Cube_Update-4

Introduction

Material about magic cubes continues to appear. This update contains  material I have received in 2004 and 2005 but not yet published.

More on Panmagic Ratios

In Cube Update-1, mention was made of Magic Ratios, and almost as an after thought, Panmagic Ratios. These ratios are a tool for comparing the ‘richness’ of magic cubes.

To recap;
Magic ratio: correct lines divided by total rows, columns, pillars, and 4 triagonals or 3m² + 4.
These are the features of a simple magic cube, so if the cube in question is magic the ratio will be 1:1 or 100%. If the cube is a class higher then a simple magic cube, the ratio will be higher then 1:1.
Panmagic ratio: divide the correct lines in the cube by the total lines as used in magic ratio plus total possible broken diagonals and broken triagonals or 13m². i.e a perfect magic cube.

In retrospect, this second term is more meaningful. It is the ratio of all correctly summing lines in the cube divided by all possible lines. We will see the value of this ratio when we look at the order 4 semi-magic cube in the next section.

The ratios for an order 4 magic cube in each of the 6 classes.
NOTE that this is an example only, because for a normal order 4 magic cube, only the first two classes are possible.

Semi-diagonal magic order 4

On Mar 4, 2004 Walter Trump of Germany sent me this order 4 cube. He asked me to not put it on my site until a pending description of it was published in a German magazine. [1]
It is only semi-magic because although all rows, columns pillars are correct, none of the four triagonals sum correctly. However all 24 of the planar diagonals also sum correctly, so the cube contains 12 order 4 magic squares parallel to the sides of the cube.

Walter refers to the cube as ‘Nearly-perfect’, referring to the definition of perfect promoted by Boyer and Trump. I would term it ‘Nearly-diagonal’ as per the Hendricks-Heinz definition.

The cube

Some comments that came from Walter with this cube:

72 lines of this non-magic cube of order 4 are magic.
Only the 4 triagonals are not magic.
The cubes magic ratio is   72 / 76 = 95%. (See Editors note below)

Properties of this cube:
It consists of all numbers from 1 to 64.
The cube is plane symmetrical
All rows, columns and pillars are magic
All diagonals are magic
The 4 triagonals are not magic:
Their sums are 100, 120, 140, 160
These sums differ only in the second digit from the magic constant 130.

The cube is perfect (Heinz=diagonal) modulo 10.
The cube is not unique, there are 64 non-trivial transformations.
There are at least 180 x 64 order-4 cubes with 72 magic lines.
There are no plane symmetrical order-4 cubes  with more than 72 magic lines.

Editors note
The original definition was for a simple magic cube, so magic ratio would be 72/52 = 138.5%
Panmagic ratio = 72/208 = 34.6%
Of course, another way to use these ratios is to use the appropriate divisor for the class of cube most nearly represented by the cube in question. This is what Mr. Trump did above.
And I apologize. My original article on this subject is rather ambiguous.

[1] Spektrum der Wissenschaft, March 2004, No. 108.

The Leibniz cube

An email received from Christian Boyer on Jan 13, 2004

About your last question, I have finally found the famous cube sent by Leibniz to the Académie des Sciences in 1715.
Not at the Académie, but in the Leibniz letters kept at the Hanover Library, Germany.
Never published before!

This cube was credited to Leibniz but was actually sent to him by Father Augustin Thomas de Saint Joseph, a professor in Horn, Austria. Leibniz was so taken with this cube that he promptly sent it on to the Académie Royal des Sciences in Paris, where it was examined (in November 1715) by two of it’s mathematicians, Pierre Varignon and Phillippe de La Hire. It was also studied by Joseph Sauveur.
Christian Boyer subsequently published a fascinating article about the history and rediscovery of this cube, in a French scientific journal. [1] He has written an English translation of the article (complete with additional details), which he hopes to publish on his web site. [2]

This cube is not magic in the current sense.
It has exactly the same features as Joseph Sauveur’s order 5 cube of 1710.

These are:
All planar diagonals are correct (sum to 42).
All four triagonals are correct.
All planes (9 cells) sum to 126.
The six oblique planes also sum to 126.
Three of these six arrays are magic squares.
1 row and 1 column of each planar array sums correctly to 42.

 [1] Christian Boyer, « Le plus petit cube magique parfait » (and « Inédit - Le cube magique de Leibniz est retrouvé »), La Recherche,
issue number 373, March 2004, pages 48-50, Paris, 2004
[2] http://www.multimagie.com/indexengl.htm

Prime Magical Cubes

In 1968 Les Card invented a new type of number square, that, while not magic in the usual sense, was still very intriguing.
The square array consists of m² cells, each of which holds a single digit. The m digits in each row form a prime number. The object is to fill all the cells with digits so that the array consists of prime numbers of length m when the digits in each line is read in either order. He expanded the concept to 3 dimensions (but with incomplete solutions). I discuss this on my Unusual cubes page.

On May 16, 2005 I received an email from Anurag Sahay responding to my challenge to provide improved models of this idea. While still a long way from a complete solution, by July 7, 2005 he was able to provide me with an order 4 and an order 5 cube with quite impressive features.

Order 4 Prime Magical Cube.
As per Les Cards specifications, the diagonals and triagonals are not required to be prime numbers. So, order 4 requires 48 distinct reversible 4-digit prime numbers (no palindromes) to be complete. That is 96 different prime numbers (3m² for the orthogonal lines times two directions).

This cube contains 65 four-digit prime numbers. 22 of these are reversible primes!

Order 5 Prime Magical Cube.

A perfect cube of order 5 would require 150 distinct primes (75 reversible non-palindrome).

This cube contains 115 five-digit prime numbers. 45 of these are reversible primes!

Anurag also sent me two order 6 cubes on July 9/05.
One had 63 reversible and 31 single primes for a total of 157 primes.
One had 62 reversible and 35 single primes for a total of 159 primes.
Required for an order 6 perfect prime magical cube: 3m² x2 or 216 distinct 6-digit primes

Can anyone do better then Anurag?

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 15, 2009
Copyright © 2005 by Harvey D. Heinz