Monster Cubes

Introduction

The subject of this page is a Word document I received from Christian Boyer (France) on May 13, 2003.
Christian has a very informative site where he discusses multimagic cubes (and squares) but his material on these large cubes is not presented in quite this way.
On Nov. 26, 2003, I reviewed his document and decided I would like to include it on my site. Christian has kindly granted me permission to do so. He also suggested I edit it as I saw fit. However, the editing I did do has been kept to a minimum, and is mostly limited to formatting.
Harvey Heinz Nov. 27, 2003

Seven new multimagic « monsters »:

tetramagic cubes, bimagic and trimagic hypercubes (tesseracts)

Christian Boyer, France, May 13^rd, 2003

I have the pleasure to announce 7 new important multimagic results:

· the first tetramagic cube, so better than my previous trimagic cubes

· the first perfect tetramagic cube, means all its diagonals and triagonals are tetramagic (and probably the biggest magic cube ever constructed !)

· the 3 first bimagic tesseracts, means four-dimensional bimagic hypercubes

· the 2 first trimagic tesseracts, one of them being also the first perfect bimagic tesseract (means

· all its diagonals, triagonals, and quadragonals are bimagic)

You will find some technical details at the end of this document.

All these huge cubes and hypercubes were constructed from February to April 2003, and have been checked(*) by other French scientists(**) from March to May 2003, the same persons that had checked my previous tetra and pentamagic squares announced two years ago. So 3 persons using 3 different programs on 3 different machines.

Both construction and checking required a lot of computing resources. Powerful P4 machines, C and assembly code, and long computing time. It is not an easy task to check an object as big as the perfect tetramagic cube of order 8192. For example if it uses each number from 0 to 549,755,813,887 only once and you are able to check only 1,000 numbers per second, you would need 17.5 years of computing time! ! !

The only complete objects that I can communicate through the Internet, for size reasons, are my two bimagic tesseracts of order 32. If you want one of them, or both, just send me a message at cboyer@club-internet.fr. But be aware that the size of these 2 zipped Excel files is already big: 4Mb each! [1]

At a first look, an order 32 seems relatively small... But in fact, not very small. A tesseract of order 32 uses exactly as many numbers as my pentamagic square of order 1024! That is because 32⁴ = 1024².

I dedicate the tetramagic cubes to Gaston Tarry and André Viricel.
Gaston Tarry, inventor of the "tetramagic" term, was the first person to have constructed a trimagic square, in 1905. It was of order 128. Also the first person to have proved the famous Euler’s conjecture of the 36 officers.

And my old friend André Viricel is the man who has invented a powerful method to construct trimagic squares of order 32. All my multimagic constructions are based on the ideas of Gaston Tarry (later improved by General Cazalas) and André Viricel. Ideas extended to work with higher orders, p^x, and higher dimensions, 3^rd and 4^th.

I dedicate the multimagic tesseracts to John R. Hendricks, the man who has done the most impressive work on tesseracts in the world. For example, the first man to publish the 58 basic tesseracts of order 3.

I will try to publish my results, perhaps again in Pour La Science. [2]
These announcements are probably the end of my research on big multimagic objects.
Becomes crazy to go further!
If one day, someone wants to construct a 5^th dimension bimagic hypercube, I am sure that he will be happy with the order 64. My conjecture is that the smallest bimagic hypercube of dimension d is of order 2^(d+1).

Best regards.
Christian Boyer.
www.multimagie.com

(*) Not yet the trimagic tesseract of order 243, more difficult to check because different of others: not a 2^n order ! Only checked by me, for the moment.
(Editors note ; It was confirmed correct by Yves Gallot (email of June 6/03).
(**) Yves Gallot, the author of the famous Proth program widely used by searchers of big prime numbers.
Look at http://www.utm.edu/research/primes/programs/gallot/. Look at his papers at http://perso.wanadoo.fr/yves.gallot/papers/
And Renaud Lifchitz, one of the main searchers of big prime numbers.
Look at http://www.primenumbers.net/prptop/prptop.php, he has found 4 among the 10 biggest PRPs known in the world!
Many thanks to Yves and Renaud.
[1] Editors note. These two files are now downloadable directly from Christian/s web pages.
[2] Editors note. Christian Boyer, Pour La Science, No. 311, Sept., 2003, pp 90-95 (about cubes only, tesseracts were not covered).

1) Tetramagic cube of order 1024

This cube is using numbers from 0 to 1073741823
The 1048576 rows, 1048576 columns and 1048576 pillars are tetramagic.
The 4 triagonals are tetramagic.
The 6144 diagonals are trimagic.
Magic sums:
S 1 = 549755813376
S 2 = 393530539689381287424
S 3 = 316912649466761540290348056576
S 4 = 272225892902925471050115525436200067584

2) Perfect tetramagic cube of order 8192

This cube is using numbers from 0 to 549755813887
The 67108864 rows, 67108864 columns and 67108864 pillars are tetramagic.
The 4 triagonals are tetramagic.
The 49152 diagonals are tetramagic.
Magic sums:
S 1 = 2251799813681152
S 2 = 825293359521335050119065600
S 3 = 340282366919700523424090353056775929856
S 4 = 149657767662003894090216275236580155584753888727040

3) Bimagic hypercube of order 32

This cube is using numbers from 0 to 1048575
The 32768 rows, 32768 columns, 32768 pillars and 32768 files are bimagic.
The 8 quadragonals are bimagic.
The 512 triagonals are magic. Only 6144 among the 12288 diagonals are magic.
Magic sums:
S 1 = 16777200
S 2 = 11728107252400

4) Bimagic hypercube of order 32

This cube is using numbers from 0 to 1048575
The 32768 rows, 32768 columns, 32768 pillars and 32768 files are bimagic.
The 8 quadragonals are bimagic.
Only 448 among the 512 triagonals are magic. The 12288 diagonals are magic.
Magic sums:
S 1 = 16777200
S 2 = 11728107252400

5) Bimagic hypercube of order 64

This cube is using numbers from 0 to 16777215
The 262144 rows, 262144 columns, 262144 pillars and 262144 files are bimagic.
The 8 quadragonals are bimagic.
The 1024 triagonals are magic. The 49152 diagonals are magic.
Magic sums:
S 1 = 536870880
S 2 = 6004798966289760

6) Trimagic hypercube of order 243

This cube is using numbers from 0 to 3486784400
The 14348907 rows, 14348907 columns, 14348907 pillars and 14348907 files are trimagic.
The 8 quadragonals are trimagic.
The 3888 triagonals are magic. The 708588 diagonals are magic.
Magic sums:
S 1 = 423644304600
S 2 = 984770901759966928200
S 3 = 2575262863742228010429792120000

7) Trimagic hypercube of order 256 (and perfect bimagic hypercube)

This cube is using numbers from 0 to 4294967295
The 16777216 rows, 16777216 columns, 16777216 pillars and 16777216 files are trimagic.
The 8 quadragonals are trimagic.
The 4096 triagonals are bimagic. The 786432 diagonals are bimagic.
Magic sums:
S 1 = 549755813760
S 2 = 1574122160406792590720
S 3 = 5070602398551734364826868121600

Editor's note: Christian Boyer prefers to use the old definition of perfect for some of these multimagic hypercubes.
In his cases all planar diagonals sum correctly, so all planar arrays are simple magic squares.
I am encouraging the use of the new definition of diagonal for this type of hypercube.
A pandiagonal magic cube then has all planar arrays pandiagonal magic squares.
A perfect hypercube is one in which all pan-n-agonals sum correctly, and all lower dimension hypercubes contained in it are perfect!
For more information, see my Perfect and Perfect-2 pages.

Due to continuing confusion with the term perfect, I am encouraging the use of the historical term nasik for hypercubes where all lines sum to the constant.
See Planck's (1905) revised definition of Frost's (1866) original nasik here.

Have you seen my original Multimagic Cubes page?
Christian Boyer's Web site is www.multimagie.com/indexengl.htm