[Solution] When 1 chess piece is placed in each of the 6 small squares shaded in the figure on the right, all the squares on the entire game board are controlled, so the minimum value we are looking for is no more than 6.

On the other hand, suppose there are at most 5 chess pieces placed on the game board, and let's assume exactly 5 are placed. In this case, among the 9 strip-like parallelograms formed by 8 lines parallel to $AB$, at least 4 do not contain any chess pieces. Similarly, among the 9 strip-like parallelograms formed by 8 lines parallel to $AD$, at least 4 also do not contain any chess pieces. Considering the points in these two sets of 4 parallelograms without chess pieces, it is easy to see that the small triangular squares belong to 8 strip-like trapezoids formed by lines parallel to the short diagonal. Note that each chess piece can only control 1 strip-like trapezoid, so at least 3 small squares are not controlled.
In summary, the minimum number of chess pieces to be placed is 6.
7.12 The shape of the game board is a rhombus containing a $60^{\circ}$ angle. Each side is divided into 9 equal parts, and through each division point, two lines are drawn parallel to the sides and the shorter diagonal, dividing the rhombus into many small equilateral triangles. If a piece is placed in one of these small triangles, three lines are drawn through the center of the triangle, each parallel to one of the triangle's sides, and it is said that all the small triangles these lines pass through are controlled by the piece. How many pieces are needed at a minimum to control all the small triangles on the game board?