Board to Pieces?
12 Mar '05 15:213 editsOn 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.
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
.
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
.
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