12 Mar '05 15:213 edits

On an empty chessboard

let the total number of moves a King can make = K

let the total number of moves a Queen can make = Q

let the total number of moves a Rook can make = R

let the total number of moves a Bishop can make = B

let the total number of moves a Knight can make = N

let the total number of moves a Pawn can make = P

eg. R = (64)(14) = 896

because, wherever you place the Rook, it always attacks 14 squares.

On an nxn chessboard

let the total number of moves a King can make = K(n) for n > 0

let the total number of moves a Queen can make = Q(n) for n > 0

let the total number of moves a Rook can make = R(n) for n > 0

let the total number of moves a Bishop can make = B(n) for n > 0

let the total number of moves a Knight can make = N(n) for n > 1

let the total number of moves a Pawn can make = P(n) for n > 3

On an nxm chessboard

let the total number of moves a King can make = K(n,m) for n,m > 0

let the total number of moves a Queen can make = Q(n,m) for n > m

let the total number of moves a Rook can make = R(n,m) for n,m > 0

let the total number of moves a Bishop can make = B(n,m) for n > m

let the total number of moves a Knight can make = N(n,m) for n,m > 1

Let the total number of moves a Pawn can make on an rxf board = P(r,f) for r > 3, f > 0

Define a Jumper as a Knight which can jump pxq (p =< q) rather than only 1x2.

Let the total number of moves a Jumper can make = J(n,m,p,q) for n,m >= q)

.

let the total number of moves a King can make = K

let the total number of moves a Queen can make = Q

let the total number of moves a Rook can make = R

let the total number of moves a Bishop can make = B

let the total number of moves a Knight can make = N

let the total number of moves a Pawn can make = P

eg. R = (64)(14) = 896

because, wherever you place the Rook, it always attacks 14 squares.

**1a)**What is the relationship between R, B, and N? Is this a coincidence?**1b)**Are there other simple relationships between any of the six pieces?On an nxn chessboard

let the total number of moves a King can make = K(n) for n > 0

let the total number of moves a Queen can make = Q(n) for n > 0

let the total number of moves a Rook can make = R(n) for n > 0

let the total number of moves a Bishop can make = B(n) for n > 0

let the total number of moves a Knight can make = N(n) for n > 1

let the total number of moves a Pawn can make = P(n) for n > 3

**2)**Find K(n), Q(n), R(n), B(n), N(n), P(n)On an nxm chessboard

let the total number of moves a King can make = K(n,m) for n,m > 0

let the total number of moves a Queen can make = Q(n,m) for n > m

let the total number of moves a Rook can make = R(n,m) for n,m > 0

let the total number of moves a Bishop can make = B(n,m) for n > m

let the total number of moves a Knight can make = N(n,m) for n,m > 1

**3)**Find K(n,m), Q(n,m), R(n,m), B(n,m), N(n,m)Let the total number of moves a Pawn can make on an rxf board = P(r,f) for r > 3, f > 0

**4)**Find P(r,f)**5)**Prove that there is only a finite number of boards such that R(i,j) = B(i,j) + N(i,j)**6)**For what size boards are there such simple relationships between the pieces?Define a Jumper as a Knight which can jump pxq (p =< q) rather than only 1x2.

Let the total number of moves a Jumper can make = J(n,m,p,q) for n,m >= q)

**7)**Find J(n,m,p,q) when (i) p = 0 (ii) p = q (iii) 0 < p < q.