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

Rob Pratt Rob.Pratt at sas.com
Thu Jul 14 01:38:53 CEST 2016

```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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```