[seqfan] Re: Nonattacking queens on a quarter-board

Rob Pratt Rob.Pratt at sas.com
Thu Jul 14 21:23:54 CEST 2016

For the side question, 12 is the largest such q.  If you force the queen in the center for n = 13, you can place only 11 queens total.

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Bob Selcoe
Sent: Wednesday, July 13, 2016 5:03 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Nonattacking queens on a quarter-board

Hi Neil and Seqfans,

I think probably the number of allowable queens (q) is< n-1, and that as n increases, n-q increases.  If so, is there a definable limit?

If I'm right, an interesting side question might be:  Starting with a queen in the center, what is the maximum q where q=n-1?  (BTW -  I find it's easier to visualize starting with the center at the top and expanding the triangle by rows downwards).

So it starts:

           O O O
       O X O O O
   O O O O X O O


           O O O
       O X O O O
   O O O O O X O

I did a few by hand and the furthest I got was 11.  I'd be very surprised if that's the actual max.

Bob S

From: "Neil Sloane" <njasloane at gmail.com>
Sent: Wednesday, July 13, 2016 11:20 AM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Nonattacking queens on a quarter-board

> Dear Seqfans,  I keep running across questions about placing queens on 
> an infinite chess-board so that they don't attack each other.
> Here is one version.
> Take the quarter-board formed from a k X k chess board by joining the 
> center square (k is odd) to the top two corners.
> If k = 21 the board looks like this:
> -----OOOOOOOOOOO-----
> ------OOOOOOOOO------
> -------OOOOOOO-------
> --------OOOOO--------
> ---------OOO---------
> ----------O----------
> Gmail is distorting the board, but the top row is the top edge of the 
> board, with 21 squares, and the bottom row contains the single square 
> at the center of the board.
> There are 11*11 = 121 squares.
> The problem is to place non-attacking queens on this board. I suspect 
> the max number is n-1, if there are n rows (and n^2 squares), and this 
> can be attained for all n > 2. (But I don't know if that is really 
> true)
> Here is a construction (I think) for the above board (X = queen):
> -----OOOOOOOOOXO-----
> ------OOXOOOOOO------
> -------OOOOXOO-------
> --------OXOOO--------
> ---------OOO---------
> ----------X----------
> What it would be nice to have is a construction for the infinite board 
> - in other words, nested solutions for all n.  So that the solution 
> for n-1 rows is obtained by dropping the top row of the solution for n 
> rows.  Is it possible?
> In the above picture the queens are in lines a knight's move apart
> This kind of question comes up in studying infinite Sudoku-like arrays 
> like
> A269526 or Zak's spiral, A274640.
> --
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