Parallelogram boards

Hex is usually played on a rhombic n×n board, but one can also try playing it on n×m parallelogram boards, where n is the number of rows, m the number of columns, and n ≠ m. For example, here is a board of size 3×7:

The problem with playing on such parallelogram boards is that the player with the shorter distance between her sides has a simple winning strategy, even when she moves second. To mitigate this, one can permit the player with the greater distance between his sides to place a certain number of stones on the board prior to the game. In particular, it has been found that Hex on a 7×9 board is a rather fair game, when Blue may start the game with two stones at once.

Size of minimal virtual connection on parallelogram boards

On a board of size n×m, one may ask what is the minimum number of stones Blue must place on the board prior to the game to guarantee a Blue win. Equivalently, one can ask what is the size of the minimal virtual connection between Blue's edges on an otherwise empty board.

The answer is known for certain small values of n and/or m:

×	1	2	3	4	5	6	7	8	9	10	11	12	Formula (if known)
1	1	2	3	4	5	6	7	8	9	10	11	12	m
2	0	1	2	2	3	4	4	5	6	6	7	8	⌈⅔m − ⅔⌉
3	0	0	1	2	3	3	4	5	5	6	7	7	⌈⅔m − 1⌉
4	0	0	0	1	2	2	3	4	4	5	6	6	⌈⅔m − 2⌉
5	0	0	0	0	1	2	2	3	3	4	5	5	⌈⅔m − 3⌉, m ≥ 7
6	0	0	0	0	0	1	2	2	3			≤5
7	0	0	0	0	0	0	1	2	2
8	0	0	0	0	0	0	0	1	2	2?

Examples of minimal virtual connections

Boards of size 1×m

For boards of size 1×m, it is obvious that Blue's virtual connection requires m pieces, because if any cell is left empty, Red will win in one move.

Boards of size 2×m

For boards of size 2×m, the size of Blue's minimal virtual connection is ⌈⅔m − ⅔)⌉. Here, ⌈x⌉ denotes the ceiling of x, i.e., the smallest integer ≥ x. Examples of such minimal virtual connections are shown for m = 2, 3, 4, 5, 6, 7. The pattern continues for larger m.

Boards of size 3×m

For boards of size 3×m, where m ≥ 3, the size of Blue's minimal virtual connection is ⌈⅔m − 1⌉. Examples of such minimal virtual connections are shown for m = 3, 4, 5, 6, 7. The pattern continues for larger m.

Boards of size 4×m

For boards of size 4×m, where m ≥ 4, the size of Blue's minimal virtual connection is ⌈⅔m − 2⌉. This is proved, using tools from combinatorial game theory, in the paper "On the combinatorial value of Hex positions". Examples of such minimal virtual connections are shown for m = 4, 5, 6, 7, 8, 9. The pattern continues for larger m.

Interestingly, in addition to a virtual connection by bridges, there is another pattern of such minimal connections when n = 4. It uses the same number of stones, but has a larger carrier.

References

• P. Selinger, "On the combinatorial value of Hex positions", Integers 22:G3, 2022. Also available from arxiv:2101.06694