Magic Cubes - Introduction

	Update-6	Feb. 2010 Much more about Compact and Complete. Small order-4 multiply cubes. Frost Whipple model order-9 nasik. First pantriagonal associated order-4? etc.
	Tesseracts	Nov. 2007 All about magic tesseracts.  A new site (because of space) consisting of 11 pages.
	Update-5	May  2007. Magic cuboids, Magic Knight Tours, transform associated to pantriagonal. Also miscellaneous items and links.
	Cube Timeline	Dec. 2006.  I expanded the cube timeline that originally appeared on my Magic Cubes - the Road to Perfect page. Then moved it to a separate page. I included a copy of  Table of First Cubes (which is on my Summary page.

Index to this page

Introduction

Several years ago John R. Hendricks introduced a coordinated set of definitions for magic cubes. It included a new definition for the ‘perfect’ magic cube, which is applicable for magic hypercubes of any dimension.

This inspired me to investigate the different definitions of ‘perfect’ magic cubes that had appeared over the years. The result was a new page that discussed the subject. However, I ended up with more questions then when I started, so developed a series of spreadsheets to investigate many characteristics of magic cubes.

After looking closely at over 200 published magic cubes of order 3 to 17, I am amazed at how few cubes I found that had all identical features. Considering the large number of possible combinations available to form a magic cube of a given order, it is not surprising I found very few duplicate cubes. However, features such as number, type, and location of included magic squares, feature variations in the oblique squares, etc., were found to be extremely varied.
The result is this new series of pages, which explores the subject in some depth. With one or two possible exceptions, all magic cubes shown on these pages will have different features (or at least will be different orders).

As is usual with the other pages on this site, I intend to keep the discussions simple and will not normally go into methods of construction. Methods, and involved mathematics, will be left to others that are more qualified to present them. These pages will be more concerned with basic principles and a survey of the history and variety of magic cubes.

Navigation
This site is simple to navigate. The pages are listed in sequence in the index (although, of course,  they do not have to be read in that order).
At the bottom of each page are left and right pointing arrows that link to the previous and next pages. The 'top' button goes to the top of the page. The 'up' button will return you to this page.
The buttons at the top of each page go to:
'Home" goes to My Magic-squares.net web site. The 'square', 'star', and '#' go to the start page of each major division of that site.
The 'cube' button returns you to this page.
There is a general links section on the index page of the Magic-squares site as well as a combined site map of of the two sites.

As usual with my Web pages, I welcome comments, both laudatory and critical. Magic cubes covers a wide field and I am sure everyone may not agree with everything I have said on these pages. Also, some may feel I have put too much emphasis on certain subjects and not enough on others. I can only respond that this is how I see it.

I would like to thank some of those who helped me with my research of magic cubes. In no particular order they are Christian Boyer, Walter Trump, Aale de Winkel, John Hendricks, Abhinav Soni, and Mitsutoshi Nakamura. Links to many of their sites are on my links page.
Some others who helped in a lesser degree are  Paul Vaderlind, Brian Alspach, Mark Swaney, Vladimír Karpenko, Jacques Sesiano, Rich Schroeppel. I apologize for any I may have missed.  Thanks to all of you.

It is almost inevitable that despite the utmost care, a work of this size will contain some errors. I apologize for any and appreciate them being brought to my attention.

6 Classes of Cubes

The following definitions will be used throughout this web site. They are presented here simply as a concise introduction to the subject.
Examples and further explanation will be presented where appropriate.
See especially: Perfect magic Cubes, Perfect Cubes - 2, and Magic Cube Definitions.

NOTE: In January, 2005, a 6th class was added to the previous 5 classes. Pantriagonal Diagonal of PantriagDiag for short.
Mitsutoshi Nakamura has an excellent site on magic hypercubes, and has extensively researched their classes. His definitions page is at http://homepage2.nifty.com/googol/magcube/en/terms.htm

Magic cubes
Minimum requirements are: All rows, columns, pillars, and 4 triagonals must sum to the same value.

Nasik
An unambiguous term that may be used in place of the term perfect (which has differing meanings). See (perfect).

Simple:
Contains NO, or less then 3m orthogonal magic squares.

Pantriagonal:
All 4m² pantriagonals must sum correctly (that is 4 one-segment, 12(m-1) two-segment, and 4(m-2)(m-1) three-segment). There may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy any other classifications.
The pantriagonal magic cube is similar to a pandiagonal magic square in this respect. A pandiagonal magic square may be transformed to another pandiagonal magic square by moving a row or column from one side of the square to the opposite side. Similarly, a pantriagonal magic cube may be transformed into another pantriagonal magic cube by moving a plane from one side of the cube to the other! Furthermore, a panquadragonal magic tesseract may be transformed to another one by moving a cube from one side to the other! Etc.

Diagonal:
All 3m planar arrays must be 'simple' magic squares (some may be pandiagonal). i.e. all planar diagonals must sum correctly.
The 6 oblique squares will then automatically be magic. The smallest normal diagonal magic cube is order 5.
These squares  were referred to as ‘Perfect’ by Gardner and others! At the same time he referred to Langman’s 1962 pandiagonal cube as ‘Perfect’.

Pantriagonal Diagonal – A magic cube that is a combination Pantriagonal and Diagonal cube. All main and broken triagonals must sum correctly, In addition, it will contain 3m order m simple magic squares in the orthogonal planes, and 6 order m pandiagonal magic squares in the oblique planes.
For short, I will reduce this unwieldy name to PantriagDiag. This is number 4 in what is now 6 classes of magic cubes. So far, very little is known of this class of cube. The only ones constructed so far  are order 8 (not associated and associated).
This cube was discovered in 2004 by Mitsutoshi Nakamura.

Pandiagonal:
ALL 3m planar arrays must be ‘pandiagonal’ magic squares. The 6 oblique squares are always magic. Several of them MAY be pandiagonal magic.
Gardner also called this (Langman’s pandiagonal) a ‘perfect’ cube, presumably not realizing it was a higher class then Myer’s cube.

 Nasik:
ALL 3m planar arrays must be ‘pandiagonal’ magic squares. In addition, ALL pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares. When Hendricks devised this classification system, he called this perfect. However, because there are different definitions of perfect, Nasik is a better choice.
Generalized:
A hypercube of dimension n is perfect if all pan-r-agonals sum correctly. Then all lower dimension hypercubes contained in it are also perfect!
Through every cell on a perfect  hypercube of dimension n there are (3ⁿ-1)/2 different routes that must sum the magic sum.

A. H. Frost (1866) referred to all but the simple magic cube as Nasik! See a quotation by C. Planck, in which he redefined nasik to mean a Hendricks perfect hypercube only.
Nasik is an unambiguous term that should be used in place of the term perfect.

Index to Magic Cube Pages on this Site

The tests

By the use of Excel spreadsheets, I examined the characteristics of about 260 (Dec./03) published magic cubes. I limited the tests to orders 3 to 17 because of the scarcity of larger published cubes, and the increased effort required to enter the larger cubes into the spreadsheets.
That is about at the practical limit using this spreadsheet approach. The file size for order 16 is over 2 MB!

 A different spreadsheet design was required for each order, but they all had the following features in common.

• An area at the top for file name, title, and a bit of relevant information
• An m x m square array of cells for each of the m horizontal planes
• Both of these areas were unprotected to admit input. The rest of the spreadsheet was protected to prevent accidental overwriting because all necessary information was automatically copied from these horizontal arrays.
• An m x m square area of cells for each of the m vertical planes parallel with the front of the cube
• An m x m square area of cells for each of the m vertical planes parallel with the side of the cube
• An m x m square array of cells for each of the six oblique squares
• The above m x m arrays were automatically filled from the contents of the horizontal arrays, and the row, column and pandiagonal totals computed.
• An m x m square array of cells for each of the four directions of triagonals. Each cell of these arrays contained the sum of the broken triagonal pair or triplet originating at that cell. The cube was pantriagonal if all m x m cells in each of  the 4 arrays contained the magic constant.
• Supplementary tests for orders 4x.

Features the spreadsheet looked at:

• Number of planar squares magic
• Number of planar squares pandiagonal magic
• Center plane in each of 3 orthogonal directions magic (odd orders)
• # of planar squares with all pandiagonals in 1 direction correct
• # of oblique squares with rows and columns sum correct (square is magic)
• # of oblique squares with rows only sum correct
• # of oblique squares with columns only sum correct
• # of oblique squares with all pandiagonals. in 1 direction correct
• # of oblique squares with all pandiagonals correct
• # of directions with all pantriagonals correct
• Compact - All 2 x 2 squares (3 orientations) sum correct (order-4),
• Compactplus -# of orders (2 to m) of cubes with all cubes have  corners summing O.K. (orders 8,12,16)
• Complete - Every pantriagonal contains m/2 complement pairs spaced m/2 apart (orders 4x).

These features were tabulated in a Word document (CubeComparison.doc) for each cube in the collection.

For each additional cube within an order, I simply made a copy of the spreadsheet, then pasted or typed in the horizontal plane numbers..

Features of this magic cube site:

• Explanation of basic principles, features,  and definitions.
• A large number of cube examples, but all within an order will have different characteristics
• Subject matter is differentiated by separate pages
• References, where applicable, will be listed at the bottom of the section. A separate page will list all references I have used to compile this site.
• With only one or two exceptions, every cube shown on these pages will be unique (i.e. no cube shown twice).
• I will use m on these pages to indicate order of the cube, and n to indicate dimension (where required).

Please send me Feedback about my Web site!
Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 31, 2010
Copyright © 2003 by Harvey D. Heinz