Шахматы и математика.

[cjelli]

Небольшое, но забавное математическое исследование о сравнительной
подвижности различных шахматных фигур.

221. 22 July 2003:
R = N + B

Richard Evans pointed out a "truly amazing relationship" between
Rook, Knight and Bishop. Counting all the moves a Rook could ever make,
you get 896 (64x14). Doing the same for Knight and Bishop, you get 336 and
560, respectively. And 336 + 560 = 896.

    Fascinating indeed and new to me - but I quickly found
that in the bible for this kind of thing, Schach und Zahl (Chess and numbers)
(Bonsdorff, Fabel & Riihimaa, 1971) this relation had already been observed.
And generalized. So here is a nice problem: for which other n x n board(s)
is it true that R = N + B?

I will give the answer in a few days.

PS 28 July: Four readers, Olaf Teschke, Pekka Karjalainen, Ulrich
Schimke and Rein Halbersma, took up this challenge and all found
the correct answer: R = N + B also (and only) for the 1 x 1 (0 = 0 + 0)
and the 3 x 3 (36 = 16 + 20) boards. Using different reasonings, they all
arrived at the formulae already given in Schach und Zahl:

R = 2*n^2*(n-1)

N = 8*(n-2)*(n-1)

B = 2*n*(2n-1)*(n-1)/3

Of course, S&Z also gives the formulae for the other pieces:

K = 4*(2n-1)*(n-1)

Q = 2*n*(5n-1)*(n-1)/3 (= R + B)

P = (3n-4)*(n-1) (for n=4 and greater)

Olaf Teschke wonders whether S&Z also gives the formulae for
n x m boards, but it doesn't. It does point to an unpublished study by
one Dr. E. von Fabrizi, titled Brett und Figur im Banne der Dualitהt
(1945) for a generalization 'in the broadest possible sense', which however 'requires
a lot of mathematical understanding'.

Ulrich Schimke mentions having sent his formulae years ago to the

On-Line Encyclopedia of Integer Sequences, giving the linked page as an example.

Pekka Karjalainen remarks that on a 0 x 0 board, the Knight is granted 16 moves
by the formula and the King 4, adding: "One could perhaps create some very
theoretical chess compositions without a board, but that is beyond my skill."

PS 7 August: Ulrich Schimke and David Bevan sent me
generalized formulae for m x n boards. They are:

K = 8*m*n - 6*(m+n) + 4 (Valid for m,n >= 1)

Q = R + B (Valid for n >= m)

R = (n+m-2)*n*m

B = (6*m^2*n - 2*m^3 - 6*m*n + 2*m) / 3 (Valid for n >= m)

N = 8*m*n - 12*(m+n) + 16 (Valid for m,n >= 2)

Bevan also gives a formula for the Pawn. On a board with f files
and r ranks, it is:

P = 3*f*r - 5*f - 2*r + 4 (Valid for r >= 4)

The formula for R = N + B on non-square boards, then, becomes:

m*n*(n-m-8) + 2/3*m^3 + 34/3*m + 12*n - 16 = 0 (if n >= m and m,n >=2)

As Schimke says, it is clear that if m is large enough,
for instance greater than 20, the left part of the equality is positive,
which means there is a finite number of boards where R = N + B.

These are (apart from the square boards 8 x 8, 3 x 3 and 1 x 1):

3 x 4 (R = 60; B = 32, N = 28)

4 x 6 (R = 192; B = 104; N = 88)

David Bevan takes this a few steps further, for one thing also giving the formula
for a (p,q)-leaper (the Knight is a 2,1 leaper) on a m x n board:

L = 4*m*n - 2*q*(m+n) (Valid for p=0)

L = 4*(m-q)*(n-q) (Valid for p=q)

L = 8*(m*n+p*q) - 4*(m+n)*(p+q) (Valid for q>p>0)

Bevan also gives a few boards with special relationships of this kind:

On 4 x 7 (files given first); 6 x 5; 3 x 17 and
16 x 4 boards, K = N + P

Not only on the normal board of 8 x 8, but also on 20 x 6
and 6 x 41 boards, B = K + P

On the 8 x 106 board, B = K + N

On the 5 x 9 board, B = K

On any 6*s x 6*s board, B = s*K

And finally, to get back to the equation everything started with: on the
8 x 8 board it is not only true that R = N + B, but also that R = K + N + P

© Tim Krabbe, 2003.
Original article (Item #221)

Comments

[diam.livejournal.com]

Интересный факт, но вообще-то это уровень продвинутых школьников.

[ifyr.livejournal.com]

Counting all the moves a Rook could ever make,
you get 896 (64x14). Doing the same for Knight and Bishop, you get 336 and 560, respectively.

Тут есть одна загвоздка: количество ходов для слона считается неправильно (это просто сумма для чернопольного и белопольного слонов). Для одного слона количество ходов равно 280-ти, что портит картину.

[cjelli.livejournal.com]

Это немного не так.

Имеется в виду-то кол-во ходов слона в принципе!