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In a Take-Away Game on hypergraphs, two players take turns to remove thevertices and the hyperedges of the hypergraphs. In each turn, a player mustremove either a single vertex or a hyperedge. When a player chooses to removeone vertex, all of the hyperedges that contain the chosen vertex are alsoremoved. When a player chooses to remove one hyperedge, only that chosenhyperedge is removed. Whoever removes the last vertex wins the game. Followingfrom the winning strategy for the Take-Away Impartial Combinatorial Games ononly Oddly Uniform or only Evenly Uniform Hypergraphs, this paper is about thenew winning strategy for Take-Away Games on neither Oddly nor Evenly UniformHypergraphs. These neither Oddly nor Evenly Uniform Hypergraphs, however, haveto satisfy the specific given requirements.

Take-Away Impartial Combinatorial Games on Hypergraphs and Other Related Geometric and Discrete Structures