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Something similar was used in this game of 1980: http://www.pathguy.com/chess/Galaches.htm
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The labels for the directions are somewhat confusing as 'oblique' usually indicates a direction such as that of the FIDE Knight, going through intervening cells off-centredly. A more accurate description for the directions of each colour are forward/backward hex-diagonal, sideways orthogonal, forward/backward orthogonal, and sideways hex-diagonal. The linepieces in these directions I term Unicoranker, Rookfiler, Rookranker, and Unicofiler. These definitions also work on a Glinsky board, but on that it is the first two that have four directions and the second two only two.