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Just to comment on the root-7 idea, note that this board is the same (as far as adjacency and connectivity go) to a cylinder that is tiled with hexagons. Then the distance in that cylinder is exactly the same as in flat hex, and so root-7 needn't even be understood as an idealization. (I've wanted a toroidal board that is actually built toroidal for a while; with magnetic pieces I think it would be amusing.)
(Also, the calculation that the distance is sqrt(7) can be shortened if one is willing to use the law of cosines.)