[seqfan] Nonattacking queens on a quarter-board

Neil Sloane njasloane at gmail.com
Wed Jul 13 18:20:54 CEST 2016


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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