.... Kanji Setsuda's New English Page
I am very sorry for my long absence, but I am now come back to this page!
I was long sick from autumn 2003 to autumn 2004. I almost lost my sight so that I could neither read nor study over a whole year. Since I became self-confident on my health, I tried to restart my study at the beginning of 2005, and in spring I found something new I would like to report of.
I have been trying to revise some old articles of mine as hard as possible.
(on November 28, 2007: Kanji Setsuda)
"About Definitions"
"About 'Solution-sets' and 'Transformation'"
"What's Hot?"
"What's New?"
"Profile: Self-Introduction"
*** Index of Revised Edition(Original in Japanese) ***
English titles(underlined) are now available as many as about 64.
Part 1. Basic Study of Magic Squares and Cubes
Chapter 1. Introduction: What are Magic Squares?
#1. Magic Squares of Order 3
#2. Magic Squares of Order 4:
... Self-Complementary Type and Pan-Magic Type
#3. Magic Squares of Order 5
#4. Magic Cubes of Order 3
#5. Magic Cubes of Order 4
#6. Let's Begin our Algebraic Study
Chapter 2. Algebraic Study of Magic Squares
#1. Study of the Basic Properties of M.S. of Order 3
#2. Self-Complementary Magic Squares of Order 4
#3. Pan-Magic Squares of Order 4
Chapter 3. Algebraic Study of Magic Squares(Part 2)
#1. Self-Complementary Magic Squares of Order 5
#2. Magic Cubes of Order 3
#3. Magic Cubes of Order 4: Self-Complementary Type
Chapter 4. Advanced Algebraic Study of Magic Cubes 4x4x4 (.pdf) [Revised]
#1. Pan-Magic Cubes of Order 4
#2. Self-Complementary Magic Cubes of Order 4
#3. 'Simultaneous' Magic Cubes 444: Both Self-complementary and Pan-triagonal
#4. Composite Pan-Magic Cubes of Order 4 (.pdf) [Revised]
Part 2. Basic Study of Magic Stars
Basic Studies of Magic Stars I (.pdf) [Hot]
Chapter 1. Introduction: What are Magic Stars?
Chapter 2. Algebraic Study of Magic Stars of Order 5
Chapter 3. Algebraic Study of Magic Stars of Order 6
Chapter 4. Algebraic Study of Magic Stars of Order 7
Basic Studies of Magic Stars II (.pdf) [Hot]
Chapter 5. Algebraic Study of Magic Stars of Order 8
Chapter 6. Algebraic Study of Magic Stars of Higher Orders
...#1. Data for Magic Stars of Order 9
...#2. Data for Magic Stars of Order 10
...#3. Data for Magic Stars of Order 11
...#4. Data for Magic Stars of Order 12
Outlines of New Triple-Ringed Magic Stars (.pdf) [Hot]
Chapter 7. New Triple-Ringed Magic Stars of Order 7
Chapter 8. New Triple-Ringed Magic Stars of Order 8
Chapter 9. New Triple-Ringed Magic Stars of Order 9
Chapter 10. Advanced Study of Some Double-Ringed Magic Stars
Part 3. New Algebraic Study of Magic Squares
Chapter 1. New Algebraic Study of Magic Squares of Order 3(.pdf) [Hot]
#1.The Standard Magic Type of Order 3
#2.Self-Complementary Magic Squares of Order 3
#3.Pan-diagonal Magic Squares of Order 3
Chapter 2. New Algebraic Study of Magic Squares of Order 4
#1. Self-Complementary Magic Squares of Order 4(.pdf) [Revised]
#2. Pan-Magic Squares of Order 4(.pdf) [Revised]
#3. 'Composite' Magic Squares of Order 4(.pdf) [Revised]
#4. 'Composite and Complete' Pan-magic Squares 4x4
... and the Meaning of Complementary Pairs(.pdf) [Hot]
Chapter 3. New Algebraic Study of Magic Squares of Order 5
#1. Study of Self-Complementary Magic Squares of Order 5(.pdf) [Revised]
#2. New Study of Pandiagonal Magic Squares of Order 5(.pdf) [Revised]
#3. Study of Simultaneous Magic Squares 5x5:
... Both Self-Complementary and Pan-Diagonal(.pdf) [Revised]
#4. Analysis on a 'Classical Composition'
... and Simultaneous 'Suzuki Squares'(.pdf) [Revised]
Chapter 4. Algebraic Study of Magic Squares of Order 6(.pdf) [Revised]
#1. Self-Complementary Magic Squares of Order 6
#2. Pan-Diagonal Magic Squares of Order 6
#3. Standard Magic Squares of Order 6
#4. 'Even-Odd Patterns' for Magic Squares 6x6 (.pdf) [Hot]
Chapter 5. Outlines of Magic Squares of Order 7(.pdf) [Revised]
#1. Self-Complementary Magic Squares of Order 7
#2. Pan-diagonal Magic Squares of Order 7
#3. Simultaneous Magic Squares of Order 7:
... Both Self-Complementary and Pan-Diagonal
Chapter 6. Outlines of Various Magic Squares of Order 8(.pdf) [Hot]
Chapter 7. Magic Squares of Order 9:
#1. Algebraic Study of Simultaneous Type:
... Both Self-Complementary and Pan-Magic Type
Chapter 8. Study of Magic Squares: Self-Complementary Types
... Symmetrical with Respect to some Lines
#1. Such Magic Squares of Order 4
#2. Such Magic Squares of Order 5
#3. Such Magic Squares of Order 6
Chapter 9. Fundamental Study of Composite Magic Squares
#1. Composite Semi-Panmagic Squares of Order 6(.pdf)[Revised]
#2. Composite and Pan-Magic Squares of Order 8(.pdf)[Revised]
... (Newer Study of Fundamental Properties)
Part 4. New Advanced Study of Magic Squares and Cubes
Chapter 1. Fundamental Study of 'Prototype' Squares and
... 'Do-it-After-the-Model' Transformation
#1. Two Types of Squares of Order 3(.pdf) [Revised]
#2. Prototype Squares of Order 4 and 'DAM Transformation'(.pdf) [Revised]
#3. Prototype Squares of Order 5 and Three Types of Magic Squares of Order 5
... (.pdf) [Revised]
Appendix: How to Compose the Models of Simultaneous Magic Squares
... of Order 5 and 7(.pdf)
Chapter 2. Fundamental Study of 'Prototype' Squares and
... 'Do-it-After-the-Model Transformation': Part 2
#4. What is the 'Do-it-After-the-Model' Transformation?
#5. Prototype Squares of Order 6
#6. Prototype Squares of Order 7(.pdf) [Revised]
#7. Prototype Squares of Order 8(.pdf) [Revised]
#8. New Study of 'Prototype Squares' of Order 8 and 'Do-it-After-the-Model
... Transformation'(.pdf) [Hot]
Chapter 3. Commentary Articles:
... Various Arts and Tools for Study of Magic Squares
#1. Various Transformations and the Solution Sets
#2. Complementary Pairs and their expressions
#3. Pan-diagonals and the Extended Form of Square
#4. Find the Fundamental Solutions
#5. Pan-Magic Squares and the Complementary Pairs
#6. Internal Transformations and the Fundamental Patterns
#7. Practice: How to make the Simultaneous Magic Squares of Order 5:
... Both Self-Complementary and Pan-Magic
Chapter 4. Commentary Articles No.2:
... Various Arts and Tools for Studying Magic Squares
#1. Notations and 'Euler Squares'(.pdf) [Hot]
#2. Apply the 'Do-it-After-the-Model' Transformation
#3-1. How to Make 'Complete Euler Squares' 5x5(.pdf) [Revised]
#3-2. How to Make 'Complete Euler Squares' 7x7(.pdf) [Revised]
#4. The Newest Study of 'Prototype Squares' of Order 7
... and 'Do-it-After-the-Model Transformation'(.pdf) [Revised]
#5. Analytical Study of 'Complete Euler Squares' of Simultaneous
... and Pan-diagonal Type 5x5(.pdf)
#6. Analytical Study of 'Complete Euler Squares' of Order 5 & 7
... and Invention of a Powerful Method of Reconstruction(.pdf) [Revised]
#7. How to Compose 'Complete Euler Squares' 4x4 by Binary Number
... System(.pdf) [Revised]
#7-2. How to Compose The Other Types of Magic Squares 4x4 by Binary
... Number System(.pdf) [Revised]
#8. How to build 'Complete Euler Squares' 8x8 by Binary Number
... System (.pdf) [Revised]
#9. How to build Various 'Euler Cubes' of Order 4 by Binary Number System
... and "New Euler's Method"(.pdf)[Revised]
#9-2. Simultaneous MC444: Both S-C and P-T Made by Binary Number
... System and "New Euler's Method"(.pdf)[Revised]
#10. How to make 'Complete Euler Squares' 9x9 by 'New Euler's Method'
... Using Positional Number System of Base 3(.pdf)[Revised]
#11. Various 'Euler Squares' of Order 3, 5 and 7 by New Euler's Method
... Using Positional Number System of Base 3, 5 and 7(.pdf)[Revised]
#12. What does New Euler's Method Mean? What else can we Apply it for?(.pdf)[Retyped]
Chapter 5. Advanced Study of 'Prototype' Cubes
#1. Prototype Cubes 3x3x3 and Self-Complementary
... Magic Cubes 3x3x3 Recomposed by 'DAMT' (.pdf)[Revised]
#2. Prototype Cubes of Order 4 and 'Composite'
... Pan-Magic Cubes of Order 4 Recomposed by 'DAMT' (.pdf)[Revised]
Chapter 6. Fundamental Study of Multi-Dimensional Extra-Cubic
... Forms and their Decomposed Ones
#1. Study of Four-Dimensional Extra-Cubic Forms of Order 3
... and their Decomposed Ones(.pdf) [Revised]
#2. Study of Four-Dimensional Extra-Cubic Objects of Order 2
... and their Decomposed Forms(.pdf) [Revised]
#2-1. Transformation and 'Fundamental Solutions' of
... Self-Complementary Magic Squares 4x4(.pdf)
#2-2. Transformations and 'Fundamental Solutions' of
... Pandiagonal Magic Squares 4x4(.pdf)
#3. Study of Six-Dimensional Extra-Cubic Objects of Order 2
... and their Decomposed Forms(.pdf) [Revised]
#3-2. 'Fundamental Solutions' and Transformation of C&C MS88(.pdf)
#3-3. The Most Fundamental Solution and Transformation Process of
... 'Composite & Complete' Magic Cubes 4x4x4(.pdf); Data Sheet(.pdf)
#3-4. The Most Fundamental Solution and Transformation Process of
... 'Composite & Complete' Magic Squares 8x8(.pdf)
#3-5. Multiple Type of 'Composite' Pan-Diagonal Magic Squares 8x8
... and Its Transformation System(.pdf); Data Sheet(.pdf)
Chapter 7. New Method of Composing High-Dimensional Extra-Cubic
... Objects and their Developed Forms
#1. About Some Fundamentals of High-Dimensional Extra-Cubic Objects
... and their Developed Forms(.pdf) [Hot]
#2. Composition of Four-Dimensional Extra-Cubic Objects of Order 2
... and their Developed Forms(.pdf) [Hot]
#3. Composition of Six-Dimensional Extra-Cubic Objects of Order 2
... and their Developed Forms(.pdf) [Hot]
#4. Composition of 'Three-Type Simultaneous' Magic Squares 8^2
... by Down-Converting from the Extra-Cubic Object 2^6(.pdf) [Hot]
#5. New Method of Composing Two Types of Magic Squares 4^2
... Using All View-Forms of Extra-Cubic Object 2^4(.pdf) [Hot]
#6. New Method of Composing 'C&C' Magic Cubes 4^3
... and 2 Types of 'C&C' MS8^2 Using All View-Forms of ECO2^6(.pdf) [Hot]
#7. New Method of Composing 'Multiple' Type of Magic Squares 9x9
... Including 9 Little Squares 3x3 within(.pdf) [Hot]
#8. Internal Relationship among the Family of 'C&C' Magic Cubes 444
... and Two Types of Magic Squares 88(.pdf) [Hot]
Profile: Self-Introduction: Kanji Setsuda
.. I was born on June 27, 1940 in Tokyo, Japan, and have grown up there.
.. When I was a boy, I liked to construct radio sets and audio amplifiers by myself. In my college years I studied history of thought, philosophy and symbolic logics.
.. When I had my first 8-bit micro-computer board, I was already over 40.
.. I was eager to learn BASIC programming and began to solve some easy problems of magic stars, squares and cubes then. I tried to count every kind of solutions by my MC6800-09 machines.
.. I was taught some basic things of magic forms from the books written by Mr. Gakuho Abe, one of the greatest senior researchers in Japan.
.. But while I ran busy engaged in my business life, I had to keep myself out of studying those things for a long time.
.. When I was retired from business scene and restarted my research for magic squares and cubes, I was almost 60 years old. I decided to make myself concentrated on my intensive study, making effort to have free time as long as I ever could.
.. On my way I also decided to concentrate my research only to a few classical topics: (1) Pan-diagonal magic type and (2) Self-complementary one (especially with all complementary pairs located symmetrically with respect to the geometric center).
.. I wanted to know the common structures and correspondences between them.
.. I studied only those two types of magic squares of every order: 3, 4, ... , 9 and magic cubes of order 3 and 4. I always kept on my algebraic analysis. I calculated a lot to get the solution set of each with my personal computer.
.. I soon noticed that complementary pairs and their locations are always important. They determine the cores, or fates, of those two types.
.. I reported my first research result to Prof. Mutsumi Suzuki in 2000. I was lucky he was so generous to read and accept my report.
.. Prof. M. Suzuki was always kind enough to introduce me to his own page, and gave me much suggestion and inspiration. I must tell you of my deepest appreciation for his understanding and his greatest favor.
.. Next I noticed about how important the transformation method is. It works well for you not only to find fundamental solutions but also to find inner relation to another type of solutions. I tried to find any one-to-one correspondence between two different sets of solutions by a certain rule of transformation.
.. Inspired by the excellent discoveries of Drs. Ollerenshaw & Brae, British researchers, I extended their ideas up to the "Do-it-After-the-Model Transformation" method and the concept of Prototype Squares of every order, that are full of transformations and one-to-one correspondences.
.. At last I invented the way how to make higher dimensional extra-cubic magic forms of low orders, and how to down-convert them into lower dimensional magic ones of higher orders. I tested and tested this theory by my original experiments and examinations.
.. I discovered many interesting one-to-one correspondences between different types of magic forms, say, even between different dimensions.
.. These days I have been studying about '(Complete) Euler Squares' of orders 3, 4, 5, 7, 8, 9 and Cubes of order 3 and 4 by the "New Euler's Method" (applied even to all pan-diagonals) using Positional Number System of Base 2, 3, 5 and 7.
.. I have found how to compose them according to their definitions literally by such a unique but powerful method that I would call "Knight's Compositions".
.. Finally I have come to the total idea of "New Euler's Method".
.. I examined it carefully by applying it to as many cases as I could.
.. Our New Euler's Method proves to be powerful enough to make almost everything. This means almost all pan-magic squares and cubes have the beautiful structure in common, what we call '(Complete) Euler Squares'.
.. We are now able to talk about common structures and even discuss about some family-tree relations between various types of magic forms, I hope.
.. (October 17, 2003: Kanji Setsuda)
.. I was sick during a whole year from autumn 2003 to autumn 2004. I could hardly see, hardly smell and hardly taste, either. I could neither study magic squares, nor read any mails. I made up my mind accepting surgical operations in hospital and finally had those horrors on the head in spring and summer 2004.
.. I have now recovered about 70 percent of what I had before.
.. As I could become self-confident on my health in the end of 2004, I restarted my study. At first I tried to read my old articles to revise them. I also bought a new personal computer and tested some old programs of mine as carefully as I could.
.. When I was making new Basic Diagrams for every magic thing, I found a new, unique way of drawing all possible 'view-forms' for high dimensional extra-cubic objects. I was surprised at the fact we could draw different view-forms of that object as many as its solutions.
.. Then I invented a new, unique method how to compose the special types of 'Complete Euler squares and cubes' that are also developed forms of any high dimensional Extra-Cubic Object at the same time. It uses all possible view forms of ECO and makes them directly into the object solutions by our 'Do-it-After-the-Model Transformation'. It is really a short program, and it runs so fast, say, faster than any other programs.
.. We can make various important magic squares of order 4, 8 and 9, and cubes of order 4 by this method. Read all the articles in Chapter 7, please.
.. I am so excited that I can feel I have touched to a great 'inner relation' among everything: 'Positional number system of the base N', 'Complete Euler Squares and Cubes', 'Prototype Squares', 'DAM Transformation', and 'High dimensional Extra-Cubic Objects'.
.. Everything seems to point at the same thing.
.----------------------.
.---->| 'DAM Transformation' |>------.
| '----------------------' |
("Do-it-After-the-Model Transformation")
| |
.---------------------. .----------v----------.
| Set of Basic Forms | | Set of Object Forms |
| (Prototype Squares | | (Solution Set of |
|and Cubes of Order N,| | Various Types of |
| High-Dimensional | | S-C and P-D Magic |
| Extra-Cubic Objects,| | Squares and Cubes, |
| Complete Euler Type)| | Complete Euler Type)|
'---------------------' '---------------------'
< One-to-one Correspondence >
.. What I am thinking now is about the relation between the two solution sets of Basic Forms and Object ones... Suppose we extend our concept of Basic Forms up to the set containing the first basic one and its variations all along, and suppose it should have as many solutions as the ones of Object Forms. Then we may well classify those Basic Forms into the same group of our 'Prototype Forms' or even of our 'High dimensional Extra-cubic Objects', and may well recognize all those three sets as the same thing, I think.
.. If we can deal with those three sets as the same one, we may well re-use and apply one concept to another, one technique to all, and one invention to any other cases.
.. For instance, we may use and apply the 'Do-it-After-the-Model Transformation' method to the sets of any Basic Forms or any 'Extra-Cubic Objects' besides our Prototype forms. When we want to make all the solutions of Basic Forms, we may well use the technique of reversible conversion of DAM Transformation, or may well apply the composing method of ECO using those multiple loops of for (...) {...} sentences.
.. Now that we know of this great concordance, we have just become more powerful and free than we had ever been, I believe.
.. The amazing 'DAMT' seems getting more important, since it makes a magic bridge between the two parallel worlds: the set of Basic Forms and the set of Object ones. It makes the one-to-one correspondence possible between them.
.. (December 25, 2005: Kanji Setsuda)
(Captured on February 4, 2007)...