###############################################################################
This Text File is Public Domain. This is available in INTERNET.
http://www.cli.di.unipi.it/~velucchi/queens.txt
This is for review/abstract in journals and magazines in all the World.
I am available for more technical details.
IF YOU PRINT THIS FILE OR REFERENCE ON, WRITE ME PLEASE.
###############################################################################
FOR ME, THIS IS THE BEST CHESS-PUZZLE
by Mario VELUCCHI
[Project Beginning: May 1995 - Last Up-Dated: 27 January 1997]
NON-Dominating Queens Problem
- Placing N Queens on an order-N board to leave a maximum number of unattacked vacant cells.
N Free(N) Prob.
1 0 100% Sure
2 0 "
3 0 "
4 1 "
5 3 "
6 5 "
7 7 "
8 11 "
9 18 "
10 22 "
11 30 "
12 36 "
13 47 "
14 56 "
15 72 "
16 82 "
17 97 probable
18 111 "
19 132 "
20 145 "
21 170 "
22 186 "
23 216 "
24 240 "
25 260 "
26 290 "
27 324 "
28 360 "
29 381 "
30 420 "
>30 others dispositions probables
[Notation: Free[N] = Maximum Known for unattacked vacant cells if i place N queens in order-N board]
======================
ROLL -
The first authors of these proposed values of Free(N) are:
N= 1 Free= 0 obvious
N= 2 Free= 0 obvious
N= 3 Free= 0 obvious
N= 4 Free= 1 in Stephen AINLEY, 1977 and Martin GARDNER, 1983 (reprint of previous Scientific American articles)
N= 5 Free= 3 in Stephen AINLEY and M. GARDNER, ibid
N= 6 Free= 5 in Stephen AINLEY and M. GARDNER, ibid
N= 7 Free= 7 in Stephen AINLEY and M. GARDNER, ibid
N= 8 Free= 11 in Walter William Rouse BALL 'Mathematical Recreations and Problems' Macmillan 1892
N= 9 Free= 18 Diego BRACAMONTE (Argentina) in 'El Acertijo' 1993
N=10 Free= 22 in Stephen AINLEY and M. GARDNER, ibid
N=11 Free= 30 Rodolfo KURCHAN (Argentina) in 'El Acertijo' 1993
Mario VELUCCHI (Italy) - [C] 1995 [new solution]
N=12 Free= 36 in Stephen AINLEY and M. GARDNER, ibid
N=13 Free= 47 Johan CLAES (Belgium), Mario VELUCCHI - [C] 1995
N=14 Free= 56 in Stephen AINLEY, ibid
Mario VELUCCHI - [C] 1995 [new solution]
N=15 Free= 72 Johan CLAES, Mario VELUCCHI - [C] 1995
N=16 Free= 82 Johan CLAES, Mario VELUCCHI - [C] 1995
N=17 Free= 97 Johan CLAES, Mario VELUCCHI - [~C] 1995
N=18 Free=111 in Stephen AINLEY, ibid
N=19 Free=132 in Stephen AINLEY, ibid
N=20 Free=145 Johan CLAES, Mario VELUCCHI - [C] 1995
N=21 Free=170 Johan CLAES, Mario VELUCCHI - [C] 1995
N=22 Free=186 in Stephen AINLEY, ibid
N>22 ever by Johan CLAES, Mario VELUCCHI, Miyoshi NAGAI (Japan) - [C] 1995
[Notation: [C] - Computer Research; [~C] - Computer and Human Research]
======================
Different Solutions [in increase]
by Johan CLAES, Mario VELUCCHI, Miyoshi NAGAI - [C] 1995
N Free # Prob.
4 1 25 100% Sure
5 3 1 "
6 5 3 "
7 7 38 "
8 11 7 "
9 18 1 "
10 22 1 "
11 30 2 "
12 36 7 "
13 47 1 "
14 56 4 "
15 72 3 "
16 82 1 "
17 97 >=1
18 111 >=1
.. ... ...
for N <= 8 the first reference is by Bernd SCHWARZKOPF (Germany) in 'Die Schwalbe' [C] 1982
======================
Generalisation:
K Queens on an order-N board
- Table Free(K,N) (best known solutions) for:
1 <= N <= 30
only for Max K-Queens near equal to Free(K,N)
in Stephen AINLEY, ibid
1 <= N <= 10 1 <= K <= (N)
incomplete
by Hiroshi OKUNO (Japan) in Martin GARDNER, ibid [C]
1 <= N <= 8 1 <= K <= (N*N)
best/number of solutions
by Bernd SCHWARZKOPF in 'Die Schwalbe' [C] 1982
1 <= N <= 30 1 <= K <= (N*N) [in increase]
best/number of solutions
by Johan CLAES, Mario VELUCCHI - [C] 1995
- Formulas and Bounds [in progress]
by Johan CLAES, Mario VELUCCHI 1995
Free(K, N) = 0 for N^2-3*N+3 <= K <= N^2
Free(K, N) = 1 for N^2-5*N+8 <= K <= N^2-3*N+2 ...
this is obvious because i have:
Free(1, N) = N^2-3*N+2
Free(2, N) = N^2-5*N+7
Free(3, N) =? N^2-6*N+9
Free(4, N) =? N^2-6*N+8 for N pair; otherwise: Free(4, N) =? N^2-6*N+9
...
Free(>4, N) = others approximate formulas
- Lower, Upper Bounds and Approximate Formulas for Free(N, N), the proposed problem [in progress]
by Johan CLAES, Mario VELUCCHI, Gustavo PI¥EIRO 1995
...
- Others theoretical considerations, conjectures and theorems [in progress]
by Johan CLAES, Mario VELUCCHI, Gustavo PI¥EIRO 1995
...
[Notation: Free(K, N) = free squares for K Queens in order-N board]
======================
References/Acknowledgements:
1. Walter William Rouse BALL 'Mathematical Recreations and Problems' Macmillan 1892
2. Stephen AINLEY 'Mathematical Puzzles' G.Bell & Sons Ltd, UK 1977
3. Martin GARDNER 'Wheels, Life and Other Mathematical Amusements' Freeman 1983
I. 'Die Schwalbe' (German journal specific on Chess Problems)
II. 'El Acertijo' (Argentinean journal specific on Recreational Mathematics)
III. 'Informazione Scacchi' (Italian journal specific on Chess)
IV. 'Jouer Jeux Math‚matiques' (French journal specific on Recreational Mathematics)
V. 'KONWAKAI NEWS' (Japanese journal specific on Recreational Mathematics)
VI. 'mathematical digest' nø 103, April 1996, p. 24 (South African magazine)
VII. 'Scientific American' Martin GARDNER's Mathematical Games Column (USA)
VIII.'SHARADA' (Russian journal specific on Recreational Mathematics)
IX. 'springaren' (Swedish journal specific on Chess Problems)
X. 'Technology Review - MIT Students special edition' Allan J. GOTTLIEB's Puzzle Corner (USA)
a. Personal communications by Allan J. GOTTLIEB (USA)
b. Personal communications by Bernd SCHWARZKOPF (Germany)
c. Personal communications by Dario URI (Italy)
d. Personal communications by Gustavo PI¥EIRO (Argentina)
e. Personal communications by HŠctor SAN SEGUNDO (Argentina)
f. Personal communications by J.H. WEBB (South Africa)
g. Personal communications by Johan CLAES (Belgium)
h. Personal communications by Martin GARDNER (USA)
i. Personal communications by Michel CRITON(France)
j. Personal communications by Miyoshi NAGAI (Japan)
k. Personal communications by Ronald L. GRAHAM (USA)
l. Personal communications by Stephen HEDETNIEMI (USA)
m. Personal communications by Toshi Junk KATO (Japan)
n. Personal communications by Vladimir N. RIBINSKY (Russia)
o. My papers (Mario VELUCCHI - Italy)
======================
Mario VELUCCHI
Via Emilia, 106
I-56121 Pisa - ITALY
e-mail: velucchi@cli.di.unipi.it
http://www.cli.di.unipi.it//~velucchi/intro.html
======================
Two nice DIAGRAMS (very small selection):
...
...
+ - + + + + - -
+ - + + + - - -
+ + + + + - - +
Q + Q + + + + +
Q + + + + + + +
Q + + + Q + + +
+ + + + + + - -
+ + Q Q Q + + +
8 : free = 11
+ Q + + + + Q +
Q + Q + + + Q +
+ Q + + + + + +
+ + + + + - + -
+ + + + - + + -
+ + + - + - + -
Q Q + + + + + +
+ + + - - - + -
8 : free = 11
...
...