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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

33.0. "Non-attacking queens" by HARE::STAN () Mon Feb 13 1984 23:48

Here are two unsolved chess-related problems from the 1982 Number
Theory conference in San Diego:

Problem 358 (Sol Golomb via Herbert Taylor):

	For some positive integer n, does there exist a
	configuration of n nonattacking queens on an n X n
	board having all vector differences distinct?

Problem 373 (Herbert Taylor):

	For every positive integer n, does there exist
	a configuration of n nonattacking rooks on an n X n
	board having all vector differences distinct?

Solutions should be sent to Richard K Guy at the University of Calgary.
T.RTitleUserPersonal
Name		DateLines
33.1METOO::YARBROUGHThu Aug 16 1984 16:0812
Not even close in either case. A more general problem is:
	For some positive integer n, does there exist an arrangement
	of n objects on a square lattice such that all vector distances
	are distinct?
The total possible number of distinct distances in such a lattice is only
(n^2+n)/2, while the number of vector distances among n objects is (n-1)!,
which is much larger for n>2, so no solution exists for either problem
except n=2.
(Oops, that should have said (n^2+n)/2-1 << n!/2 . Sorry about the inexactness,
but the differing growth rates are the key.)

Lynn Yarbrough
33.2TURTLE::GILBERTThu Aug 16 1984 17:0010
For some positive integer n, does there exist an arrangement
of n objects on a square nxn lattice such that all vector distances
are distinct?

	O - -		O - O -
	- O -		- O - -
	- O -		- - - -
			- - - O

					- Gilbert
33.3METOO::YARBROUGHThu Aug 16 1984 18:5919
I was too quick in my first response; I miscounted the number of vector
distances by confusing products with sums. Tsk, Tsk. Gilbert's first
diagram is a counterexample, but his second is not; he included two with
distance = sqrt(2). However, this a sound example for n=4:
	0 - - -
	- 0 - -
	- - - 0
	- - - 0
and here's one for n=5:
	0 - - 0 -
	- 0 - - -
	- 0 - - -
	- - - - -
	- - - - 0
It's tougher than I thought. Above n=5 you run into problems with pythagorean
triangles.

Since all the examples so far are failures for Rooks, it looks as if the
original answer is still NO.
33.4TURTLE::GILBERTThu Aug 16 1984 21:368
The "vector distance" includes the direction.  Thus, the vector distance
between (1,3) and (2,2) is (1,-1); the Euclidian distance is sqrt(2).

I'd originally thought the Euclidian distance was intended.  However,
Golomb and Taylor would not have proposed these problems if the simple
case of n=3 provided a counter-example.

					- Gilbert
33.5Ooops - I goofed again on this one.METOO::YARBROUGHFri Apr 18 1986 13:534
    This is a tougher problem than it appears: it's hard to count
    everything. My example of a 5x5 in .-2 is incorrect, having two
    distances = sqrt(5). So we have a correct 3x3 and a correct 4x4
    so far...
33.6By the way...LDP::HAFEZAmr A. Hafez 'On the EVE of Destruction'Thu May 07 1987 23:03548
    As a non-mathmatecian I had to write a program to solve this problem.
    My program has 2 limitations (1) it does not check the uniquness
    of solutions (2) it may not necessarily find all soutions. 

	    There are however more than 1 solution for a given n>4,
    but many solutions can be shown to be the same by symmetry since
    it is a square lattice. I tried running the program with n=17
    and got very many solutions. Now if we are looking for a counting
    technique for the number of solutions for a given n, we can safely
    assume that the number will increase with n.
    
    A friend of mine and I were trying to find some rules for the solutions
    and kept being frustrated. We know that the queens must be a knight's
    move apart. It seems that the rules are different for odd and even
    n values.
    
    If it is of interest to anyone, send mail to LDP::hafez and I can
    mail you the source. SMG and print versions are availlable, the
    program is written in C and currently you must recompile to try
    a new grid size. This is a non-recursive version, I would be willing
    to discuss writting a recursive version, but I don't see the benefit.
    
    Following the <FF> I will give the solutions for the n=8 case. 
    


Initial position (0,0)

Number of queens ========> 8 

                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . Q . . . . . 


Initial position (0,1)

Number of queens ========> 8 

                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                . . . Q . . . . 

Initial position (0,1)

Number of queens ========> 8 

                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . . . . . . Q . 
                                . . . Q . . . . 

Initial position (0,2)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . Q . . . . . . 

Initial position (0,2)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 

Initial position (0,2)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . Q . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 

Initial position (0,2)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . Q . . . . 
                                . . . . . . . Q 
                                . . . . Q . . . 

Initial position (0,2)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . Q . . . . . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . . Q . . 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . . Q . 
                                . . Q . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                Q . . . . . . . 
                                . . . . . Q . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . . Q . . 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 

Initial position (0,3)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 

Initial position (1,0)

Number of queens ========> 8 

                                . . . Q . . . . 
                                Q . . . . . . . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                . . Q . . . . . 
                                . . . . . . Q . 
                                . Q . . . . . . 

Initial position (1,0)

Number of queens ========> 8 

                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 

Initial position (1,0)

Number of queens ========> 8 

                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 

Initial position (1,0)

Number of queens ========> 8 

                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 

Initial position (1,0)

Number of queens ========> 8 

                                . . . . Q . . . 
                                Q . . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                . . Q . . . . . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . . Q . . . 
                                . Q . . . . . . 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . Q . . . . 
                                . . . . . . . Q 
                                . . Q . . . . . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . . . Q . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . . . . . . Q . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . . . . . . Q . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . . . Q . . . . 

Initial position (1,1)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . . . Q . . . . 

Initial position (1,2)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . Q . . . . . 
                                . . . . . . Q . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . Q . . . 
                                Q . . . . . . . 
                                . . . Q . . . . 

Initial position (1,2)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . Q . . . . . 
                                . . . . . . Q . 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . . . Q . . . 

Initial position (1,3)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . Q . . . . . 

Initial position (1,3)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 

Initial position (1,3)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . Q . . . . . 

Initial position (2,0)

Number of queens ========> 8 

                                . . . . Q . . . 
                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . . Q . 

Initial position (2,0)

Number of queens ========> 8 

                                . . . . Q . . . 
                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . . Q . 

Initial position (2,0)

Number of queens ========> 8 

                                . . . . Q . . . 
                                . . Q . . . . . 
                                Q . . . . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . . Q . 

Initial position (2,1)

Number of queens ========> 8 

                                . . . . . . Q . 
                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . . Q . . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . . 

Initial position (2,1)

Number of queens ========> 8 

                                . . . . Q . . . 
                                . . . . . . Q . 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . . . Q 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . . Q . . 

Initial position (2,3)

Number of queens ========> 8 

                                . . Q . . . . . 
                                . . . . . Q . . 
                                . . . Q . . . . 
                                . Q . . . . . . 
                                . . . . . . . Q 
                                . . . . Q . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 

Initial position (3,0)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . Q . . . . . 

Initial position (3,0)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . Q . . . . . 

Initial position (3,0)

Number of queens ========> 8 

                                . . . . . Q . . 
                                . . . Q . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . Q . . . 
                                . . Q . . . . . 

Initial position (3,1)

Number of queens ========> 8 

                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . . . . Q . 
                                Q . . . . . . . 
                                . . Q . . . . . 
                                . . . . Q . . .
33.7CLT::GILBERTeager like a childFri May 08 1987 00:30121
No solutions to the queens problem were found for n = 2 thru 12.

Here are some solutions to the rooks problem for n = 2 thru 14.

*** 2 ***
	O - 
	- O 
*** 3 ***
	O - - 
	- - O 
	- O - 
*** 4 ***
	O - - - 
	- O - - 
	- - - O 
	- - O - 
*** 5 ***
	O - - - - 
	- - - O - 
	- O - - - 
	- - O - - 
	- - - - O 
*** 6 ***
	O - - - - - 
	- O - - - - 
	- - - - - O 
	- - - O - - 
	- - O - - - 
	- - - - O - 
*** 7 ***
	O - - - - - - 
	- O - - - - - 
	- - - - - O - 
	- - - O - - - 
	- - - - - - O 
	- - O - - - - 
	- - - - O - - 
*** 8 ***
	O - - - - - - - 
	- O - - - - - - 
	- - - - - - - O 
	- - - - - O - - 
	- - O - - - - - 
	- - - - O - - - 
	- - - O - - - - 
	- - - - - - O - 
*** 9 ***
	O - - - - - - - - 
	- O - - - - - - - 
	- - - - - - - - O 
	- - - O - - - - - 
	- - - - - - O - - 
	- - O - - - - - - 
	- - - - - - - O - 
	- - - - - O - - - 
	- - - - O - - - - 
*** 10 ***
	O - - - - - - - - - 
	- O - - - - - - - - 
	- - - - - - - - O - 
	- - O - - - - - - - 
	- - - - O - - - - - 
	- - - - - - - - - O 
	- - - - - - - O - - 
	- - - O - - - - - - 
	- - - - - - O - - - 
	- - - - - O - - - - 
*** 11 ***
	O - - - - - - - - - - 
	- O - - - - - - - - - 
	- - - - - - - - O - - 
	- - - - O - - - - - - 
	- - O - - - - - - - - 
	- - - - - - - - - - O 
	- - - - - - - O - - - 
	- - - - - - - - - O - 
	- - - O - - - - - - - 
	- - - - - - O - - - - 
	- - - - - O - - - - - 
*** 12 ***
	O - - - - - - - - - - - 
	- O - - - - - - - - - - 
	- - - - O - - - - - - - 
	- - O - - - - - - - - - 
	- - - - - - - - - O - - 
	- - - - - O - - - - - - 
	- - - - - - - - - - - O 
	- - - O - - - - - - - - 
	- - - - - - - - O - - - 
	- - - - - - - - - - O - 
	- - - - - - - O - - - - 
	- - - - - - O - - - - - 
*** 13 ***
	O - - - - - - - - - - - - 
	- O - - - - - - - - - - - 
	- - - - - - - - - - - O - 
	- - O - - - - - - - - - - 
	- - - - - - - - - O - - - 
	- - - - - O - - - - - - - 
	- - - - - - - - O - - - - 
	- - - - - - - - - - O - - 
	- - - O - - - - - - - - - 
	- - - - - - - - - - - - O 
	- - - - - - - O - - - - - 
	- - - - - - O - - - - - - 
	- - - - O - - - - - - - - 
*** 14 ***
	O - - - - - - - - - - - - - 
	- O - - - - - - - - - - - - 
	- - - - - - - - - - - - O - 
	- - - - - - - - - O - - - - 
	- - O - - - - - - - - - - - 
	- - - - - - - - O - - - - - 
	- - - O - - - - - - - - - - 
	- - - - - O - - - - - - - - 
	- - - - - - - - - - - - - O 
	- - - - - - - - - - - O - - 
	- - - - - - - O - - - - - - 
	- - - - - - O - - - - - - - 
	- - - - - - - - - - O - - - 
	- - - - O - - - - - - - - -
33.8There are solutions for n=4...n=17LDP::HAFEZAmr A. Hafez 'On the EVE of Destruction'Mon May 11 1987 19:17237
    First let me apologize for .6, I used a version of the program that
    has too many solutions, most non-unique.
    
    
    <re .7>
    anything less than n=4 has no solution. n=4...n=17 have solutions,
    many solutions, in the queens problem. The rooks problem is much
    simpler and should have more solutions.
    	Below, I will include solutions for n=4...n=17. This time I
    promise, only one solution per n. 
    
    	I would like someone to help me develope a counting technique
    for the number of solutions for a given n.
    
    solutions :
    


Initial position (0,1)

Number of queens ========> 4 

                                . Q . . 
                                . . . Q 
                                Q . . . 
                                . . Q . 

Initial position (0,0)

Number of queens ========> 5 

                                Q . . . . 
                                . . Q . . 
                                . . . . Q 
                                . Q . . . 
                                . . . Q . 

Initial position (0,1)

Number of queens ========> 6 

                                . Q . . . . 
                                . . . Q . . 
                                . . . . . Q 
                                Q . . . . . 
                                . . Q . . . 
                                . . . . Q . 

Initial position (0,0)

Number of queens ========> 7 

                                Q . . . . . . 
                                . . Q . . . . 
                                . . . . Q . . 
                                . . . . . . Q 
                                . Q . . . . . 
                                . . . Q . . . 
                                . . . . . Q . 

Initial position (0,0)

Number of queens ========> 8 

                                Q . . . . . . . 
                                . . . . . . Q . 
                                . . . . Q . . . 
                                . . . . . . . Q 
                                . Q . . . . . . 
                                . . . Q . . . . 
                                . . . . . Q . . 
                                . . Q . . . . . 

Initial position (0,0)

Number of queens ========> 9 

                                Q . . . . . . . . 
                                . . . Q . . . . . 
                                . . . . . Q . . . 
                                . . . . . . . Q . 
                                . Q . . . . . . . 
                                . . . . Q . . . . 
                                . . Q . . . . . . 
                                . . . . . . . . Q 
                                . . . . . . Q . . 

Initial position (0,2)

Number of queens ========> 10 

                                . . Q . . . . . . . 
                                . . . . . Q . . . . 
                                . . . . . . . . Q . 
                                . . . . . . Q . . . 
                                Q . . . . . . . . . 
                                . . . Q . . . . . . 
                                . Q . . . . . . . . 
                                . . . . Q . . . . . 
                                . . . . . . . Q . . 
                                . . . . . . . . . Q 

Initial position (0,0)

Number of queens ========> 11 

                                Q . . . . . . . . . . 
                                . . . . . . Q . . . . 
                                . . . . . . . . Q . . 
                                . . . . . . . . . . Q 
                                . . Q . . . . . . . . 
                                . . . . Q . . . . . . 
                                . . . . . . . . . Q . 
                                . Q . . . . . . . . . 
                                . . . Q . . . . . . . 
                                . . . . . Q . . . . . 
                                . . . . . . . Q . . . 

Initial position (0,2)

Number of queens ========> 12 

                                . . Q . . . . . . . . . 
                                . . . . . . . Q . . . . 
                                . . . . . . . . . Q . . 
                                . . . . . . . . . . . Q 
                                . . . Q . . . . . . . . 
                                . . . . . . . . . . Q . 
                                Q . . . . . . . . . . . 
                                . . . . . Q . . . . . . 
                                . Q . . . . . . . . . . 
                                . . . . Q . . . . . . . 
                                . . . . . . Q . . . . . 
                                . . . . . . . . Q . . . 

Initial position (1,2)

Number of queens ========> 13 

                                . . . . . . Q . . . . . . 
                                . . Q . . . . . . . . . . 
                                . . . . . . . . . . Q . . 
                                . . . . . . . . Q . . . . 
                                . . . Q . . . . . . . . . 
                                . . . . . . . . . . . . Q 
                                . . . . Q . . . . . . . . 
                                . Q . . . . . . . . . . . 
                                . . . . . . . . . . . Q . 
                                Q . . . . . . . . . . . . 
                                . . . . . Q . . . . . . . 
                                . . . . . . . Q . . . . . 
                                . . . . . . . . . Q . . . 

Initial position (0,3)

Number of queens ========> 14 

                                . . . Q . . . . . . . . . . 
                                . . . . . . Q . . . . . . . 
                                . . . . . . . . Q . . . . . 
                                . . . . . . . . . . Q . . . 
                                . . . . . . . . . . . . Q . 
                                . . . . . . . Q . . . . . . 
                                Q . . . . . . . . . . . . . 
                                . . Q . . . . . . . . . . . 
                                . . . . . . . . . Q . . . . 
                                . . . . . . . . . . . . . Q 
                                . Q . . . . . . . . . . . . 
                                . . . . Q . . . . . . . . . 
                                . . . . . . . . . . . Q . . 
                                . . . . . Q . . . . . . . . 

Initial position (0,1)

Number of queens ========> 15 

                                . Q . . . . . . . . . . . . . 
                                . . . . . . Q . . . . . . . . 
                                . . . . . . . . Q . . . . . . 
                                . . . . . . . . . . Q . . . . 
                                . . . . . . . Q . . . . . . . 
                                . . . . . . . . . Q . . . . . 
                                . . . . . . . . . . . . . . Q 
                                . . Q . . . . . . . . . . . . 
                                Q . . . . . . . . . . . . . . 
                                . . . Q . . . . . . . . . . . 
                                . . . . . . . . . . . . Q . . 
                                . . . . Q . . . . . . . . . . 
                                . . . . . . . . . . . Q . . . 
                                . . . . . . . . . . . . . Q . 
                                . . . . . Q . . . . . . . . . 

Initial position (0,0)

Number of queens ========> 16 

                                Q . . . . . . . . . . . . . . . 
                                . . . . . . . Q . . . . . . . . 
                                . . . . . . . . . Q . . . . . . 
                                . . . . . . . . . . . Q . . . . 
                                . . . . . . . . Q . . . . . . . 
                                . . . . . . . . . . Q . . . . . 
                                . . . . . . . . . . . . . . . Q 
                                . . Q . . . . . . . . . . . . . 
                                . . . . . Q . . . . . . . . . . 
                                . Q . . . . . . . . . . . . . . 
                                . . . . . . . . . . . . Q . . . 
                                . . . . . . . . . . . . . . Q . 
                                . . . . . . Q . . . . . . . . . 
                                . . . Q . . . . . . . . . . . . 
                                . . . . . . . . . . . . . Q . . 
                                . . . . Q . . . . . . . . . . . 

Initial position (0,0)

Number of queens ========> 17 

                                Q . . . . . . . . . . . . . . . . 
                                . . . . . . . Q . . . . . . . . . 
                                . . . . . . . . . Q . . . . . . . 
                                . . . . . . . . . . . Q . . . . . 
                                . . . . . . . . Q . . . . . . . . 
                                . . . . . . . . . . Q . . . . . . 
                                . . . . . . . . . . . . . . . Q . 
                                . . Q . . . . . . . . . . . . . . 
                                . . . . . Q . . . . . . . . . . . 
                                . Q . . . . . . . . . . . . . . . 
                                . . . . . . . . . . . . Q . . . . 
                                . . . . . . . . . . . . . . Q . . 
                                . . . . . . Q . . . . . . . . . . 
                                . . . Q . . . . . . . . . . . . . 
                                . . . . . . . . . . . . . Q . . . 
                                . . . . . . . . . . . . . . . . Q 
                                . . . . Q . . . . . . . . . . . .
33.9What are we talking about?AKQJ10::YARBROUGHWhy is computing so labor intensive?Tue May 12 1987 12:3810
I'm confused:
                                . Q . . 
                                . . . Q 
                                Q . . . 
                                . . Q . 

is certainly not a solution to the original problem; so what problem is it 
the solution of?

-Lynn-
33.10Queens?LDP::HAFEZAmr A. Hafez 'On the EVE of Destruction'Tue May 12 1987 16:336
    My understanding was that one of the 2 problems presented in .0
    is to place n non-attacking queens on an nXn board. By that token
    the picture in .9 is a solution for n=4, it is also a solution for
    the rooks problem since a rook is a subset of a queen. I will re-read
    the original note to double check.
33.11CLT::GILBERTeager like a childThu May 14 1987 17:1083
Re: Amr and Lynn's comments

The problems in 33.0 have the additional proviso that all the vector
distances between the pieces are distinct.


Re: Other solutions

There are no solutions to the queens problem for n = 2 thru 18.

Here are some solutions to the rooks problem for n = 15 thru 18
(solutions for n = 2 thru 14 were given in note 33.7).

*** 15 ***
	O - - - - - - - - - - - - - - 
	- O - - - - - - - - - - - - - 
	- - - - - O - - - - - - - - - 
	- - - - - - - - - - - - - O - 
	- - - - - - - - O - - - - - - 
	- - O - - - - - - - - - - - - 
	- - - - - - - - - - - - - - O 
	- - - - - - - - - - - - O - - 
	- - - - O - - - - - - - - - - 
	- - - - - - - - - O - - - - - 
	- - - - - - - - - - - O - - - 
	- - - - - - - - - - O - - - - 
	- - - - - - - O - - - - - - - 
	- - - O - - - - - - - - - - - 
	- - - - - - O - - - - - - - - 
*** 16 ***
	O - - - - - - - - - - - - - - - 
	- O - - - - - - - - - - - - - - 
	- - - - - - - - - - - - - O - - 
	- - - - - - - O - - - - - - - - 
	- - - - O - - - - - - - - - - - 
	- - O - - - - - - - - - - - - - 
	- - - - - - - - - - - - - - - O 
	- - - - - - O - - - - - - - - - 
	- - - - - - - - - - - O - - - - 
	- - - - - - - - - - - - - - O - 
	- - - O - - - - - - - - - - - - 
	- - - - - - - - - - - - O - - - 
	- - - - - O - - - - - - - - - - 
	- - - - - - - - - O - - - - - - 
	- - - - - - - - O - - - - - - - 
	- - - - - - - - - - O - - - - - 
*** 17 ***
	O - - - - - - - - - - - - - - - - 
	- O - - - - - - - - - - - - - - - 
	- - - - - - - - - - - - - - - - O 
	- - - - - - - - - - - O - - - - - 
	- - - - - - - - - - O - - - - - - 
	- - - - - - - - O - - - - - - - - 
	- - - - - - - - - - - - - - O - - 
	- - - - - - O - - - - - - - - - - 
	- - O - - - - - - - - - - - - - - 
	- - - - - - - - - - - - - O - - - 
	- - - - O - - - - - - - - - - - - 
	- - - - - - - - - O - - - - - - - 
	- - - O - - - - - - - - - - - - - 
	- - - - - - - - - - - - O - - - - 
	- - - - - - - - - - - - - - - O - 
	- - - - - O - - - - - - - - - - - 
	- - - - - - - O - - - - - - - - - 
*** 18 ***
	O - - - - - - - - - - - - - - - - - 
	- O - - - - - - - - - - - - - - - - 
	- - - - - - - - - - - - - O - - - - 
	- - O - - - - - - - - - - - - - - - 
	- - - - - - - - - - - - - - - - O - 
	- - - - - - - - - - - - - - O - - - 
	- - - - - - O - - - - - - - - - - - 
	- - - O - - - - - - - - - - - - - - 
	- - - - - - - - O - - - - - - - - - 
	- - - - - - - - - - - - - - - - - O 
	- - - - - - - - - - - - O - - - - - 
	- - - - - - - - - - - - - - - O - - 
	- - - - - O - - - - - - - - - - - - 
	- - - - - - - O - - - - - - - - - - 
	- - - - - - - - - - - O - - - - - - 
	- - - - O - - - - - - - - - - - - - 
	- - - - - - - - - - O - - - - - - - 
	- - - - - - - - - O - - - - - - - -