Chess and the "Golden Ratio"...

[Steve Maughan]

Recently I was asked if the Golden Ratio or Mean (http://en.wikipedia.org/wiki/Golden_ratio) can be found in chess or computer chess. The value is approximately 1.618:1

My first reaction was "no". Later it did occur to me the branching factor of modern engines must be quite close to the ratio. A coincidence? Could the Golden Ratio be a theoretical lower bound of the branching factor (I doubt it, but who knows).

Can anyone think of any other occurrences of the Golden Ratio in chess?

Steve

[elpapa]

Pawn = 1.00
Minor = 3.25
Rook = 5.25
Queen = 8.50

Queen / Rook = 1.619
Rook / Minor = 1.615
Minor / 2 Pawns = 1.625

I don't see how this has anything to do with geometry and the golden ratio, though.

[Ajedrecista]

Hello Steve:

Recently I was asked if the Golden Ratio or Mean (http://en.wikipedia.org/wiki/Golden_ratio) can be found in chess or computer chess. The value is approximately 1.618:1

My first reaction was "no". Later it did occur to me the branching factor of modern engines must be quite close to the ratio. A coincidence? Could the Golden Ratio be a theoretical lower bound of the branching factor (I doubt it, but who knows).

Can anyone think of any other occurrences of the Golden Ratio in chess?

Steve

I found nearly golden ratios in perft... I admit that it is farfetched. I took a look to this link (thank you very much to Steven Edwards):

http://dl.dropbox.com/u/31633927/Perft/InitialPosition

I calculated each ratio of different moves of the same depth with this Fortran 95 programme that I quickly wrote for this purpose:

Code: Select all

program Golden_ratio_in_perft

implicit none

integer :: i, j
real(KIND=3) :: m(1:20), d(1:19,2:20), phi, r(1:19,2:20), s(1:19,2:20)

phi = 5d-1*(1d0 + sqrt(5d0))  ! By definition.

open(unit=11, file='Input.txt', status='unknown', action='read')
do i = 1, 20
  read(11,*) m(i)  ! Perft of i-th move.
end do
close(11)

open(unit=10, file='Output.txt', status='unknown', action='write')
do i = 1, 19
  do j = i+1, 20
    d(i,j) = m(i)/m(j)
    r(i,j) = abs(d(i,j)*phi - 1d0)
    s(i,j) = abs(d(i,j)/phi - 1d0)
    write(10,'(F9.6,A,I2.2,A,I2.2)') r(i,j), '     ', j, '/', i
    write(10,'(F9.6,A,I2.2,A,I2.2)') s(i,j), '     ', i, '/', j
    write(10,*)
  end do
end do
close(10)

end program Golden_ratio_in_perft

I choosed r(i,j) and s(i,j) in a wise form, in the sense that I only had to search numbers near zero in the Notead. I finally picked up r(i,j) < 0.01 or s(i,j) < 0.01.

Code: Select all

Na3 63,558,937,554,457,795
Nc3 84,663,535,534,493,517
Nf3 82,432,607,453,980,338
Nh3 64,658,238,653,217,709
a3 54,239,338,583,061,004
a4 79,097,912,720,718,078
b3 72,470,867,205,869,240
b4 73,447,518,515,970,566
c3 85,630,662,795,901,357
c4 97,580,419,694,352,415
d3 141,516,447,571,471,634
d4 213,234,089,428,603,755
e3 241,123,823,129,738,679
e4 246,890,754,665,609,637
f3 43,376,546,076,090,990
f4 61,027,789,644,649,681
g3 75,907,561,722,826,536
g4 65,566,769,916,823,116
h3 53,765,991,918,774,270
h4 80,876,962,213,785,922 
Depth: 13   Count: 1,981,066,775,000,396,239

Move 1 is 1.- Na3, ..., move 20 is 1.- h4. For exampe, here is Input.txt file for depth 13:

Code: Select all

63558937554457795
84663535534493517
82432607453980338
64658238653217709
54239338583061004
79097912720718078
72470867205869240
73447518515970566
85630662795901357
97580419694352415
141516447571471634
213234089428603755
241123823129738679
246890754665609637
43376546076090990
61027789644649681
75907561722826536
65566769916823116
53765991918774270
80876962213785922

I did these calculations up to depth 13. Here are the two numbers I found enough close to &#934; = [1 + sqrt(5)]/2:

Code: Select all

Perft(9):

d3 176,976,245,463
Nc3 109,418,317,145

(176,976,245,463)/(109,418,317,145) ~ 0.9996254535018362639733146888474 * Φ

------------------------

Perft(11):

e4 245,841,494,675,197
d3 151,857,971,385,067

(245,841,494,675,197)/(151,857,971,385,067) ~ 1.0005296275760005698065976755521 * Φ

I hope that someone can find a less farfetched golden ratio occurrence in chess (I also found rook/knight ~ 5/3.09 or 5/3.1, for example).

Regards from Spain.

Ajedrecista.