Queens can sometimes be vain. Place 7 Queens on the intersections of this triangular chess board so that none can look down a line and see another Queen.

This is not a solution because the arrows point to lines that have more than one queen.

How many ways are there to do this?

Place 6 queens so that there is no room for a seventh queen without one of the original 6 queens moving.

Extensions:

Prove that it is impossible to place 8 vain queens. (Hint: label the lines as follows:)

Create your own problem on a chess board.

Credits:

Second extension and its beautiful proof by Richard Guy