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Math Magic Archive

Problem of the Month (September 2016)

On an N×N chessboard, when we place Q queens, what is the maximum number of squares that can be attacked exactly A times? For example, the solutions for a 3×3 board are shown below. What are the best solutions for larger N? What patterns exist?

3×3 Board
Q \ A012345678
2
1 square
2 solutions

5 squares
1 solution

5 squares
1 solution
3
3 squares
1 solution

2 squares
1 solution

4 squares
2 solutions
4
1 square
3 solutions

4 squares
4 solutions

5 squares
2 solutions

2 squares
3 solutions
5
3 squares
1 solution

4 squares
4 solutions

2 squares
3 solutions

2 squares
1 solution
6
2 squares
2 solutions

3 squares
2 solutions

2 squares
4 solutions

1 square
3 solutions

1 square
6 solutions
7
2 squares
2 solutions

2 squares
1 solution

2 squares
1 solution

1 square
2 solutions
8
1 square
1 solution

1 square
1 solution

1 square
1 solution


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