Some counting questions with the Rules of Chess

Chessboard and Chess pieces The game of chess is played on an 8-by-8 grid of alternately colored squares using 32 named pieces, half Black and half White, belonging to the game's two players. Pieces are named 'pawn', 'knight', 'bishop', 'rook', 'queen' and 'king', based on their shapes, with special rules which govern their initial positions on the 8x8 chessboard. Players take turns selecting one of their pieces to be moved, the rules-of-movement being different for each type of chess piece. By moving to a square which already is occupied by a piece the opposing player owns, that piece is 'captured' and removed from the chessboard. The game ends when a player captures his opponent's 'King'.

Chess moves Each player owns 8 pawns. Initially a pawn can move one or two spaces forward; afterward can only move one space forward; it can 'capture' by moving one space forward in a diagonal direction. Each player has 2 knights, 2 bishops, and 2 rooks. A rook can move any number of spaces along a column or row of the chessboard, until it encounters a piece on an occupied square, whereupon it 'captures' an opponent's piece if located there. A bishop's moves are along diagonals of the chessboard, while a knight's moves are to squares at the diagonally opposite corner of a 2-by-1 rectangle. Each player has 1 queen and 1 king. A queen's are along any row, or column, or diagonal, as many spaces as desired, until an occupied square in encountered; a King's are limited to any adjacent square that is unoccupied.

Exam Question If the Black queen and the White queen are randomly placed on an empty chessboard, what is the probability that one can capture the other on the next move?

Easier questions first If a Black rook and a White rook are randomly placed on an empty chessboard, then what is the probability that they will be in striking position? If a Black bishop and a White bishop are randomly placed on an empty chessboard, Then what is the probability that they will be in striking position?

Visualization The number of different ways of placing one Black piece and one White piece on this empty chessboard will be 64 x63, according to the Fundamental Counting Principle. So in what fraction of these ways will the two pieces be in striking position?

Alumnus Grant Steer's idea Separate the chessboard's squares into four distinct groups, based on their distances from the chessboard's outer edges, as shown above. For all the squares lying within the same one of these groups, the number of other squares in 'striking position' for a rook, or for a bishop, or for a queen, will be the same, and so the possibilities can be easily counted. Adding probabilities for four separate, mutually-exclusive sub-events will give the probability of the 'union' event that we are interested in.