A337722 -id:A337722 - cp's OEIS frontend

A337746 Maximum number of bishops within an n X n chessboard, where each bishop has a path to an edge.

1, 4, 8, 12, 19, 28, 37, 48, 57, 72, 87

Offset: 1

Nicholas Tran, Sep 17 2020

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).

Nicholas Tran, Solution in ASP

Cf. A337722 (knights), A335445 (rooks).

A375762 Maximum number of knights within an n X n chessboard, where each knight has a path to an edge.

Original entry on oeis.org

1, 4, 8, 14, 20, 30, 41, 55

Offset: 1

Walter Robinson, Aug 26 2024

Each knight must be either already on an edge square, or have a path of unoccupied squares which reach an unoccupied edge square (and without any other knights moving).

			For n=3, the following board, with X for each knight, is the unique solution a(3) = 8 and which cannot be 9 since the central square has no move to anywhere within the board.
  XXX
  X-X
  XXX
For n=4, the following is a solution for a(4) = 14, with each of the 4 central knights able to make a single move to one of the unoccupied corner squares.
  -XX-
  XXXX
  XXXX
  XXXX
For n = 8, one 55 knight solution is:
  XXXXXXXX
  XXXXXXXX
  XX-X-XXX
  XX-X-XXX
  -XX---XX
  XXXX-XXX
  XXXXXXXX
  XXXXXXXX

Cf. A335445 (rooks), A337746 (bishops), A337722 (knights moving off the board).

cp's OEIS Frontend

A337746 Maximum number of bishops within an n X n chessboard, where each bishop has a path to an edge.

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