# Board to Pieces?

THUDandBLUNDER
Posers and Puzzles 12 Mar '05 15:21
1. 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.

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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2. 13 Apr '05 22:24
These are good questions...I am trying to chip them away slowly.

For 1A:

By my calculations I get B = 280 and N = 336, so together they satisfy

2B + N = R.

Not a coincidence.
3. 14 Apr '05 02:09
Nope, your Bishop formula must be wrong.

By the way, P(n,m) includes captures and en passant.
4. 14 Apr '05 03:03
Originally posted by THUDandBLUNDER
Nope, your Bishop formula must be wrong.

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

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...
5. 14 Apr '05 03:171 edit
Originally posted by davegage
If that's not right, then I think I am missing something...
Yeah, that's right.
Why do you think it is not a coincidence?
6. 14 Apr '05 04:22
Originally posted by THUDandBLUNDER

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

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