Magic Cube Update-2

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Introduction

Material about magic cubes continues to appear. This is material I have received during February, March, and April of 2004.

The first magic cubes?

The Kurushima Cubes

Order 6 Projection Cubes

More about Pantriagonal Magic Cubes

The first magic cubes?

On April 18 2004, I received an email from Mitsutoshi Nakamura [1], advising me of two magic cubes dated about 1757. Both are order 4, and because all 48 orthogonal lines and the 4 triagonals sum to 130, are magic by the modern definition.
These are probably the earliest fully magic cubes ever constructed!
Fermat’s order 4 cube of 1640 was only semi-magic because the 4 triagonals are incorrect.
Sauveur’s order 5 cube of 1710 had all correct triagonals (and diagonals) but was not magic because most orthogonal lines summed incorrectly.

These Kurushima cubes were described in a book on magic squares, published in Japan in 1983 [2].

Title page

Pages 154 and 155 from Researches in Magic Squares

Some relevant passages from this book were kindly translated by Mr. Nakamura and are shown here (quoted from
his email of April 23, 2004). He also supplied the scanned page images shown above. 

The cube (1) on page 154 is the Fermat semimagic cube, the cube (2) is the Yoshizane Tanaka 
semimagic cube published in 1683. (See the second section following. hh)
The cubes (3) and (4) are Kurushima's magic cubes. (See the following section. hh)
The book explains the Kurushima cubes on pages 155 and 156.
The following is the description translated by me.
(I am not a translator, so I am afraid that the translation may not be suitable.)
----------
The cubes (3) and (4) have been found in a manuscript written by Yoshihiro Kurushima (?-1757).
Kurushima wrote the cubes only as series like the figure(*), so it is possible that people did not
notice that the series expressed magic cubes.
It can be said that after Kurushima there was no study of magic cubes until the 20th century in 
Japan.
The cubes (3) and (4) have the feature that all 48 orthogonals sum to the constant and also the 4 
triagonals 
1+46+31+52, 4+47+30+49, 61+18+35+16, 64+19+34+13 (for the cube 3;
and 49+27+38+16, 1+43+22+64, 61+23+42+4, 13+39+26+52 for the cube 4.
They are the first order 4 magic cubes in the true sense.
Furthermore, they partially have the feature of compactness.
The cube (3) has the feature like the cube (1) that all horizontal planes are associated, and the 
cube (4) 
is an associated magic cube.
 (*)The figure is shown on page 155. The following is its translation.
(This series expresses the cube (4).)
An order 4 magic cube:
A1 60 56 13 B63 6
10 51 A48 21 25 36
B18 43 39 30 A32
37 41 20 B34 27 23
46 A49 12 8 61 B
15 54 58 3 C62 7
11 50 D4 57 53 16
C19 42 38 31 D45 2-
4 28 33 C35 26 22
47 D29 40 44 17
C14 55 59 2 D52 9
5 64
----------
The book also discusses Kurushima on page 204.
Yoshihiro Kurushima (?-1757) was an exceptional genius in mathematics in Japan.
He succeeded in mathematics by studying it by himself,
and had few interest in anything but drinking and mathematics.
The following story about him has been handed down.
When Yoshihiro had not known the game "go", he saw a master of "go"
and was taught the rule of "go" by the master.
The next day, Yoshihiro made a problem of "go" and brought it to the master.
The master watched it and was surprised that Yoshihiro became an expert at "go".
Yoshihiro also made checkmate problems of "shogi" (Japanese chess),
and the problems are now appreciated by shogi players.
Yoshihiro served Masaki Naito and stayed for a long time at Kyushu Nobeoka in west Japan.
When Yoshihiro came back to Edo (Tokyo), he used his fair-copied manuscript for making a clothes 
box.
Yoshihiro did not publish any books because he was indifferent to doing as the story.
Only manuscripts copied by his pupils have been handed down as "Kushi Ikou" and so on.
He made magic squares by probably the easiest way in the world.
The book describes the following on page 297.
1. Yoshinao Katagiri made an order 12 "pantriagonal and diagonal" magic cube in 1977.
2. Gakuho Abe made an order 10 pantriagonal magic cube in 1960.
However, I do not have access to these cubes.

[1] A special thanks to Mitsutoshi Nakamura for advising me of these cubes. He also sent the two images and the translation of pages 154-155 shown
above.
[2] Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.

The Kurushima Cubes

From the information we have, it is impossible to determine which cube was actually the first one constructed by Yoshihiro.
We will refer to the cube (3) from the above book as Kurushima 1 and cube (4) as Kurushima 2.

Kurushima 1 is a simple magic cube. The four horizontal planes are each an associated magic square. 2 of the 6 diagonal squares are simple magic squares. Of the other 4 oblique squares, 2 have all rows correct, 2 have all columns correct.

I                         II                        III                      IV      
01  62  63  04     60  07  06  57     56  11  10  53     13  50  51  16
44  23  22  41     17  46  47  20     29  34  35  32     40  27  26  37
24  43  42  21     45  18  19  48     33  30  31  36     28  39  38  25
61  02  03  64     08  59  58  05     12  55  54  09     49  14  15  52

 

Kurushima 2 is an associated magic cube. It contains no magic squares.

I                         II                        III                       IV
49  32  48  01     15  34  18  63     14  35  19  62     52  29  45  04
12  37  21  60     54  27  43  06     55  26  42  07     09  40  24  57
08  41  25  56     58  23  39  10     59  22  38  11     05  44  28  53
61  20  36  13     03  46  30  51     02  47  31  50     64  17  33  16

Both these cubes were constructed before 1757. Both are taken from page 154 of Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.
Are these the only magic cubes produced in Japan before the 20th century?

Early Semi-magic Cubes

Page 154 of the above book also shows the 2 earliest (?) semi-magic cubes. Both are also order 4. Both cubes had all 48 orthogonal lines summing correctly, but incorrect sums for the triagonals. Fermat’s and Tanaka’s cubes both appeared earlier then Kurushima’s magic cubes!

The Fermat semi-magic cube of 1640 is shown on my Early Cubes page. 8 of the 12 orthogonal planes are magic squares (the 4 horizontal are associated). Rows and columns of 4 of the 6 oblique planes were correct but are not magic squares because the diagonals (cube triagonals) are incorrect. Columns only are correct on the other 2 oblique squares.

The Tanaka cube of 1683 had identical features to the Kurushima 1 cube, except that the triagonals were incorrect. i.e. the 4 horizontal planes are magic squares (but not associated), 2 oblique squares have rows and columns summing correct, 2 have rows only correct and 2 have columns only correct. I                         II                        III                      IV
14  54  43  19     20  44  53  13     33  25  08  64     63  07  26  34
59  03  30  38     37  29  04  60     24  48  49  09     10  50  47  23
22  46  51  11     12  52  45  21     57  01  32  40     39  31  02  58
35  27  06  62     61  05  28  36     16  56  41  17     18  42  55  15

Order-6 Projection Cubes

A projection cube [1] consists of zeros or ones in each of the m3 cells. The m-digit binary number that each line represents, ‘projects’ a decimal integer from 0 to m2 on the appropriate cube face, depending on which of the two directions the line is being read.
I show an order 4 projection cube on my Unusual Cubes page.

The original idea was proposed by K. S. Brown and answered by Dan Cass [2].

[1] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, page 25.
[2] The Sci.math newsgroup Dec. 2, 1996 and Dec. 10, 1996.

On Feb. 10, 2004, Peter Manyakhin (who was obviously unaware of the above mentioned page) proposed an order 6 cube of this type. A spirited discussion subsequently took place between him, Walter Trump, Guenter Stertenbrink and Aale de Winkel.
See the encyclopedia for a lot of information on order 4 and 6 cubes of this type.

One of the projection cubes listed
on the encyclopedia page is:

Top      II       III    
111111   111000   110000 
111110   111101   110111 
110101   110001   001010 
100101   111001   100100 
100010   101101   001100 
100000   011000   110110 
IV       V        Bottom
111100   110011   101100
010010   000000   011110
010111   011001   010000
101010   011101   001011
000101   100001   001110
110011   010110   000100

The projections on the faces of the above cube, reading from:
Left                Front               top
63 56 48 60 50 44   63 07 03 15 19 13   63 62 53 37 34 32
62 61 55 18 00 30   31 47 59 18 00 30   56 61 49 57 45 24
53 49 10 23 25 16   43 35 20 58 38 02   48 55 10 36 12 54
37 57 36 42 29 11   41 39 09 21 46 52   60 18 23 42 05 51
34 45 12 05 33 14   17 45 12 40 33 28   50 00 25 29 33 22
32 24 54 51 22 04   01 06 27 51 26 08   44 30 16 11 14 04
Right               Back                Bottom
63 07 03 15 19 13   63 56 48 60 50 44   63 31 43 41 17 01
31 47 59 18 00 30   62 61 55 18 00 30   07 47 35 39 45 06
43 35 20 58 38 02   53 49 10 23 25 16   03 59 20 09 12 27
41 39 09 21 46 52   37 57 36 42 29 11   15 18 58 21 40 51
17 45 12 40 33 28   34 45 12 05 33 14   19 00 38 46 33 26
01 06 27 51 26 08   32 24 54 51 22 04   13 30 02 52 28 08

Thanks to Aale de Winkel for this listing.

Peter Manyakhin reported on April 28, 2004 that he had already found over 200,000 order 6 cubes of this type.
Guenter Stertenbrink estimates that there are over 20,000,000 different order 6 projection cubes.
See the dialog on this subject at http://groups.yahoo.com/group/magiccubes/

More about Pantriagonal Magic Cubes

On April 28, 2004 I received an order 15 diagonal magic cube in an email attachment from Mitsutoshi Nakamura with help from Abhinav Soni [1].
I was excited about this because it filled the last vacant cell in my table of “First cube of each order for each class”.

Reminder
A proper Diagonal magic cube has m2 rows, m2 columns, m2 pillars, 6m diagonals and 4 triagonals = S.
The above sentence indicates that all 3m orthogonal arrays are order m simple magic squares.
All six oblique planes are also magic squares.

This is the Hendricks/Heinz designation for this type of  magic cube. Boyer and Trump refer to this as a 'perfect' magic cube. The Hendricks definition for perfect (nasik) is here. And a discussion and 1905 quotation on nasik is here.

When I pasted it into my test spreadsheet, I quickly realized there was something different about this cube. It is not ‘proper’ because only 12 of the orthogonal planes are diagonal magic squares. The other 33 planes are pandiagonal magic squares. However, it must still be classed as ‘Diagonal’ because that is the highest class it qualifies for.

What caught my interest though, was the fact that all 6 oblique squares are pandiagonal magic!

The fact that these 6 squares are pandiagonal magic indicates that the pandiagonals of these squares (which are the pantriagonals of the cube) all sum correctly. I have inspected 56 other cubes of orders 3 to 17 that had all pandiagonals of these 6 oblique squares summing correctly. In ALL cases, these cubes were pantriagonal magic. (Side note: these squares in most cases are NOT pandiagonal magic because all rows or all columns of these squares did not sum to S. Under the basic definition of a magic cube, all rows OR all columns of each of these oblique squares are required to sum to S.

Besides the 56 cubes mentioned above, I have many other cubes with this feature, but they also contain 3m pandiagonal magic squares in the orthogonal planes. Because they are a combination pandiagonal and pantriagonal cube, they are classed as perfect (the modern definition). In this case, all 6m oblique squares are also pandiagonal magic. These are also called nasik to avoid the confusion with the definition for perfect.

Pandiagonal magic cube     = m2 rows, m2 columns, m2 pillars, 6m2 pandiagonals and 4 triagonals = S
Perfect (nasik) magic cube = m2 rows, m2 columns, m2 pillars, 6m2 pandiagonals and 4m2 pantriagonals = S

A close inspection of the pantriagonal test for this order 15 cube showed that for each of the 4 directions, 8 of the 225 pantriagonals sum to 24902 instead of the correct 25320, and 8 sum to 25738.

The explanation:
Cube_15-Diagonal.xls is the first example I have seen of a magic cube where all 6 oblique squares are pandiagonal magic, but the cube is NOT pantriagonal!
This is because some of the pantriagonals that start on interior cells do not sum correctly.
Another way to explain this is that not ALL pandiagonals of some of the broken oblique squares sum correctly!
i.e. It is necessary, but not sufficient, that all pandiagonals of all 6 oblique squares sum correctly for the cube to be pantriagonal magic.
However, these oblique squares may not be pandiagonal magic because of incorrect sums for the rows OR columns!.

Above is just a clarification. The definition for a pantriagonal magic cube has NOT changed.
Pantriagonal magic cube =
m2 rows, m2 columns, m2 pillars and 4m2 pantriagonals = S

[1] Mitsutoshi Nakamura is a 40 year old computer programmer living in Morioka, Japan. He majored in mathematics at university, but has studied magic cubes only since 2000. He is unmarried and is an admirer of Yoshihiro Kurushima (? – 1757). His website is http://homepage2.nifty.com/googol/magcube/en/
Abhinav Soni is a graduate student in the Bachelor of Technology degree at the Indian Institute of Technology in Roorkee, India. His interest in mathematics led him to write a program to generate magic cubes.

This page was originally posted May 2004
It was last updated October 16, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz