induction hypothesis.

Move 2 below gives a 3rd row ladder, which P′′ escapes.

To be continued!

Sixth row ladders and up

A similar issue arises with 6th row ladders. It is currently unknown how much space must be guaranteed under the ladder stone to form a 6th row ladder, of if this is even possible at all. And for 7th row ladders the situation is even worse. As explained in open problems about edge templates, no amount of space under the ladder (even if we demand that the entire 5th row is clear) is known to guarantee a red connection if Blue just ignores the ladder and plays elsewhere. Thus, it is possible that 7th row ladders do not even exist in theory. Of course they do not occur in practice either.

To push the theory of ladder escapes to the 5th and 6th row, one first needs to come up with a rigorous definition of a ladder (the issue being how much space one needs under the ladder stone to stop blue from cutting off the ladder immediately). Once this is done one needs to come up with a finite list of templates analogous to, say, A to D in the 4th row argument above. This may not be as easy as it looks. For example B above is A plus one extra row, and when I tried to prove the result initially I only had A, C and D, and I could not get the argument to work; the extra space was essential. More issues of this nature surely await anyone who tries to extend these results to 5th row ladders or higher. Because I am not sure that 5th row ladder escapes really ever come up in practice, I am not particularly motivated to pursue this.