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 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?

• 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, 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 Little Golem forum.)

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:

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 tall variant of Tom's move:

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

The corresponding template of height 9 requires this much space:

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.