a study in optimization
My command line program and source code can be downloaded here.
My command line executable can be downloaded here.
A version which prints out the board positions for each solution is
Note that printing out the board positions slows down the program quite
a bit, so you may want to pipe the output to a text file (nqprint 16
> output.txt) & then examine the results after the program has
The N Queens Problem
(chess graphics from this site)
How many ways are there to arrange N non-attacking queens on an N x N chessboard?
For a regular-sized board (8 x 8), there are 92 distinct solutions, one of which is shown above.
Note that for each solution, no two queens can occupy the same column, row or diagonal.
For a 1 x 1 board, there is one trivial solution:
For 2 x 2 and 3 x 3 boards, there are no solutions. For a 4 x
4 board, there are two:
These are considered distinct solutions, even though the solutions are mirror images of each other.
There is no quick and easy way to calculate the total number of N queen solutions for an N x N board.
My program is a heavily-optimized C program, yet it still takes over a week for an 800 MHz PC to calculate the number of solutions for a 21 x 21 board. (In case you're curious, there are 314,666,222,712 solutions.)
2x - 10x speedup over other solutions
My program is almost twice as fast as the program used to calculate the "world record" N queens solution (the 23 x 23 board). Source for that program, written by Sylvain Pion and Joel-Yann Fourre, can be found here.
It's also up to 10 times as fast the N-queens
program written by Dr. Timothy Rolfe.
Rolfe is a computer science professor who has written Algorithm
Alley columns for Dr. Dobb's Journal.
His paper on Queens on a Chessboard: Making the Best of a Bad Situation
can be found here.
To compile the "world record" program in Visual C++, just add algo.c and apart.c to your project. The board size is hard-coded in algo.h.
Why is this important?In and of itself, the N queens problem is not important. However, the problem works as an interesting optimization testbed -- there are many, many solutions to this problem on the web, but few of them are fast. Also, techniques used to speed up this code can be used in other areas, such as a chess-playing program or a unit path-finding algorithm for a game.
History and Algorithm
The 8 queens problem was studied by Gauss in the mid-1800's. It is often described in many undergraduate programming texts, since solving it requires students to understand how to program a backtracking algorithm.
The backtracking algorithm used to solve this problem
I became interested in this problem when I was considering taking an Artificial Intelligence course which used LISP as its programming language. I downloaded a LISP compiler and a sample program which found one solution for the 8 queens problem. This program was s....l....o....w. I figured I could write a program in C which found all solutions to the 8 queens problem faster than this LISP program could find one solution. I was right. Inspired by the writings of Michael Abrash, I've been trying to optimize that program ever since.
Postscript: To be fair, the LISP program was not compiled, and since then I've taken that A.I. course and learned to appreciate LISP's strengths.
Home page of Sylvain Pion, who co-wrote the "world record" N queens
Home page of Timothy Rolfe, Ph.D.:
Eric Weisstein's World of Mathematics:
Information on the n Queens problem:
Interesting chess problem page which appears to be in Czech:
return to my programming demos page.