Magic Cube Update2

Material about magic cubes continues to appear. This is material I have received during February, March, and April of 2004.
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 Kurushima cubes were described in a book on magic squares, published in Japan in 1983 [2]. Some relevant passages from this book were kindly
translated by Mr. Nakamura and are shown here (quoted from 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 faircopied 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 154155 shown
The Kurushima CubesFrom the information we have, it is
impossible to determine which cube was actually the first one constructed by
Yoshihiro.
Early Semimagic Cubes Page 154 of the above book also shows the 2 earliest
(?) semimagic 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!
Order6 Projection Cubes A projection cube [1] consists
of zeros or ones in each of the m^{3} cells. The mdigit
binary number that each line represents, ‘projects’ a decimal integer from
0 to m^{2} on the appropriate cube face, depending on which of the
two directions the line is being read. 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, Selfpublished, 2000,
0968798500, page 25. 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.
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. 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].
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.
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:
[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/
