View source for Open problems

* Are there cells other than a1 and b1 which are theoretically losing first moves?

* Is it true that for every cell (defined in terms of direction and distance from an [[Board#Corners|acute corner]]) there is an ''n'' such that for any [[Board]] of size at least ''n'' that cell is a losing [[opening|opening move]]?

* Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5?

* Is the [[center opening|center hex]] on every Hex board of [[Board_size|odd size]] a winning opening move?

* On boards of all [[board size|sizes]], is every opening move on the [[Board#Diagonals|short diagonal]] winning?

* Is the following true? Assume one player is in a winning position (will win with [[optimal play]]) and the opponent plays in a hex X. Let the set A consist of all empty hexes that are members of any path between opponents edges that uses the stone at X. If A is non-empty, A contains a winning move. Otherwise any move is winning, even [[passing]] the turn. (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].)


== Formerly open problems ==

=== [[Sixth row template problem]] ===

Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the sixth row?

'''Answer:''' Yes, [[edge template VI1a]] is such a template.

=== Triangle template problem ===

Are the templates below valid in their generalization to larger sizes?  (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].) 

<hexboard size="1x1"
  float="inline"
  edges="bottom"
  coords="none"
  contents="R a1"/><hexboard size="2x2"
  float="inline"
  edges="bottom"
  coords="none"
  visible="area(b1,a2,b2)"
  contents="R b1"/><hexboard size="3x3"
  float="inline"
  edges="bottom"
  coords="none"
  visible="area(c1,a3,c3)"
  contents="R c1 a3"/><hexboard size="4x4"
  float="inline"
  edges="bottom"
  coords="none"
  visible="area(d1,a4,d4)"
  contents="R d1 b3"/><hexboard size="5x5"
  float="inline"
  edges="bottom"
  coords="none"
  visible="area(e1,a5,e5)"
  contents="R e1 c3 a5"/><hexboard size="6x6"
  float="inline"
  edges="bottom"
  coords="none"
  visible="area(f1,a6,f6)"
  contents="R f1 d3 b5"/> 

'''Answer:''' No. The first one in the sequence that is not connected is the one of height 8. 

In fact, using a variant of [[Tom's move]], it is easy to see that even the following triangle, which has more red stones, is not an edge template:
<hexboard size="8x8"
 float="inline"
  edges="bottom"
  coords="none"
  visible="area(h1,a8,h8)"
  contents="R h1 f3 d5 b7 c7,a8--e8"/>

To see why, imagine that the right edge is a blue edge and that all cells outside the carrier are occupied by Blue. Note that Blue gets a 2nd-and-4th row [[parallel ladder]]. Blue wins by playing the [[Tom's_move#Tall_variant|tall variant of Tom's move]]:
<hexboard size="8x8"
 float="inline"
  edges="bottom right"
  coords="none"
  visible="area(h1,a8,h8) g1--a7"
  contents="R h1 f3 d5 b7 c7,a8--e8 B g1--a7 B 1:g2 R 2:h2 B 3:f5 R 4:g4 B 5:g3 R 6:h3 B 7:d6 R 8:g5 B 9:f7"/>

There is in fact a template of height 8 continuing the above sequence, but it requires slightly more space:

<hexboard size="8x9"
 float="inline"
  edges="bottom"
  coords="none"
  visible="area(h1,a8,i8,i6)"
  contents="R h1 f3 d5 b7"/>
The corresponding template of height 9 requires this much space:

<hexboard size="9x11"
 float="inline"
  edges="bottom"
  coords="none"
  visible="area(i1,a9,k9,k7,i5)"
  contents="R i1 g3 e5 c7 a9"/>

=== Seventh row template problem ===

Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the seventh row?

'''Answer:''' Yes. See [[Seventh row edge templates]].

[[category: Open problems]]
[[category: Forums]]