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

[Bob Selcoe]

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:

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

or

               X
           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.

Cheers,
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:
>
> OOOOOOOOOOOOOOOOOOOOO
> -OOOOOOOOOOOOOOOOOOO-
> --OOOOOOOOOOOOOOOOO--
> ---OOOOOOOOOOOOOOO---
> ----OOOOOOOOOOOOO----
> -----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):
>
> OOOOOOOOOOOOOOOXOOOOO
> -OOOOOOOOOOOXOOOOOOO-
> --OOOOXOOOOOOOOOOOO--
> ---OOOOOOOOOOXOOOO---
> ----OOOXOOOOOOOOO----
> -----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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>