ThorAvaTahr: **error margins on estimates?**
(2009-12-14 14:30) [#6700]

Tromp and Farneback do not provide any error margins on the numbers of positions for square boards larger than 17x17. However, they do give a significancy of 4 digits for the 19x19 as opposed to up to 2 for the other square boards.

I am curious whether the heuristic used realy gives an error margin of the order of less then 1 promile. Since it is a heuristic, i would like more evidence of the error margin.

HermanHiddema: **Re: error margins on estimates?**
(2009-12-14 14:38) [#6702]

They actually give a more accurate estimate on page 28, where they say:

*For 19x19, the formula gives 2.08168199382 x 10^170, of which we can expect all digits to be correct.*

This number was arrived at not through simulation, but through a converging ratio of legal positions between square and almost square boards. The number:

L(n,n) x L(n+1, n+1) -------------------- L(n, n+1)^2

(where L(X,Y) is the number of legal positions on an X x Y board)

converges very rapidly for growing n, and can be used to accurately estimate the number of legal positions.