Talk:Eight queens puzzle

message: : I have found a modular solution for extremely big boards (e.g. N>1000). It is an extension of the Klove algorithm, but for arbitrary board sizes(the Klove
solution is only for gcd(N, 6)=1(odd numbers bigger than 6). A Matlab program is in the file-exchange repository. I will not include a link, but type in 
Matlab fileexchange: N-Queen problem modular. It will flash there. Mihail Georgiev
 195.75.72.179 (talk) 16:01, 27 September 2017 (UTC)
thread_title:  Mihail Georgiev's solution 

message: How about some attribution or reference for the simple algorithm? When was it
discovered? Given the number of published algorithms using various kinds of search
even for finding single solutions, it would seem that this has to be a fairly recent
discovery? Or?
: Recent?  Not really. The first proof of a simple algorithm for producing a solution to the n-queens problem for every n>=4 can be found here: Wilhelm Ahrens. Mathematische Unterhaltungen und Spiele. B.G. Teubner, 1910.
: A proof in English can be found here: E.J. Hoffman, J.C. Loessi and R.C. Moore. Constructions for the Solution of the m Queens Problem. National Mathematics Magazine, March-April:66-72, 1969.
: PittBill 13:28, 20 Apr 2005 (UTC)
replies: text:  I don't think any of those are on the Internet, so I can't check whether any of the published algorithms are simpler to follow or remember than the one in the article.  If so, can you please replace the one in the article and give the source.  If not, please leave it be. — 131.230.133.185 08:31, 13 Jun 2005 (UTC)
thread_title:  Algorithm Attribution 

message: :In general, the number of possible solutions of placing N queens on a N by N board, considering the symmetries as distinct solutions, is:
And then the formula is not given, only the values.  
-- WhiteDragon 06:26, 10 Feb 2005 (UTC)
thread_title:  Number of Solutions 

message: For N=2 and N=3 there are no solutions. For all other values of N<600 there is at least one solution. This I know because For each of these values of N I computed a solution. I conjecture that solutions exist for all values of N excep N=2 and N=3. Does somebody know something about this conjecture, may be even a proof or a counterfact?
Thanks Jos.koot (talk) 18:00, 9 July 2010 (UTC)
thread_title: Question

message: What is CLP(FD)? PittBill 01:23, 6 Jun 2005 (UTC)
replies: text: Constraint logic programming over finite domains (http://www.google.com./search?btnI&q=CLP+FD).  Fixed in article.  — 131.230.133.185 08:31, 13 Jun 2005 (UTC)
thread_title:  What is CLP(FD)? 

message: I am missing either link or listing of brilliant sollution posted to Usenet.
Philipz 09:07, 11 Sep 2005 (CEST)
#include <stdio.h> 
static int count = 0; 
void try(int row, int left, int right) { 
   int poss, place; 
   if (row == 0xFF) ++count; 
   else { 
      poss = ~(row|left|right) & 0xFF; 
      while (poss != 0) { 
         place = poss & -poss; 
         try(row|place, (left|place)<<1, (right|place)>>1); 
         poss &= ~place; 
      } 
   } 
} 
void main() {
   try(0,0,0); 
   printf("There are %d solutions.\n", count); 
}
It's not brilliant, though it is compact. Similar code that works for N other than 8 and takes half as long to run by counting vertically symmetric solutions only once can be found at https://rosettacode.org/wiki/N-queens_problem#C -- Jibal (talk) 09:39, 7 June 2022 (UTC)
thread_title:  C Program 

message: The eight-queens problem doesn't seem to be exact cover.  A solution for a square larger than 1x1 can not use all the nontrivial diagonals (2x2 and 3x3 have no solutions, and for n > 3, 2n-3 > n).  Should "such that every column has a 1 in precisely one of the chosen rows" be restated to "such that each of the first 2n columns has a 1 in precisely one of the chosen rows, and each other column has a 1 in at most one of the chosen rows."?
Ralphmerridew 20:12, 17 January 2006 (UTC)
thread_title:  Exact Cover 

message: Here's a simple implementation in Python that generates all 8-castles solutions by a permutations() generator, then filters diagonals by checking that the set of diagonals represented has 8 elements.  I think this is a more clear solution of the problem in Python, but the article's current recursive solution is probably better for readers because it represents how the problem would be solved in most languages.
# Print all solutions to 8-queens problem.  Public domain, Connelly Barnes 2006.
def permutations(L):
  if len(L) <= 1:
    yield list(L)
  else:
    for i in range(len(L)):
      for p in permutations(L[:i]+L[i+1:]):
        yield [L[i]] + p
def queens(n):
  for L in permutations(range(n)):
    if (len(set(i+L[i] for i in range(n))) == n and   # Main diagonals
        len(set(i-L[i] for i in range(n))) == n):     # Skew diagonals
      yield L
if __name__ == '__main__':
  for sol in queens(8):
    for col in sol:
      print '.'*col + 'Q' + '.'*(8-1-col)
    print
You can shorten the code significantly at the expense of clarity by using generator expressions.
- Connelly 00:29, 11 May 2006 (UTC)
thread_title:  Solution by Permutations in Python 

message: ..is it possible to cover a 8x8 chesstable with 4 queens?
The number of solutions for covering an n by n chessboard with 4 queens are
as follows (including rotational and reflectional variants):
     n Solutions
 5 5230
 6 2744
 7 86
 8 0
Computed by a J program. Roger Hui 06:10, 19 May 2006 (UTC)
thread_title:  4-Queens Puzzle 

message: The statement 
 The GNU Prolog program below resolved a 100 queens problem in less than a tenth of a second. 
is meaningless without a frame of reference. Giving at least the processor used for the test and the time of a slower algorithm would help matters greatly. Andrew J. MacDonald 03:51, 30 November 2006 (UTC)
thread_title:  Complexity 

message: The article seems to be arbitrarily using the words "distinct" and "unique" with its own meanings. While it does help to make it less verbose, "unique" already has the usual meaning of being the only one of something, and it's absurd to speak of "unique solution 1, unique solution 2", etc. If at all there is a unique solution, there can be only of it. Does anyone have any suggestions on what to use in place of "unique"? ("Truly distinct", "distinct even  after considering symmetries", "nonisomorphic",...) --Shreevatsa 17:49, 6 January 2007 (UTC)
replies: text: Durango Bill's website uses "unique", but the standard term seems to be "fundamental". (See for example, Watkins' book.  Or Google 'fundamental solution n-queens') PittBill (talk) 12:14, 17 June 2008 (UTC)
replies: text: I have fixed the abuse of language concerning "unique".  Shreevatsa is correct; the word is wholly inappropriate.  There are two kinds of solution counts: those that consider symmetrically related configurations to be the same, and those that consider them to be distinct.  The correct term for the former is "fundamental".  The correct term for the latter is not so clear, but "all" would be appropriate.  I've allowed the article to call them "distinct", for lack of a better term.  Zaslav (talk) 04:43, 12 March 2014 (UTC)
thread_title:  Distinct&ndash;Unique 

message: Isnt this in some way similiar to solving sudoku? Each row/column may only have one queen, is like every row may only have the same value once. Sould we include Sudoku in the Related Related Problems section? 81.221.196.70 14:17, 4 May 2007 (UTC)
replies: text: Both this problem and Sudoku are examples of Exact cover problems.  I've added a sentence to that effect to the Related Problems section. Oliphaunt 09:28, 5 May 2007 (UTC)
thread_title:  Similar to Sudoku? 

message: I came to this page to learn about the problem and try to solve it myself (and to find out if a solution was possible).  The solution appears right at the top, spoiling the puzzle for those who just want to try it.  —Preceding unsigned comment added by 70.171.221.203 (talk) 22:44, 27 September 2007 (UTC)
replies: text: Wikipedia is about providing information, not hiding it. Also, 8 queens isn't really a puzzle and isn't "spoiled" in any way by knowing any or all of the 92 solutions. Jibal (talk) 09:49, 7 June 2022 (UTC)
thread_title:  Move solution to the bottom of the page and add a spoiler warning 

message: Can anyone post the pseudocode version of the fastest algorithm that can find all unique solutions for n queens? 75.87.86.76 (talk) 23:05, 17 April 2008 (UTC)
replies: text: fascinating contribs from that IP ... numerous screws loose. -- Jibal (talk) 10:11, 7 June 2022 (UTC)
thread_title: fastest algorith

message: Currently under the section "Constructing a solution", it states:
:"there are 4,426,165,368 (64×63×...×58×57/8!, [...]) possible arrangements"
I wonder if it would be better to write
:"there are 4,426,165,368 ((64!–56!)/8!, [...]) possible arrangements"
Or would this just render it more obscure to a non-mathematical reader (especially one unfamiliar with the factorial)? − Twas Now ( talk • contribs • e-mail ) 01:23, 27 July 2008 (UTC)
replies: text: Not, of course. You probably mean \frac{64!}{56!}8!. 16:38, 15 July 2009 (UTC)

text: Mr. Matei seems to like to drop irrelevant comments all over Wikipedia. Jibal (talk) 10:12, 7 June 2022 (UTC)
thread_title:  Possible change 

message: In the aforementioned paragraph, it says that there are around 4 billion possible arrangements when in fact an 8x8 board has ~16 mil arrangements. Am I missing something?  —Preceding unsigned comment added by ThisThat911 (talk • contribs) 17:35, 5 February 2009 (UTC)
replies: text: There are 4 billion arrangements of 8 queens on the board.  But it is easy to see that you can restrict the possibilities to 1 per row (or column) and that reduces the possibilites to 16 million.  I'll try to clarify that in the article later.  Bubba73 (talk), 18:18, 5 February 2009 (UTC)
thread_title:  Problem with paragraph "Constructing a solution" 

message: The related article Eight queens puzzle solutions is being discussed for deletion here. Johnuniq (talk) 03:47, 12 August 2010 (UTC)
thread_title:  Eight queens puzzle solutions 

message: to give u trouble, lets say we got only small mem but we seek 4 bigger than mem possibilities n-queens solution, we might find usefull random generators or similaries to produce the brute solution of a slice of the board (virtual board)... in mem we might find usefull to note corrections from brute solutions.these brute solutions might b formed evn by markov sources like generators of numbers 93.118.212.93 (talk) 05:34, 24 August 2012 (UTC)
thread_title:  N-Queens; largest N 

message: It should be noted somewhere inside that the N-Queens problem has a polynomial (and very quick) solution. I found it by accident, see http://doi.acm.org/10.1145/101340.101343
Neználek (talk) 13:00, 7 April 2011 (UTC)
replies: text: There is no polynomial f(n) that gives the number of solutions of the n-Queens Problem.  Zaslav (talk) 04:39, 12 March 2014 (UTC)

text: I believe that paper provides an algorithm to find a solution to an N-queens problem for large N, not to calculate the number of solutions. Jibal (talk) 10:17, 7 June 2022 (UTC)
thread_title:  Polynomial solution 

message: Why would the 'Explicit' solution shown for n=8 not appear to be among the 'Unique' or 'fundamental' solutions?Cander0000 (talk) 03:51, 31 December 2012 (UTC)
replies: text: The explicit soluition for n=8 is a variation on #12 of the fundamental solutions.  Bubba73 You talkin' to me? 04:12, 31 December 2012 (UTC)
thread_title:  Explicit Solutions - not included in the unique solutions 

message: The tabular in this article mentions for N=10 that there are 92 fundamental solutions.
I just implemented this myself and can confirm all numbers from 1 to 9, but I get 96 fundamentals for N=10.
Can someone confirm one or the other?  — Preceding unsigned comment added by 94.218.43.105 (talk) 21:08, 28 November 2016 (UTC)
replies: text: This is not the right forum for this type of question, but you can follow the links (as I did) to see that all published values that I could find give 92. --Bill Cherowitzo  (talk) 21:55, 28 November 2016 (UTC)
thread_title:  Fundamental Solutions for N=10 wrong? 

message: There should not be too much computer code added to the article, see MOS:CODE.  In particular, "Multiple source code implementations are not appropriate ...".  There is already one example, referenced to a classic book.  No more should be added. Bubba73 You talkin' to me? 15:35, 17 July 2013 (UTC)
thread_title:  Computer code 

message: "Solution 10 has the additional property that no three queens are in a straight line.". Maybe I'm just missing something, but at least solution 9 has that property aswell.79.253.51.222 (talk) 22:56, 6 November 2015 (UTC)
replies: text: A5, E3 and G2 are a straight line. --JBL (talk) 23:07, 6 November 2015 (UTC)
thread_title:  3 Queens on a straight line 

message: "Solution 10 has the additional property that no three queens are in a straight line."
Isn't that obvious, and shared by all solutions? If there were three queens in a straight line, they'd be threatening each other, and it wouldn't be a solution.
2605:A000:EF84:2C00:10C:6B90:8406:330B (talk) 23:20, 18 April 2017 (UTC)
replies: text: Not all lines are rows, columns, or diagonals.  (See also the talk page section directly above this one.)  --JBL (talk) 23:50, 18 April 2017 (UTC)

text: "line" means "line" in the usual geometric sense.  In the section above, I have identified a set of three queens that lie in a straight line; does that help? --JBL (talk) 12:38, 9 January 2019 (UTC)

text: The meaning of "straight line" is well known and well documented elsewhere; there is no need to add it to this article. Ranks (rows), files (columns), and diagonals on a chessboard are lines at 0 (or 180), 90 (or 270), and 45 (or 135) degrees ... but that of course does not exhaust all the possibilities for straight lines. Jibal (talk) 10:30, 7 June 2022 (UTC)
thread_title:  No Three In A Line 

message: In the solution to the Eight Queens section, it says that if you allow symmetry operations of rotations and reflections to be reduced to equivalence classes, you get 12 distinct solutions. I would like to point out that if, in addition, you allow translations, you can reduce the solution set to (at most) six equivalence classes. See this Problem of the Week at Math Help Boards that I posted for an explanation. We may or may not want to allow translations to reduce the solution set further, but I thought I would point this out. Ackbeet (talk) 17:18, 13 June 2019 (UTC)
thread_title:  Solutions Section: Further Duplications 

message: The solutions have a certain order, but wouldn't it make more sense to show them in sequential order? I.e., decide on an order for the Quuens, say first from bottom to top, second from left to right [following the coordinate system]. Then solutions that have a Queen in the corner A1 appear first, then solutions with a Queen on B1, etc. If we use a coordinate system, we can use (c1,c2,c3,c4,...) to indicate that the n-th Queen is in the n-th row and cn-th column. The solutions should then be ordered as follows (using current numbering): 3 (1,2,4 - flipped), 2 (1,2,5 - flipped); 1 (2,4,6 - flipped), 4 (2,5,7,1), 5 (2,5,7,4), 6 (2,6,1), 7 (2,6,8), 8 (2,7,3), 9 (2,7,5); 12 (3,5,2), 10 (3,5,8), 11 (3,6,2). Considering that only 3 solutions are flipped and 10 solutions appear in the correct order, I assume that the shown order was not the one originally envisioned. Bcurfs (talk) 03:56, 16 November 2020 (UTC)
replies: text: Yep, lexicographic ordering (of the minimal elements of the groups) does make sense, particularly as it's the order in which the solutions would be enumerated by a standard backtracking solver. The current ordering seems to be a holdover from a different, fairly OR-ish discussion of the structure of solutions. Sneftel (talk) 11:19, 18 November 2020 (UTC)
replies: text:  I don't care about what order they are listed in, but anyone who goes to change this should be aware that there are several different places in the article where solutions are referred to by number, so care is needed in re-ordering them. --JBL (talk) 12:37, 18 November 2020 (UTC)
thread_title:  Order of Solutions 

message: What of these recent set of articles on a "solution" by Harvard mathematician Michael Simkin?
* https://phys.org/news/2022-01-harvard-mathematician-year-old-chess-problem.html
* https://scitechdaily.com/harvard-mathematician-resolves-150-year-old-chess-problem/
* https://www.sciencealert.com/a-harvard-mathematician-has-solved-an-epic-150-year-old-chess-problem
* https://www.iflscience.com/editors-blog/dont-slay-queen-chess-puzzle-solved-after-150-years-thanks-to-mathematics/
This development should be in the article.  I see Simkin mentioned only twice in the notes. TuckerResearch (talk) 18:48, 27 January 2022 (UTC)
replies: text: It is already covered in the article.  The real question is why there are a bunch of new churnalist rehashings of the Quanta piece (also cited in the article) from September. --JBL (talk) 21:31, 27 January 2022 (UTC)

text: I have restructured the article to better highlight the recent work, and also rewritten the lead section. --JBL (talk) 17:09, 29 January 2022 (UTC)
thread_title:  What of these recent set of articles on a solution? 

message: So back in February  removed the sourced sample code as "outdated and provid[ing] no insight".  Subsequently,  has added it back -- except not actually, because what they've added is a rewritten version in a different language and with various changes made to the cited version.  I was under the impression that this was something that's been discussed a lot here (in light of WP:OR and WP:CODE) but I'm either confused or incompetent because I don't see it on this page or in the article edit history.  So let me try to kick off a discussion about it.  Personally, I'm inclined towards the view that if we're going to include sample code, I would rather include a code sample directly cited to something, rather than reworked versions of it.  --JBL (talk) 11:55, 17 April 2022 (UTC)
replies: text: It would be desirable to have a citation for the code, but Python is essentially a runnable pseudocode which is certainly more useful to the reader as a high-level overview of one of the algorithms than Wirth's Pascal code which required reader to carefully examine to figure what it does. I'm not entirely sure it adds anything to the article more than what is covered in the previous section, but few lines of code sometimes read better than prose so I would tolerate it. We certainly don't want to have two versions differing only in insignificant details as in the current version of the article, though! If someone wants more, the External Links section already has a link to Rosetta Code (and two GitHub links of questionable usefulness; there are thousands of implementations written as an exercise every year). – MwGamera (talk) 16:24, 17 April 2022 (UTC)

text: In fact, I reintroduced the code for several reasons: First, I was intrigued by Niklaus Wirth's idea of ​​representing the chessboard by three lists (a, b, and c). I think that idea alone is worth presenting in some form. Second, to document that as languages ​​continue to evolve, it is possible to represent Wirth's basic idea without low-level operations. And third, because there are so many implementations that just take care of such low-level details. While there are over a hundred versions on Rosetta code, it takes a long time to find one that's optimized for readability and still works in the real world. That's exactly why I put Wirth's code in exactly this form. I also wanted to demonstrate the trick mentioned in the previous section of combining recursion and permutation to reduce the number of investigations. But because this is exactly what damages readability, I'm not sure if that was a good idea. --Majow (talk) 08:21, 18 April 2022 (UTC)
replies: text: You may want to read through WP:CODE for potential reasons *not* to include certain types of sample code. I'm not sure whether Wirth's representation of the chess board is notable enough, or relevant enough to the eight-queens puzzle, to be captured in the article, but if it is then it should be discussed directly and supported by independent sources, not just hinted at in the sample code. But something being "worth preserving" doesn't necessarily make it worth including in an article. WP does not strive to be a repository of source code, even carefully hand-picked "best" source code. Sneftel (talk) 11:14, 19 April 2022 (UTC)
replies: text: It may indeed be a personal problem, but I only understand some things because I can try them and change them in the process. When we discussed the eight queens puzzle at school, I could understand it only because of the explanation and code of N. Wirth (I still have a copy of the book "Algorithms and Data Structures"). I had a lot of trouble understanding the principle of recursion and kept adding print statements to the code and recompiling it again and again until I finally got the idea behind it. That's why I simply missed the sample code here and I personally think N. Wirth's solution is excellent. However, since the array data structure is used in the Pascal language, there are some low-level things you just don't need to do when using lists in Python. Therefore, code readability in Python is better, but memory consumption and execution speed are significantly worse than in Wirth's original. Wirth hardly uses the stack because he allocates the arrays globally and not locally, and the operations on arrays are of order O(1), while on lists they are significantly slower with order O(n). --Majow (talk) 09:45, 20 April 2022 (UTC)
replies: text: None of these personal facts are relevant. Please review WP:OR Jibal (talk) 10:48, 7 June 2022 (UTC)

text: Looking through the sample code, it's a great illustration of the difference between a pseudocode description of an algorithm and an "example implementation" of that algorithm, and is firmly in the latter camp. If one came to the page not knowing the backtracking algorithm to solve the eight-queens puzzle, that code is not formulated to help them. It involves generators, a relatively advanced feature of Python and one which would confuse someone with only a passing familiarity with the language; the variable names are opaque; and it gratuitously uses terse list operators which, again, would confuse someone with limited Python exposure. (It fails as an implementation as well: the state structures are copied rather than mutated, and there's an unnecessary iteration over the row, leading to an additional factor of n in the time complexity. But that's a comparatively minor concern.) I might suggest, Majow, that that is the sort of code which looks beautiful to one who has spent a long time looking at it, yet is opaque to one who is seeing it for the first time. And we really need to prioritize that latter group when writing an encyclopedia. Sneftel (talk) 09:29, 25 April 2022 (UTC)
thread_title:  Sample code: how do we feel about it? 

message: Just suggesting an extra column, symmetrical and fundamental solutions. For some reason I could not find this particular sequence in OEIS!
n 	fundamental and symmetrical	fundamental           	all
1 	1                          	1                     	1
2 	0                          	0                     	0
3 	0                          	0                     	0
4 	1                          	1                     	2
5 	1                          	2                     	10
6 	1                          	1                     	4
7 	2                          	6                     	40
8 	1                          	12                    	92
9 	4                          	46                    	352
10	3                          	92                    	724
11	12                         	341                   	2,680
12	22                         	1,787                 	14,200
13	36                         	9,233                 	73,712
14	105                        	45,752                	365,596
15	310                        	285,053               	2,279,184
16	766                        	1,846,955             	14,772,512
17	2,070                      	11,977,939            	95,815,104
18	4,526                      	83,263,591            	666,090,624
19	11,046                     	621,012,754           	4,968,057,848
20	36,275                     	4,878,666,808         	39,029,188,884
21	94,092                     	39,333,324,973        	314,666,222,712
22	312,673                    	336,376,244,042       	2,691,008,701,644
23	895,310                    	3,029,242,658,210     	24,233,937,684,440
24	2,919,602                  	28,439,272,956,934    	227,514,171,973,736
25	8,533,964                  	275,986,683,743,434   	2,207,893,435,808,352
26	28,929,567                 	2,789,712,466,510,289 	22,317,699,616,364,044
27	80,100,756                 	29,363,495,934,315,694	234,907,967,154,122,528
28	312,733,584                	???                   	???
29	900,029,716                	???                   	???
30	3,647,359,506              	???                   	???
=== Some Examples ===
==== 7×7 (all) ====
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AndrejPodzimek (talk) 00:22, 19 May 2022 (UTC)
thread_title:  Symmetrical Solutions 

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==== 10×10 (all) ====
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==== 20×20 (“first” and “last”) ====
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AndrejPodzimek (talk) 00:22, 19 May 2022 (UTC)
thread_title:  Some Examples 

message: █████████████████████████████████████████████████████████████████████████████████████
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AndrejPodzimek (talk) 00:22, 19 May 2022 (UTC)
thread_title:  20×20 (“first” and “last”)