Code for the "8 Queens problem"
; REN-C solution to the N Queens problem
;
; Place all N queens on the NxN chessboard where every queen is to be placed
; on a field that is not taken nor covered by any of the other queens.
;
; We know the solutions are not symmetrical themselves but the solutions can be
; mirror images of other solutions or turned solutions.
; This solution works as follows
; By symmetry we only need to test the first half of the row or column with
; the first queen. On an odd sized board we only cannot double the number of
; results for the middle position.
; We do not know or even care if we work horizontally or vertically and even if we are
; starting at the first or last row or column is irrelevant.
; We start with adding our first queen to the board. This is a number in a
; solution block. The we go to the next level and determine the free choices
; for our next queen. This is a collection of all fields minus the ones other
; queens cover. This propagates through all rows / columns. At the end we
; reach an empty set so no solution or a set of solutions.
; This global variable is a quick and dirty fix.
number-of-solutions: 0
reset-number-of-solutions: does [
number-of-solutions: 0
]
add-one-to-number-of-solutions: does [
number-of-solutions: number-of-solutions + 1
]
get-number-of-solutions: does [
number-of-solutions
]
; This variable is to signal that results should not be duplicated for
; this value.
odd-symmetry-limit: 0
reset-symmetry-limit: does [
odd-symmetry-limit: 0
]
add-to-symmetry-limit: does [
odd-symmetry-limit: odd-symmetry-limit + 1
]
;
logic-countonly: 0
set-countonly-true: does [
logic-countonly: 1
]
set-countonly-false: does [
logic-countonly: 0
]
; Function to print the board for a solution
print-board: function [n [integer!]
board-values [block!]
][
a: copy ""
for-each b board-values [
repeat t b - 1 [ append a ". "]
append a "Q "
repeat t n - b [ append a ". "]
append a newline
]
print a
]
; Function for recursion
add-queen: function [n [integer!]
solution [block!]
free-places [block!]
][
; Determine the free choices for this queen
forbidden-places: copy []
can-see: 1
rsolution: reverse copy solution
for-each sol rsolution [
append forbidden-places sol
append forbidden-places sol + can-see
append forbidden-places sol - can-see ; too lazy to check for out of bounds
can-see: can-see + 1
]
free-choices: exclude free-places forbidden-places
if not empty? free-choices [
either n = 1 [
; now check for a solution, no more recursion possible
for-each place free-choices [
append solution place
if logic-countonly = 0 [print-board length-of solution solution]
add-one-to-number-of-solutions
if any [odd-symmetry-limit > first solution
odd-symmetry-limit = 0]
[
if logic-countonly = 0 [print-board length-of solution reverse copy solution]
add-one-to-number-of-solutions
]
clear skip solution ((length-of solution) - 1)
]
][
for-each place free-choices [
append solution place
add-queen n - 1 solution free-places
clear skip solution ((length-of solution) - 1)
]
]
]
]
; Function for main solution
solve-n-queens: function [
n [integer!] "The number queens on to place the board of size nxn"
/countonly "Only print the number of solutions found"
][
; We need to know this within our recursive function
either "/countonly" = mold countonly [
set-countonly-true
][
set-countonly-false
]
; make a basic block of the row / column numbers
places-block: copy []
count-up i n [append places-block i]
reset-number-of-solutions
; We need only do half of the first row / column for reasons of symmetry
half: either odd? n [(n + 1) / 2][n / 2]
reset-symmetry-limit
if odd? n [
loop half [add-to-symmetry-limit]
]
either n > 1 [
count-up first-queen half [
solution: copy []
append solution first-queen
add-queen n - 1 solution places-block
clear solution ; this is okay because we are in the outermost loop here
]
][
add-one-to-number-of-solutions
print-board n [1]
]
; print the number of solutions now
print spaced ["Total number of solutions for n =" n "is" get-number-of-solutions]
]
; Examples
solve-n-queens 1
solve-n-queens 2
solve-n-queens 3
solve-n-queens 4
solve-n-queens 5
solve-n-queens/countonly 5
solve-n-queens/countonly 8
; 92
If we want to know how slow REN-C is on my machine
>> delta-time [solve-n-queens/countonly 5]
Total number of solutions for n = 5 is 10
== 0:00:00.00145
>> delta-time [solve-n-queens/countonly 8]
Total number of solutions for n = 8 is 92
== 0:00:00.041275
>> delta-time [solve-n-queens/countonly 12]
Total number of solutions for n = 12 is 14200
== 0:00:08.126828
>> delta-time [solve-n-queens/countonly 14]
Total number of solutions for n = 14 is 365596
== 0:06:21.679977
