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Single-stone templates to connect a 7th row ladder

Say that Red is laddering along the 4th row and Blue suddenly decides to play elsewhere (move 1 in the position below).

Red can now instantly connect to the bottom by playing on the third row, because of the Ziggurat 3rd row single-stone edge template (marked with plus signs).

But because no single-stone 7th row template is known, if Red is laddering along the 8th row and Blue decides to ignore the ladder and play elsewhere, Red might not be able to immediately connect! Red can play on the 7th row but perhaps Blue can block the connection to the edge, even on a very large board.

In fact, this issue is even a problem for 7th row ladders. Because at the time of writing (May 2016) the only known minimal 6th row single-stone template is edge template VI1a, and this template has vacant hexes to both the left and the right of the stone, the template is not suitable for connecting a 6th row stone placed adjacent to a 7th row ladder. Is Red's stone 1 connected to the bottom edge here?

Our 6th row single stone edge template cannot be used. This sort of problem seems very difficult at the time of writing, even using the publically available computer programs for Hex analysis, although one might imagine that custom computer code could be used to analyse this sort of position, and of course computers get faster over time so perhaps this problem will one day soon be accessible.

Because this problem seems to be unsolved, this provides a theoretical barrier for analysing 7th row ladder escape templates, as the defender can play in the escape before the ladder has reached the escape. For example it may be theoretically possible that no 7th row ladder escape template exists, although such a possibility does not seem likely.

We finish this section with the remark that this is not an issue for 6th row ladders, because an appropriate 5th row template is known: the hexes marked with a plus and the piece marked 1 form a single-stone 5th row template.

In particular black has to keep defending against a 6th row ladder until the template no longer fits (e.g. because we are approaching a ladder escape).

Answer: yes

Edge template VI1b is such a template. Therefore, given enough space, the defender in a 7th row ladder cannot just ignore the ladder. (However, this does not imply that 7th row ladders can always be pushed. It it not known whether it is possible for the defender to stop the ladder.)

Single-stone templates with a 120 degree corner

One reason that the standard 4th row edge template

is more useful than the standard 5th row edge template

is because the stone in the 4th row template has three "liberties", i.e. adjacent hexes on the board but not in the template, whereas the stone in the 5th row template only has two. However there is a single-stone 5th row template where the stone has three liberties:

(I learnt this template from shalev at http://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=669 ).

Is there a single-stone 6th row template where the stone has three liberties outside the template?

Answer: yes

Edge template VI1b is such a template.

Is there a single-stone 7th row edge template?

Among the single-stone edge templates on this site, there seem to be templates where the single stone is getting further and further away from the edge. One might then jump to the conclusion that (as long as there are enough columns) one can create templates with one stone as high up as one likes. This may be true, but at the time of writing (May 2016) this is an open problem -- nobody knows how to do construct these templates. In fact, when Cameron Browne wrote his book "Connection games" in 2005, he gave what at the time was apparently a complete list of known minimal single-stone edge templates (see the edge templates page for an explanation of the terminology used here), and there were only 12 such templates (counting reflections as different templates). There were some 5th row single-stone templates, but no 6th row template seemed to be known at the time.

In 2008 Art Duval announced the existence of edge template VI1a, a single-stone 6th row template on the Littlegolem forum here: https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339. Unfortunately, many of the links and the pictures in that thread no longer work; however the 6th row template lives on:

For simple 3rd row templates like the ziggurat, the stone to be connected is on the 3rd row and there are no hexes on the 4th or higher rows in the template. However, for a general single-stone nth row template, one could allow for the possibility that there are vacant hexes in the template on rows higher than n. For example, this 4th row single-stone edge template has vacant hexes on the 5th row in the template.

But even allowing for the possibility of vacant template hexes in higher rows than the stone, it still seems at the time of writing (May 2016) that no 7th row single stone edge template is known. Is there such a template? If so, is there a single-stone 8th row edge template? Is there a single-stone 1000th row edge template? Experts seem to be divided on this, if the Littlegolem forum thread is anything to go by.

Answer: yes

See seventh row edge templates.