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A grid on a cell of a game board attacks all neighboring cells. The domination number counts the minimum number of grids such that each cell of a board is occupied or attacked by a grid.
For square boards (chess boards), the domination number has been determined in a series of papers. Here, we start to consider grids on hexagon boards \( B_n \) as parts of the Euclidean tessellation by congruent regular hexagons, where \( B_1 \) is one hexagon, \( B_2 \) consists of the three hexagons around one vertex, and \( B_n \) for \( n \geq 3 \) consists of \( B_{n-2} \) together with all hexagons having at least one hexagon in common with \( B_{n-2} \).
An upper bound is presented for the grid domination number, and exact values are determined by computer for small \( n \).