[seqfan] Re: Question re: A250000

[Bob Selcoe]

Hi again Rob an others,

And  two more 42-queen variations for n=17:

Alternative 1:

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



Alternative 2:

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


The symmetry of the W-blocks and the quasi-symmetry of the B-blocks for the 
second example surprises me.

I wonder if there are any solutions for a(n) > floor_(9n^2/64) for any n?

Cheers,
Bob S


--------------------------------------------------
From: "Rob Pratt" <Rob.Pratt at sas.com>
Sent: Saturday, February 07, 2015 9:43 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Question re: A250000

> 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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