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.

### The Math in This Problem:

This math problem challenges students to analyze and study the properties of a triangular chess board. With the constraint of not allowing two Queens to be on the same line, students will come up with the many ways of placing 7 Queens on various intersections. They are also challenged to place 6 Queens in a certain way to refrain from allowing a 7^{th} to enter the chess board.