cp's OEIS Frontend

Given an n X n chessboard, this sequence is the maximum number of bishops that can be placed on the board, such that each bishop has a free path to the edge of the board without another bishop moving.

Bishops on even and odd squares don't interfere and can be maximized separately. When n is even, the two types are equal via symmetry and thus the maximum number of bishops will be 2 * (maximum number of one type of bishop).

Given an n X n chessboard, this sequence is the maximum number of knights that can be placed on the board, such that each knight has a free path to jump "off" the board without another knight moving.

An example for a(3):

XXX

XXX

XXX

Here, all knights can jump off the board via knight movement. The middle knight is not blocked by any pieces.

An example for a(5):

XXXXX

XXXXX

XX-XX

XXXXX

XXXXX

Here, a knight can occupy all spaces except the central space, where a knight would not be able to jump off the board and would not be able to jump to another free square.

The puzzle can be visualized on an n X n chessboard. The goal is to maximize the number of rooks within the chessboard that can leave the grid without requiring another rook to move. An example invalid chessboard for a(3):

XXX

XXX

XXX

Here, the top and bottom rows of rooks can leave the chessboard. The middle row is invalid as the middle rook would not be able to leave the chessboard without one of the surrounding rooks having to move. A valid example chessboard for a(3):

XXX

X.X

XXX

Here, all rooks have free access to the outside of the grid.

This game is isomorphic to the Ship City problem proposed by Ilmer.