The N-Queens Puzzle In Three
My son Martin and I have extended the well-tested and
known 2-dimensional "n queens on an n x n chessboard" puzzle, into
the third dimension, and have used permutations of
solved 2-dimensional boards to find fully N-Queens
rule-compliant 3D solutions.
A key feature of these solutions is that they are 'maximally
populated'. Each solution contains exactly N2
queens, and no more
un-contested queens can be added. While there are multiple solutions
for each of the cubes we show, within each solution, every
available queen's position is occupied. Here are two examples
from our project.
interactive 3D cubes are displayed using the free JavaView
Lite 3D geometry viewer by Konrad Polthier.
See the links at the bottom of this page for details on JavaView
the cube on the right to open a new browser page which shows
interactive 3-dimensional view of just one solution for
an 113 'board'.
This will load a
Java applet into your browser and you'll be able to
rotate the cube
and examine the pattern of 121 queens that
comprise this solution.
'board' or cube, showing all 169 queens
that comprise just one solution to a 133
sized version of this
|Here is a significantly
cube containing 961 ( 312 ) queens,
in a solution to the 313
sized version of this
In spite of this high queen population
each queen sits unchallenged in her own 'square' in the cube.
larger interactive cubes on these linked pages, your computer must
working version of Java installed.
You do NOT need to have JavaView
Lite installed because the JavaView Lite applet is
downloaded to your computer.
You may check the System
below to confirm that your computer will display the full
sized JavaView Lite panels.
About Our 3D Extension To This 2D Puzzle
The 3-dimensional N-Queens solutions above were
searched for and discovered in a joint project between myself and my
- Martin describes his work on our joint project
on his own website Queens On A Chessboard.
- In his Beyond
The 2nd Dimension page Martin
also discusses the prospect
of taking his work further into 4
dimensions and higher!
- My own (long!) historical notes on this 3D extension to
2-dimensional project are here.
| About JavaView
- The web-integration and display of our 3-dimensional
is made possible
to the free and (quite) easy to use JavaView
Lite 3D geometry viewer offered
to us by Konrad Polthier.
- This clever web-integration means that your computer
NOT need to have JavaView or JavaView Lite installed (although
do need to have a Java Virtual Machine installed. See
- You can develop your own JavaView applets by
installing the full version of JavaView on your own computer.
- The JavaView
Lite library file (or applet)
downloaded to your computer from my website
is Copyright ©
1999-2003 by Konrad Polthier.
JavaView used here was last
--- Konrad Polthier
Requirements for JavaView
- To view these 3D Java applets, your computer does
NOT need to
have JavaView or JavaView Lite installed.
- But to allow the JavaView applets to
browser, your computer must have
a version of Java
I recommend the free Java
Runtime Environment (JRE)
downloadable from Sun Microsystems, Inc..
- Installing this JRE will automatically prepare your
browser to run Java and JavaView applets.
(You can check if your computer has a working Java
visiting this Sun Java 'Test' website.)
- The linked web
pages display our 3D cube by telling your browser
to download a small JavaView
Lite applet (about 200KB in
size) from my CSP Queens
website. The applet is stored temporarily in the Java / JRE
cache on your
- As you rotate a 3D model in the JavaView
Lite panels, your computer's CPU
and graphics card will work quite hard, so obviously a more powerful
computer will give you a smoother experience.
to return to this site's Home page.
updated: 27th November 2010