Difference between revisions of "Open problems"

Revision as of 19:55, 5 October 2021

• 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 acute corner) there is an n such that for any Board of size at least n that cell is a losing opening move?

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

• Seventh row template problem: Does there exist an edge template which guarantees a secure connection for a piece on the seventh row?

• Is the center hex on every Hex board of odd size a winning opening move?

• On boards of all sizes, is every opening move on the 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 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, and edge template VI1a is such a template.

• Are the templates below valid in their generalization to larger sizes? (This problem was posed by Jory in the Little Golem forum.)
  
  
  Answer: No. The first one in the sequence that is not connected is the one of height 8. Instead, it requires this much space:
  
  
  The corresponding template of height 9 requires this much space: