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

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

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..8.

Example

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

Crossrefs

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

Sequence in context: A343880 A121896 A368610 * A173290 A312686 A312687

Adjacent sequences: A375759 A375760 A375761 * A375763 A375764 A375765

Keyword

nonn,hard,more

Author

Walter Robinson, Aug 26 2024

Status

approved