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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.
|
Class of
magic cube |
Formula for
Panmagic ratio |
Ratio
(order 4) |
Percentage of
correct lines |
Smallest
actual
order possible |
|
Simple |
3m2
+ 4/13m2 |
52:208 |
25 |
3 |
|
Pantriagonal |
7m2/13m2 |
112:208 |
53.8 |
4 |
|
Diagonal |
3m2
+ 6m + 4/13m2 |
76:208 |
36.5 |
5 |
|
PantriagDiag |
7m2
+ 6m/13m2 |
136:208 |
65.4 |
8 ? |
|
Pandiagonal |
9m2
+ 4/13m2 |
148:208 |
71.2 |
7 |
|
Perfect |
13m2/13m2 |
208:208 |
100 |
8 |
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
|
59 |
30 |
40 |
1 |
|
23 |
2 |
60 |
45 |
|
42 |
63 |
5 |
20 |
|
6 |
35 |
25 |
64 |
|
21 |
4 |
58 |
47 |
|
14 |
43 |
17 |
56 |
|
51 |
22 |
48 |
9 |
|
44 |
61 |
7 |
18 |
|
34 |
55 |
13 |
28 |
|
57 |
32 |
38 |
3 |
|
8 |
33 |
27 |
62 |
|
31 |
10 |
52 |
37 |
|
16 |
41 |
19 |
54 |
|
36 |
53 |
15 |
26 |
|
29 |
12 |
50 |
39 |
|
49 |
24 |
46 |
11 |
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).
|
1 |
3 |
7 |
2 |
|
2 |
5 |
9 |
3 |
|
3 |
1 |
4 |
3 |
|
1 |
1 |
9 |
3 |
|
7 |
6 |
9 |
9 |
|
7 |
8 |
7 |
9 |
|
7 |
3 |
9 |
3 |
|
2 |
3 |
1 |
1 |
|
8 |
9 |
7 |
1 |
|
2 |
4 |
2 |
6 |
|
3 |
0 |
1 |
1 |
|
1 |
9 |
9 |
9 |
|
7 |
7 |
5 |
7 |
|
4 |
3 |
1 |
5 |
|
3 |
1 |
0 |
9 |
|
3 |
3 |
7 |
1 |
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).
|
3 |
9 |
9 |
7 |
9 |
|
7 |
7 |
9 |
7 |
7 |
|
1 |
5 |
7 |
3 |
3 |
|
3 |
1 |
1 |
3 |
7 |
|
9 |
1 |
3 |
9 |
3 |
|
9 |
9 |
1 |
3 |
9 |
|
9 |
1 |
9 |
6 |
7 |
|
7 |
9 |
1 |
1 |
9 |
|
9 |
3 |
9 |
9 |
0 |
|
3 |
9 |
3 |
1 |
3 |
|
7 |
3 |
3 |
3 |
1 |
|
7 |
3 |
3 |
1 |
3 |
|
3 |
3 |
3 |
3 |
2 |
|
6 |
7 |
2 |
9 |
9 |
|
9 |
1 |
9 |
5 |
1 |
|
1 |
9 |
3 |
3 |
3 |
|
3 |
4 |
1 |
4 |
1 |
|
9 |
7 |
5 |
7 |
1 |
|
3 |
1 |
1 |
8 |
3 |
|
3 |
8 |
1 |
1 |
9 |
|
3 |
7 |
1 |
9 |
9 |
|
3 |
7 |
7 |
7 |
2 |
|
3 |
7 |
9 |
9 |
7 |
|
1 |
1 |
1 |
9 |
7 |
|
7 |
1 |
1 |
1 |
9 |
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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