Open problems about edge templates

Is there a single-stone 7th row edge template?
If one looks at the single-stone edge templates on this site, one might jump to the conclusion that if you allow yourself more and more space, you can create templates with one stone as high up as you like. This might be true, but this is an open problem -- nobody knows how to do it. 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). There were only 12 such templates, and this included reflections. 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 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: R6 C14 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Rj1 Sl1 Sm1 Sn1 Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sm2 Sn2 Sa3 Sb3 Sc3 Sd3 Sn3 Sa4 Sb4 Sa5 The template is also recorded on this site's page about edge templates with one stone.

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 at all. However for a general single-stone n'th 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 R5 C9 Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2 Sa2 Sb2 Sc2 Vd2 Sa3 Sb3 Sa4 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.

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 (modulo reflection) the one shown in the previous section above, 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.

<hexboard size="7x16" coords="hide" contents="R a1 B b1 B c1 B d1 B e1 B f1 B g1 B h1 B i1 B j1 B k1 B l1 B m1 B n1 B o1 B p1 R a2 R b2 R c2 R d2 R e2 E +:f2 E +:g2 E +:h2 B a3 B b3 B c3 B d3 R 1:e3 E +:f3 E +:g3 E +:h3 E +:i3 E +:j3 E +:k3 E +:l3 E +:d4 E +:e4 E +:f4 E +:g4 E +:h4 E +:i4 E +:j4 E +:k4 E +:l4 E +:m4 E +:c5 E +:d5 E +:e5 E +:f5 E +:g5 E +:h5 E +:i5 E +:j5 E +:k5 E +:l5 E +:m5 E +:n5 E +:b6 E +:c6 E +:d6 E +:e6 E +:f6 E +:g6 E +:h6 E +:i6 E +:j6 E +:k6 E +:l6 E +:m6 E +:n6 E +:a7 E +:b7 E +:c7 E +:d7 E +:e7 E +:f7 E +:g7 E +:h7 E +:i7 E +:j7 E +:k7 E +:l7 E +:m7 E +:n7" /> 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).

Single-stone templates with a 120 degree corner
One reason that the standard 4th row edge template R4 C7 Sa1 Sb1 Sc1 Sd1 Ve1 Sg1 Sa2 Sb2 Sa3 is more useful than the standard 5th row edge template R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2 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: <hexboard size="6x16" coords="hide" contents="E +:h1 E +:i1 E +:j1 R f2 E +:g2 E +:h2 E +:i2 E +:j2 E +:k2 E +:l2 E +:m2 E +:n2 E +:e3 E +:f3 E +:g3 E +:h3 E +:i3 E +:j3 E +:k3 E +:l3 E +:m3 E +:n3 E +:o3 E +:c4 E +:d4 E +:e4 E +:f4 E +:g4 E +:h4 E +:i4 E +:j4 E +:k4 E +:l4 E +:m4 E +:n4 E +:o4 E +:p4 E +:b5 E +:c5 E +:d5 E +:e5 E +:f5 E +:g5 E +:h5 E +:i5 E +:j5 E +:k5 E +:l5 E +:m5 E +:n5 E +:o5 E +:p5 E +:a6 E +:b6 E +:c6 E +:d6 E +:e6 E +:f6 E +:g6 E +:h6 E +:i6 E +:j6 E +:k6 E +:l6 E +:m6 E +:n6 E +:o6 E +:p6" /> (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?