1. Can you place N queens on an NxN chessboard such that no queen is attacking any other? Can you find solutions for N <= 7? How many?
2. Consider a hexagon with a side-length of 3. Each side is marked off at unit intervals, then lines are drawn parallel to the sides to connect these unit marks. This results in 6 rows of triangles in each of the three directions. Assuming now that queens can move in any of the three directions, how many queens can be placed, one per triangle, such that no queen attacks any other?
3. Consider a triangle with a side-length of 9. Each side is marked off at unit intervals, then lines are drawn parallel to the sides to connect these unit marks. (This looks basically like the hexagon with its sides extended.) This results in 9 rows of triangles in each of the three directions. Now, how many of the 3-directional queens can be placed, one per triangle, such that no queen attacks any other?

Source: 1. Many sources. 2,3. Original.