login
Maximum number of safe (i.e., not attacked) squares when placing n queens on a standard 8 X 8 chessboard.
1

%I #34 Mar 26 2021 12:48:02

%S 42,31,25,24,17,15,13,11,10,9,8,7,7,6,6,5,5,4,4,4,4,4,4,4,3,2,2,2,2,2,

%T 2,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Maximum number of safe (i.e., not attacked) squares when placing n queens on a standard 8 X 8 chessboard.

%C On a standard chessboard, one can place from 1 to 64 queens. By arranging them in specific patterns, one can maximize the number of "safe" squares, i.e., squares that are not attacked by any queen.

%e a(8)=11 because the maximum number of safe squares when 8 queens are placed on a regular 8 X 8 chessboard is 11. In the diagram below, each Q represents a queen and each s is a safe square:

%e .

%e Q Q Q . . . . .

%e . Q Q Q . . . .

%e . . . . . s s s

%e . . . . . . s s

%e . . . . . . . s

%e Q Q . . . . . .

%e . . . . s s . .

%e . . . . s s s .

%e .

%e This is in fact the "first" solution (when all possible ones are sorted), with the queens nearest the top-left corner.

%Y Cf. A001366, which answers the same question but on boards of different sizes and same number of queens.

%K nonn,fini,full

%O 1,1

%A _Alexios Zavras_, Mar 02 2021