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Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.
4

%I #35 Oct 30 2024 08:07:20

%S 16,25,36,49,62,76,92,104,120,136,152,168,184,200,216

%N Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.

%e 4 X 4:

%e x Q x x

%e x x x Q

%e Q x x x

%e x x Q x

%e 5 X 5 there are several arrangements:

%e x Q x x x

%e x x x x x

%e x x x x Q

%e Q x x x x

%e x x x Q x

%e 6 X 6 and 7 X 7 (add a row and column) pattern as 4 queens knight-1,3 and 1,4 separation (not symmetric):

%e . . . . . . .

%e x x x x Q x .

%e Q x x x x x .

%e x x x x x x .

%e x x x x x Q .

%e x Q x x x x .

%e 8 X 8: queens all knight-1,4 apart;

%e 8 X 8 has 2 o/s;

%e 9 X 9 has 5 o/s;

%e 10 X 10 has 8 o/s;

%e o x x x x x x x x o

%e x o x x x x x x o x

%e x x x Q x x x x x x

%e x x x x x x x Q x x

%e x x x x x x x x x x

%e x x x x x x x x x x

%e x x Q x x x x x x x

%e x x x x x x Q x x x

%e x o x x x x x x o x

%e o x x x x x x x x o

%e beyond 10 X 10, the 4 queens separated as 1,2 knights begins to be the best layout; at 15 X 15, the pattern is clear.

%e o x x o o x x x x o o x x o x

%e x o x x o x x x x o x x o x x

%e x x o x x x x x x x x o x x o

%e o x x o x x x x x x o x x o o

%e o o x x o x x x x o x x o o o

%e x x x x x x Q x x x x x x x x

%e x x x x x x x x Q x x x x x x

%e x x x x x Q x x x x x x x x x

%e x x x x x x x Q x x x x x x x

%e o o x x o x x x x o x x o o o

%e o x x o x x x x x x o x x o o

%e x x o x x x x x x x x o x x o

%e x o x x o x x x x o x x o x x

%e o x x o o x x x x o o x x o x

%e x x o o o x x x x o o o x x o

%Y Column 4 of A376732.

%Y Cf. A017113, A047461, A374933, A375116, A374935, A374936, A374937, A374938.

%K nonn,more

%O 4,1

%A _John King_, Aug 08 2024

%E a(18) added using data from _Mia Muessig_ by _Andrew Howroyd_, Oct 05 2024