Solving N-Queens Problem… How far can we go?

Question

The N-Queens Problem:

This problem states that given a chess board of size N by N, find the different permutations in which N queens can be placed on the board with

Answer 1

As to what is the largest N solved by computers there are references in literature in which a solution for N around 3*10^6 has been found using a conflict repair algorithm (i.e. local search). See for example the classical paper of [Sosic and Gu].

As to exact solving with backtracking,there exist some clever branching heuristics which achieve correct configurations with almost no backtracking. These heuristics can also be used to find the first-k solutions to the problem: after finding an initial correct configuration the search backtracks to find other valid configurations in the vicinity.

References for these almost perfect heuristics are [Kale 90] and [San Segundo 2011]

Answer 2

Which Prolog system are you using? For example, with recent versions of SWI-Prolog, you can readily find solutions for N=80 and N=100 within fractions of a second, using your original code. Many other Prolog systems will be much faster than that.

The N-queens problem is even featured in one of the online examples of SWI-Prolog, available as CLP(FD) queens in SWISH.

Example with 100 queens:

?- time((n_queens(100, Qs), labeling([ff], Qs))).
Qs = [1, 3, 5, 57, 59 | ...] .
2,984,158 inferences, 0.299 CPU in 0.299 seconds (100% CPU, 9964202 Lips)

SWISH also shows you nices image of solutions.

Here is an animated GIF showing the complete solution process for N=40 queens with SWI-Prolog:

Answer 3

a short solution presented by raymond hettinger at pycon: easy ai in python

#!/usr/bin/env python
from itertools import permutations
n = 12
cols = range(n)
for vec in permutations(cols):
    if (n == len(set(vec[i] + i for i in cols))
          == len(set(vec[i] - i for i in cols))):
        print vec

computing all permutations is not scalable, though (O(n!))