View source for User:Bobson8

<hexboard size="11x11"
   coords="hide"
   contents="R e4 R h4 R c7 R g7 R c8 R d8 R e8 R f8"
   />

=== 7th row single-stone pre-template ===

The following pattern guarantees a stone on 7th row to connect to the edge. It was verified by me using HexProver developped by mirefek. Whether this connection is minimal is not known. This partially answers the single-stone 7th row edge template problem (The template exists but the shape is unknown).

<hexboard size="7x23"
  coords="none"
  edges="bottom"
  visible="area(a7,w7,w5,u3,r3,p1,n1,j3,g3,c5)"
  contents="R o1"
/>

To start with, one notice that Red can connect to the edge with a variant of [[edge template V1b]] from both sides.

<hexboard size="7x23"
  coords="none"
  edges="bottom"
  visible="area(a7,w7,w5,u3,r3,p1,n1,j3,g3,c5)"
  contents="R o1 1:n3 S blue:area(j7,w7,w5,u3,r3,p1,o1,n2,m4)"
/>

Thus Blue has these possible blocks (a~j, mirrored positions are removed).

<hexboard size="7x23"
  coords="none"
  edges="bottom"
  visible="area(a7,w7,w5,u3,r3,p1,n1,j3,g3,c5)"
  contents="R o1 E a:n2 b:n3 c:m4 d:l5 e:m5 f:k6 g:l6 h:j7 i:k7 j:l7 X:o2 Y:m2 Z:n4"
/>

For Blue playing a,b,c,d, Red can respond with X,Y,Y,Z respectively. For the rest Red can respond with b.

=== A conjecture on Trapezoid boards ===

Conjecture. For an isosceles trapezoid board of edge lengths a (Red), b (Blue), a+b-1 (Red), b (Blue), 

(1) "Red always loses regardless of who goes first" if and only if b >= 2a+1
(In other words, Red's average edge length <= Blue's-1).

(2) "Red always wins regardless of who goes first" if and only if b <= 2a-3
(In other words, Red's average edge length >= Blue's+1).

Indeed, this conjecture is an analog of the results on parallelogram boards. I have verified this conjecture for a=1,2,3, and KataHex suggested that a=4,5,6 are possibly true.


To prove this conjecture one need to verify

(1) When b=2a-3, Red wins even if Blue goes first.

(2) When b=2a-2, Red loses if Blue goes first.

(3) When b=2a, Red wins if Red goes first.

(4) When b=2a+1, Red loses even if Red goes first.


We demo some small cases (a=1,2,3).

(a,b)=(1,2): Red wins with a bridge.

<hexboard size="3x2"
   coords="hide"
   edges="none"
   visible="area(a3,b2,b1,a2)"
   contents="R 1:b1 a3"
   />

(a,b)=(1,3): Blue edges form a span template.

<hexboard size="3x4"
   coords="hide"
   edges="none"
   visible="area(a2,a3,c3,d2,d1,b1)"
   contents="B b1 d1 a2 d2"
   />

(a,b)=(2,1): Red edges form a bridge template.

<hexboard size="3x2"
   coords="hide"
   edges="none"
   visible="area(a3,b2,b1,a2)"
   contents="R b1 a3"
   />

(a,b)=(2,2): Blue wins with a chain of two bridges.

<hexboard size="2x4"
   coords="hide"
   edges="none"
   visible="area(a1,a2,c2,d1)"
   contents="B a1 d1 1:b2"
   />

(a,b)=(2,4): Red wins with a chain of a bridge and a ziggurat.

<hexboard size="6x5"
   coords="hide"
   edges="none"
   visible="area(a5,a6,d6,e5,e1)"
   contents="R e1 a6 b6 c6 d6 1:d3 S gray:area(e1,a5,a6,c6,d5,e2)"
   />

(a,b)=(2,5): Blue edges form a template. One possible defence for Red:

<hexboard size="5x7"
   coords="hide"
   edges="none"
   visible="area(a4,a5,f5,g4,g1,d1)"
   contents="B a4 b3 c2 d1 g1 g2 g3 g4 2:d4 R 1:e2"
   />

(a,b)=(3,3): Red edges are connected by the edge template IV2a.

<hexboard size="5x5"
   coords="hide"
   edges="none"
   visible="area(a4,a5,d5,e4,e1,d1)"
   contents="R a5 b5 c5 d5 e1 d1 S gray:area(a4,a5,c5,d4,e2,e1,d1)"
   />

(a,b)=(3,4): Blue wins with a chain of two ziggurats.

<hexboard size="4x7"
   coords="hide"
   edges="none"
   visible="area(a3,a4,f4,g3,g1,c1)"
   contents="B a3 b2 c1 g1 g2 g3 1:d3 S gray:area(a3,a4,f4,g3,g1,f1,e2,d2,d1,c1)"
   />

(a,b)=(3,6): Red can win with this template:

<hexboard size="8x8"
   coords="hide"
   edges="none"
   visible="area(a7,a8,g8,h7,h1,g1)"
   contents="R h1 g1 a8 b8 c8 d8 e8 f8 g8 1:f4 S gray:area(a7,a8,f8,g7, g4,h3,h1,g1)"
   />

(a,b)=(3,7): Blue edges form a pre-template. There are two template within it, removing cells either with * or with + (due to Quasar). One possible defence for Red:

<hexboard size="7x10"
   coords="hide"
   edges="none"
   visible="area(a6,a7,i7,j6,j1,f1)"
   contents="+:(d7 e7 f7) *:h1 B a6 b5 c4 d3 e2 f1 j6 j5 j4 j3 j2 j1 2:f5 R 1:g3"
   />