Edge template for half-infinite board?

[malcolm....@gmail.com]

Hi there.

My friend and I were discussing Hex the other day when an interesting
question arose. We weren't able to resolve it at the time (although it
may be obvious). Perhaps it would interest some of you too.

--------8<-------------8<-------------8<-----------

Consider an infinite half-plane of hexes, arranged so that there is a
bottom row which has no hexes underneath.

Number that row 0 and the rows above 1, 2, 3, etc.

Let there be a single black stone in row n.

I've used some ascii art here, so I apologise if it doesn't render
correctly.

/\
|##|
. \/ .
. . .
. . .
. . .
/\ /\ /\ /\
.... | | | | | ....
\/ \/ \/ \/
bottom edge

Black's goal is to reach the bottom edge, and White's goal is to stop
Black doing so.

Assuming White plays next, is there a strategy for White that prevents
Black from winning?

For no n, some n or all n?

--------8<-------------8<-------------8<-----------

From a few quick glances at some hex theory on the web, I think it's
related to the existence of a certain class of edge template. For
example, it may be relevant that there is no edge template with a
single black stone in the sixth row here:
http://www.drking.plus.com/hexagons/hex/templates.html
Of course, that page only considers finite boards.

Malcolm

[OHex]

On Feb 10, 6:13 am, "malcolm.r.tyrr...@gmail.com"

It's a really good quesion.

The tallest template I know if is on row 4 (your count; I would call
it row 5)
and needs a size-10 board to work. See http://hex.kosmanor.com/hex/t/t.html
for a full workup.

Looking at the templates that exist, it seems the edge length needed
grows pretty fast
with the height of the template, and there may be a height for which
no template exists
at all. Taller templates may exist but have not been worked out
because of both
difficulty and that play is not common on the required board size.

Feel free to investigate. Let us know. :o)

++ kevin

[malcolm....@gmail.com]

Thanks. To avoid any confusion, I've rewritten the problem so that the
bottom row is row 1.

--------8<-------------8<-------------8<-----------

Consider an infinite half-plane of hexes, arranged so that there is a
bottom row which has no hexes underneath.

Number that row 1 and the rows above 2, 3, etc.

Let there be a single black stone in row n.

I've used some ascii art here, so I apologise if it doesn't render
correctly.

/\
|##|
. \/ .
. . .
. . .
. . .
/\ /\ /\ /\
.... | | | | | ....
\/ \/ \/ \/
bottom edge

Black's goal is to reach the bottom edge, and white's goal is to stop
black doing so.

Assuming white plays next, is there a strategy that white can use that
prevents
black from winning?

For no n, some n or all n?

--------8<-------------8<-------------8<-----------

Known edge-templates mean that black wins for n <= 5.

Malcolm

[twixter]

Could a valid win for black involve a path which is higher than row n
at some point?
If so, there are issues of relevance to Hex theory, since most
templates involve a
presumption that the rows above could be part of other battles which
should not
be considered.

Perhaps another way of wording the problem which avoids the use of a
half infinite
plane is, is there some width K of a trapezoidal shaped grid of
height N such that
black can force connection from a single stone in the middle of the
top row to the
bottom? By trapezoidal, I mean the bottom row is longer than the top,
the two
bottom corners have interior angles 60 degrees, and the two top
corners have
interior angles of 120 degrees. The width K is the number of cells in
the bottom row.

Maybe you could allow a path higher than row N, as long as you keep
some region
above the stone off limits. Instead of a flat top trapezoid, the point
where the stone lies
would be below a concave (60 degree? 300 degree?) angle. I hope this
makes sense.

The smaller this forbidden region above the stone is, arguably the
less relevant the
problem is for someone whose task is to find the best move on the
whole board. After
all, there is an easy way to win for the player with less distance to
cross on a rhomboid
grid. Of course that proof shows that you can force a connection from
somewhere on
the bottom to somewhere on the top, not to a specific stone.

On Feb 10, 9:13 am, "malcolm.r.tyrr...@gmail.com"

[na_wspak]

There was similar discusion at littlegolem forum (see:
http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 )

On hexwiki there are templates up to 6th row (
http://www.hexwiki.org/index.php?title=Edge_templates_with_one_stone )

What is interesting, F14 (using standard notation) is connected to
bottom on 19x19 board (of course for vertical player) and this is the
minimal size of board required for connection.

It is very propable that there are single stone templates up to 8th
row due to parallel ladder trick (
http://www.hexwiki.org/index.php?title=Parallel_ladder#A_parallel_ladder_trick_puzzle
) that allows to go down from rows 7,5 to 6,4 and then 6,4 to 5,3 and
so on. but still there might exist some 'far' defense techniques for
horizontal player to block higher row one-stone-templates.

[OHex]

I think any template would be interesting to some degree, so long as
the
"allowed" region is finite. If the shapes are odd or remarkable, then
that fact becomes part of the conversation. I suppose one could judge
one template "better" than another if its "allowed" region is a subset
of
the other's, but if not then the two templates may each be useful in
different circumstances.

On the other hand, if there's some sort of fight that is unbounded,
whether it's recognizably a ladder or has some other character, then
I would not call it a connection template at all. Ladder templates
are already interesting, of course, and a discussion of those could
also be interesting on large spaces from a theoretical point of view.

++ kevin