- 12 Mar '05 15:21 / 3 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

. - 14 Apr '05 03:03

Hmmm...maybe i misunderstood initially. By B do you mean the number of moves a bishop can make, assuming he is not confined to one color (I was assuming one color only)?*Originally posted by THUDandBLUNDER***Nope, your Bishop formula must be wrong.**

By the way, P(n,m) includes captures and en passant.

In that case it would be 2 X 280 = 560, and then

B + N = R.

Still not a coincidence.

If that's not right, then I think I am missing something... - 14 Apr '05 04:22

Not sure yet. I was hoping that if I stated it matter-of-factly you would just buy that I knew what I was talking about...my plan seems to have failed...*Originally posted by THUDandBLUNDER*

**Why do you think it is not a coincidence?**

it seems a reasonable relationship, but still not quite sure how to rationalize it yet...