Galloping Queens #1

A galloping queen is also known as an amazon . Notice that an amazon attacks a 5x5 square centered at itself (in addition to other squares).

First, as an amazon includes a rook's movement, clearly each row and column can only contain one amazon. Since there are eight amazons and eight rows/columns, each row/column must have exactly one amazon.

Consider the amazon in the third row and suppose it is on column $c$ . Without loss of generality, suppose $c \le 4$ , otherwise reflect the board. This amazon attacks rows 1-5, columns $c+1$ and $c+2$ . Thus the two amazons in column $c+1$ , $c+2$ can only be among rows 6-8. But an amazon on any of these six squares attacks all the other five squares, so there cannot be two amazons here, contradiction.

Thus it's impossible to place eight non-attacking amazons on a regular chessboard, and so the answer is $\boxed{0}$ .

I also checked to see if any of the solutions with 8 normal queens would work for galloping queens (I correctly thought they wouldn't). Some further investigation showed me that the maximum number of galloping queens to place is actually 6.

Nice set of problems!

if this problem has a solution, this solution must necessarily be a solution for [the eight queens problem] .

We can see that no solution from eight queens solve the present problem. The problem has no solutions.