CSP Queens - On The 3-Dimensional N-Queens Puzzle

The N-Queens Puzzle In Three Dimensions

My son Martin and I have extended the well-tested and well 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.

These 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 Lite.

Click on the cube on the right to open a new browser page which shows
an 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.

A larger 'board' or cube, showing all 169 queens
that comprise just one solution to a 13³ sized version of this puzzle.

Here is a significantly larger cube containing 961 ( 31² ) queens,
in a solution to the 31³ sized version of this puzzle.

In spite of this high queen population
each queen sits unchallenged in her own 'square' in the cube.

To view the larger interactive cubes on these linked pages, your computer must have a working version of Java installed.
You do NOT need to have JavaView Lite installed because the JavaView Lite applet is automatically downloaded to your computer.
You may check the System Requirements 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 son Martin.

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 my 2-dimensional project are here.

About JavaView And JavaView Lite:

The web-integration and display of our 3-dimensional N-Queens solutions is made possible thanks 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 does NOT need to have JavaView or JavaView Lite installed (although you do need to have a Java Virtual Machine installed. See below.)
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) which is downloaded to your computer from my website
is Copyright © 1999-2003 by Konrad Polthier.
The version of JavaView used here was last modified: 06.08.03 --- www.javaview.de --- Konrad Polthier

System Requirements for JavaView Lite:

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 appear in your browser, your computer must have a version of Java installed.
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 installation by 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 computer.
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.

Click here to return to this site's Home page.

Page updated: 27th November 2010

With the exception of any downloaded JavaView Lite components, content on this CSP Queens site
by Colin S Pearson and Martin S Pearson is licensed under
a Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales License.
Permissions beyond the scope of this license may be available via the feedback page.

These 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 Lite.
Click on the cube on the right to open a new browser page which shows an 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.
A larger 'board' or cube, showing all 169 queens that comprise just one solution to a 13³ sized version of this puzzle.
Here is a significantly larger cube containing 961 ( 31² ) queens, in a solution to the 31³ sized version of this puzzle. In spite of this high queen population each queen sits unchallenged in her own 'square' in the cube.
To view the larger interactive cubes on these linked pages, your computer must have a working version of Java installed. You do NOT need to have JavaView Lite installed because the JavaView Lite applet is automatically downloaded to your computer. You may check the System Requirements below to confirm that your computer will display the full sized JavaView Lite panels.