| 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 |
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.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 33.1 | METOO::YARBROUGH | Thu Aug 16 1984 12:08 | 12 | ||
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 | |||||
| 33.2 | TURTLE::GILBERT | Thu Aug 16 1984 13:00 | 10 | ||
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 | |||||
| 33.3 | METOO::YARBROUGH | Thu Aug 16 1984 14:59 | 19 | ||
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. | |||||
| 33.4 | TURTLE::GILBERT | Thu Aug 16 1984 17:36 | 8 | ||
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 | |||||
| 33.5 | Ooops - I goofed again on this one. | METOO::YARBROUGH | Fri Apr 18 1986 09:53 | 4 | |
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...
| |||||
| 33.6 | By the way... | LDP::HAFEZ | Amr A. Hafez 'On the EVE of Destruction' | Thu May 07 1987 19:03 | 548 |
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 . . .
| |||||
| 33.7 | CLT::GILBERT | eager like a child | Thu May 07 1987 20:30 | 121 | |
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 - - - - - - - - - | |||||
| 33.8 | There are solutions for n=4...n=17 | LDP::HAFEZ | Amr A. Hafez 'On the EVE of Destruction' | Mon May 11 1987 15:17 | 237 |
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 . . . . . . . . . . . .
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| 33.9 | What are we talking about? | AKQJ10::YARBROUGH | Why is computing so labor intensive? | Tue May 12 1987 08:38 | 10 |
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-
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| 33.10 | Queens? | LDP::HAFEZ | Amr A. Hafez 'On the EVE of Destruction' | Tue May 12 1987 12:33 | 6 |
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.
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| 33.11 | CLT::GILBERT | eager like a child | Thu May 14 1987 13:10 | 83 | |
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 - - - - - - - - | |||||