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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 |
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. The cubes (3) and (4) are Kurushima's magic cubes. 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.
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
|
 |
Kurushima 2 is an associated magic cube. It contains no magic squares. I II
III IV 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. |
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 |
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 (06) encyclopedia page is:
Top II III IV V Bottom 111111 111000 110000 111100 110011 101100 111110 111101 110111 010010 000000 011110 110101 110001 001010 010111 011001 010000 100101 111001 100100 101010 011101 001011 100010 101101 001100 000101 100001 001110 100000 011000 110110 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/
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”.
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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. |
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.
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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. |
[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).
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.
However, he no longer has a web page
(Nov./09)
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
November 02, 2009
Copyright © 2004 by Harvey D. Heinz