What is big X? Nobody knows.

Rooks

  1. What is the largest number of rooks that can be placed on a chessboard in such a way that no rook may capture another? In how many ways can the rooks be placed?
  2. What is the least number of rooks that can be placed on a chessboard in such a ways that they control every square? In how many ways can the rooks be placed?

Bishops

  1. What is the largest number of bishops that can be placed on a chessboard in such a way that no bishop may capture another? In how many ways can the bishops be placed?
  2. What is the least number of bishops that can be placed on a chessboard in such a ways that they control every square? In how many ways can the bishops be placed?

Kings

  1. What is the largest number of kings that can be placed on a chessboard in such a way that no king may capture another? In how many ways can the kings be placed?
  2. What is the least number of kings that can be placed on a chessboard in such a ways that they control every square? In how many ways can the kings be placed?

Queens

  1. What is the largest number of queens that can be placed on a chessboard in such a way that no queen may capture another? In how many ways can the queens be placed?
  2. What is the largest number of queens that can be placed on a chessboard in such a way that every queen threatens exactly one other queen?
  3. What is the largest number of queens that can be placed on a chessboard in such a way that every queen threatens exactly two other queens?
  4. Is there a way to place a number of queens on a chessboard in such a way that each queen threatens exactly five other queens?

Knights

  1. Prove that, independently of his initial position, a knight may move to all the squares of a chessboard.
  2. What is the largest number of knights that can be placed on a chessboard in such a way that no knight may capture another? In how many ways can the knights be placed?
  3. What is the least number of knights that can be placed on a chessboard in such a ways that they control every square? In how many ways can the knights be placed?
  4. Can a knight move from the lower left corner of the chessboard to the upper right corner without landing twice in the same square?

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