[seqfan] Re: Question re: A250000

[Rob Pratt]

Here it is with unoccupied squares indicated with dots (view with a fixed-width font):

....WWWWW........
....WWWWW........
....WWWWW.......W
....WWWW.......WW
....WWW.......WWW
.....W.......WWWW
.............WWWW
.............WWW.
.............WW..
..BB.............
.BBB.............
BBBB.............
BBBB.......B.....
BBBB......BBB....
BBBB.....BBBB....
BBB......BBBB....
BB.......BBBB....

From: Rob Pratt
Sent: Saturday, February 07, 2015 5:18 PM
To: Sequence Fanatics Discussion list
Subject: RE: [seqfan] Question re: A250000


Here's a 17x17 solution with 42 queens of each color:



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1

W

W

W

W

W

2

W

W

W

W

W

3

W

W

W

W

W

W

4

W

W

W

W

W

W

5

W

W

W

W

W

W

6

W

W

W

W

W

7

W

W

W

W

8

W

W

W

9

W

W

10

B

B

11

B

B

B

12

B

B

B

B

13

B

B

B

B

B

14

B

B

B

B

B

B

B

15

B

B

B

B

B

B

B

B

16

B

B

B

B

B

B

B

17

B

B

B

B

B

B






-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Bob Selcoe
Sent: Saturday, February 07, 2015 4:02 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Question re: A250000





Hello Seqfans,



The sequence A250000 (maximum number of peacefully coexisting equal-sized "armies" of queens on chess boards of varying n X n sizes) poses some fascinating problems.



The length of the sequence is small; only up to a(13) = 24.



The "known" lower bound for a solution is a(n) = 9/64*n^2. There is a link to a paper by Prestwich and Beck referenced in the sequence which expands on this idea.  I can't follow the paper, but it apparently provides an upper bound, as well.



For all n = 4m+1 m>=0, I can show a pattern of quasi-symmetric queen placement such that a(n) = 2m(m+1).  For m = {0..3}, this is indeed the maximum number of queens possible.  For m>=4, these solutions are < the known lower bound of (9/64)*n^2.



I have proposed for A250000 examples of solutions using this queen pattern for n=9 and n=13.  Please refer to the sequence history to see the pattern.

Since the pattern yields 40 for n=17, and a(17)=42 is the known lower bound, it (apparently) does not provide a solution for a(17).



Can anyone show an actual example of a 17 X 17 chessboard where the number of queens > 40, even if it can't be proven to be a solution (maximum number of queens) for a(17)?



Best Wishes,

Bob Selcoe







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