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

Bob Selcoe rselcoe at entouchonline.net
Thu Jul 14 01:50:57 CEST 2016

```Rob,

Your results are consistent with my conjecture from a previous reply (has it
posted yet?), but are you sure these are maximums?  I've found several other
configurations of 13 queens for n=15, but I'm not 100% certain they're
maximums.

Is there a proof?

Cheers,
Bob S.

--------------------------------------------------
From: "Rob Pratt" <Rob.Pratt at sas.com>
Sent: Wednesday, July 13, 2016 6:38 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Nonattacking queens on a quarter-board

> After that, the n - 2 formula then holds up through n = 30.  But n = 31
> yields only 28 queens:
>
> OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOO
> -OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOO-
> --OOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO--
> ---OOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO---
> ----OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOO----
> -----OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOO-----
> ------OOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO------
> -------OOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO-------
> --------OOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOOO--------
> ---------OOOOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOO---------
> ----------OOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOOO----------
> -----------OOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOOOOO-----------
> ------------OOOOOOOOOOOOOOOOOOOOOOOOOOOXOOOOOOOOO------------
> -------------OOOOOOOOOOOOOOOOOOXOOOOOOOOOOOOOOOO-------------
> --------------OOOOOOOOOOXOOOOOOOOOOOOOOOOOOOOOO--------------
> ---------------OOOOOXOOOOOOOOOOOOOOOOOOOOOOOOO---------------
> ----------------OOOOOOOXOOOOOOOOOOOOOOOOOOOOO----------------
> -----------------OOOOOOOOOOOOOOOOOOOXOOOOOOO-----------------
> ------------------OOOOXOOOOOOOOOOOOOOOOOOOO------------------
> -------------------OOOOOOOOOOOOOOOOXOOOOOO-------------------
> --------------------OXOOOOOOOOOOOOOOOOOOO--------------------
> ---------------------OOOOOOOOOOOOOXOOOOO---------------------
> ----------------------OOOOOOOOXOOOOOOOO----------------------
> -----------------------OOOXOOOOOOOOOOO-----------------------
> ------------------------OOOOOOOOOXOOO------------------------
> -------------------------XOOOOOOOOOO-------------------------
> --------------------------OOOOOOXOO--------------------------
> ---------------------------OOXOOOO---------------------------
> ----------------------------OOOOO----------------------------
> -----------------------------OOO-----------------------------
> ------------------------------O------------------------------
>
> -----Original Message-----
> From: Rob Pratt
> Sent: Wednesday, July 13, 2016 5:21 PM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: RE: [seqfan] Nonattacking queens on a quarter-board
>
> n = 15 yields only 13 queens:
>
> OOOOOOXOOOOOOOOOOOOOOOOOOOOOO
> -OOOOOOOOOOOOOOOOOOXOOOOOOOO-
> --OOOOOXOOOOOOOOOOOOOOOOOOO--
> ---OOOOOOOOOOOOOOOXOOOOOOO---
> ----OOOOOOOOOOOXOOOOOOOOO----
> -----OOOOOOOXOOOOOOOOOOO-----
> ------OOOXOOOOOOOOOOOOO------
> -------OOOOOOOOOXOOOOO-------
> --------OOXOOOOOOOOOO--------
> ---------OOOOOOOOXOO---------
> ----------OOOOXOOOO----------
> -----------XOOOOOO-----------
> ------------OXOOO------------
> -------------OOO-------------
> --------------O--------------
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Wednesday, July 13, 2016 12:21 PM
> 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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>
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> Seqfan Mailing list - http://list.seqfan.eu/
>
```