.... Kanji Setsuda's New English Page
I am now come back again to this page! But I am very sorry for my long absence.
I have suffered from my various break-downs:
I was shocked by the horrible earthquake in the eastern part of Japan in spring, 2011.
My PC broke down. And my heart also broke down as much.
I have lost half of my eye-sight now because of my age.
I had been in long mourning for my late family in 2009-11.
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.
I wrote about my new study of various types of 'Composite' Pan-magic Squares. Please, read my new articles in Chapter 9, Part 3.
I have written about 'Complete Euler Squares' of order 11 and 13 for the Simultaneous Type: Self-complementary and Pan-diagonal. Please, read my new articles in Section 13, Chapter 4, Part 4 and Section 14, Chapter 4, Part 4.
I have written about 'Complete Euler Cubes' of order 5 for the Self-complementary type, Pan-triagonal type and Simultaneous type. Please read my new articles in Section 15, Chapter 4, Part 4.
The newer articles in 2012 report about my recent experiments:
I reconstructed all the solutions of typical magic squares of order 4 only by the 4-th Increment Number System and New Euler's Method, without using 'Greco-Latinian Method'. Please read the article in the Section 7-4, Chapter 4, Part 4.
I reconstructed all the 48544 solutions of Self-complementary MS55 by the 5-th Increment System and by our 'New Euler's Method'. Please read the article in the Section 11-2, Chapter 4, Part 4.
I have also reconstructed two solution sets of Pan-magic Cubes of Order 4 only by the 4-th Increment System and New Euler's Method. Please read the recent article of mine in the Section 9-3, Chapter 4, Part 4.
I have recently composed the 150552 solutions of "Euler Type" of Simultaneous MS77: Self-complementary and Pan-diagonal; by the 7-th Increment System and New Euler's Method. Please read the newest report of mine in the Section 11-3, Chapter 4, Part 4.
I have also revised the Section 11 into the newest version according to the original ideas I have recently got.
I have now revised the Section 8 into the newest version according to the results of my recent experiments. Read the following new article in the Section 8-2, please.
The hottest articles in 2013 report about my newest experiments:
I reconstructed 2 types of magic squares of order 4 using the 'selected' Composite Conditions of 2x2 Little Units. Please read the article in the Section 5, Chapter 2, Part 3.
I have also challenged to classify the Standard MS44 by the 'Composite Conditions' of 2x2 Little Squares within. Please read the Section 6, Chapter 2, Part 3.
I recently made the 3 types of 'Semi-Composite' MS66 using the 'selected' Composite Conditions of 2x2 Little Units. Please read the Section 6, Chapter 4, Part 3.
I also made the 'Semi-Composite' Type of Magic Squares of Order 10 using the 'selected' Composite Conditions. Please read the report in the Section 7, Chapter 9, Part 3.
In 2014 I revised the Part 3: Chapter 2: Section 4 into the newest version according to the results of my new ideas and recent experiments. And I wrote the following new article in the Section 4-2. Read them, please.
In 2015 I also revised the Part 1: Chapter 4: Section 4 into the newest version and wrote the following new articles in the Section 4-2, 4-3 and 4-4, according to the recent results of my study and new experiments. Will you read the 'Composite' Pan-magic Cubes of Order 4: Revised and the following new articles: Composite PMC444 and the Meaning of Complementary Pairs, Study on the Unique Structures of 'C&P' MC4, and Seven Subsets of Standard Solutions of 'C&P' MC4, please.
I revised all the old articles of Part 3: Chapter 3: Study of Magic Squares of Order 5; into the newest versions. Please, read them.
I have recently put the newest article of mine to Part 4: Chapter 5: Section3: 'Composite & Pan-triagonal' Magic Cubes of Order 4 Reconstructed by Prototype Cubes and 'DAMT' Again Please, read it, will you?
(on September 20, 2015: Kanji Setsuda: E-Mail Address: jag12001@nifty.com)
"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 more than 70.
Part 1. Basic Studies 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. Basic Study of Magic Squares (.pdf) [New]
#1. Standard Magic Squares of Order 3
#2. Standard Magic Squares of Order 4
Chapter 3. Algebraic Study of Magic Squares and Cubes (.pdf) [New]
#1. Standard Magic Squares of Order 5
#2. Standard Magic Cubes of Order 3
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]
#4-2. 'Composite' PMC444 and the Meaning of Complementary Pairs (.pdf) [Hotter]
#4-3. Study on the Unique Structures of 'Composite & Pantriagonal' MC444 (.pdf) [Hottest]
#4-4. Seven Sub-Sets of Standard Solutions of 'Composite & Pantriagonal' Magic Cubes of Order 4 (.pdf) [Hottest]
Part 2. Basic Study of Magic Stars
Basic Studies of Magic Stars I (.pdf) [New]
Chapter 1. Introduction: What are Magic Stars?
Chapter 2. Algebraic Study of Double-Ringed Magic Stars 5
Chapter 3. Algebraic Study of Double-Ringed Magic Stars 6
Chapter 4. Algebraic Study of Double-Ringed Magic Stars 7
Basic Studies of Magic Stars II (.pdf) [New]
Chapter 5. Algebraic Study of Double-Ringed Magic Stars 8
Chapter 6. Algebraic Study of Magic Stars of Higher Orders
...#1. Data for Double-Ringed Magic Stars 9
...#2. Data for Double-Ringed Magic Stars 10
...#3. Data for Double-Ringed Magic Stars 11
...#4. Data for Double-Ringed Magic Stars 12
Basic Studies of Magic Stars III (.pdf) [New]
Chapter 7. Special Type of Triple-Ringed Magic Stars 7
Chapter 8. Special Type of Triple-Ringed Magic Stars 8
Chapter 9. Special Type of Triple-Ringed Magic Stars 9
Basic Studies of Magic Stars IV (.pdf) [New]<
Chapter 10. Advanced Study of Some Double-Ringed Magic Stars
...#1. Special Type of Double-Ringed Magic Stars 7
...#2. Special Type of Double-Ringed Magic Stars 9
...#3. Special Type of Double-Ringed Magic Stars 11
...#4. Special Type of Double-Ringed Magic Stars 12
...#5. Special Type of Double-Ringed Magic Stars 13
Part 3. New Algebraic Study of Magic Squares
Chapter 1. New Algebraic Study of Magic Squares of Order 3(.pdf) [New]
#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. New Study of 'Composite' Squares of Order 4(.pdf) [Revised]
#4. 'Composite and Complete' Pan-magic Squares 4x4
... and the Meaning of Complementary Pairs(.pdf) [Revised Edition]
#4-2. 'Self-Complementary' Magic Squares 4x4
... and the Meaning of 'Complementary Pairs'(.pdf) [Hottest]
#5. Make 2 Types of Magic Squares of Order 4
... by the Selected ‘Composite Conditions’ of 2x2 Little Squares(.pdf) [Newest]
#6. Classification of Standard Magic Squares of Order 4
... by the ‘Composite Conditions’ of 2x2 Little Squares within(.pdf) [Newest]
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 the 'Classical Composition' of Order 5
... 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) [New]
#5. About 'Classical Composition' of Magic Squares 6^2 (.pdf) [New]
#6. 'Semi-Composite' Type of Magic Squares 6x6 (.pdf) [Newest]
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) [New]
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' Squares
#1. Study of 'Composite' Squares of Order 6(.pdf)[Revised]
#2. Composite and Pan-Magic Squares of Order 8(.pdf)[Revised]
... (Newer Study of Fundamental Properties)
#2-2. Discover Correspondence between Two Types of 'Composite' Pan-magic Squares of Order 8(.pdf)[New]
#3. Three-Type Simultaneous Magic Squares of Order 12: 'Composite', Self-complementary and Pan-diagonal(.pdf)[New]
#3-2. 'Composite and Complete' Pan-magic Squares of Order 12(.pdf)[New]
#3-3. Discover Correspondence between 2 types of 'Composite' Pan-magic Squares of Order 12(.pdf)[New]
#4. Two Types of 'Composite' Pan-magic Squares of Order 16(.pdf)[New]
#4-2. Correspondence between the Two Types of 'Composite' PMS16: (1) 3-T Simultaneous Type and (2) Multiple 'C&C' PMS16(.pdf)[New]
#5. 'Diamond Composite' Pan-diagonal Squares of Order 6(.pdf)[New]
#6. 'Diamond Composite' Pan-diagonal Squares of Order 10(.pdf)[New]
#7. 'Semi-Composite' Type of Magic Squares 10x10(.pdf)[Hottest]
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]
#2-2: About Fundamental Solutions of Object Squares, Prototype Squares 4^2 and 'DAM Transformation'(.pdf) [New]
#2-3: Prototype Squares of Order 4 for the Standard Magic Type and its DAM Transformation(.pdf) [Newer]
#3. Prototype Squares of Order 5 and Reconstruction of Three Types of Magic Squares
... of Order 5 by DAMT (.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) [New]
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) [New]
#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) [Revised]
#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]
#7-3. Standard Magic Squares of Order 4 and 'Euler Squares'(.pdf) [Revised]
#7-4. Compose All the Solutions of 3 Types of Magic Squares of Order 4 only by the 4-th Increment Number System (.pdf) [Newest Revision]
#8. How to build 'Complete Euler Squares' 8x8 by Binary Number System [Revised](.pdf) [Newest Revision]
#8-2. How to build various 'Euler Squares' 8x8 by the N-th Increment Number System(.pdf) [Newest]
#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]
#9-3. Reconstruct Two Types of Magic Cubes 4x4x4 by the 4-th Increment Number System and N.E.M.(.pdf)[Newest]
#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)[Newest Revision]
#11-A: Appendix to the Section 11: Program Sheets for Order 5 and 7 (.pdf) [Newest]
#11-1: 'Complete Euler Squares' of Order 7 for the Simultaneous Type
...and the "Classic Composition"(.pdf)[Newer]
#11-2: Compose Self-complementary MS55 by the 5-th Increment Number System and our "New Euler's Method"(.pdf)[Newest]
#11-3: Compose "Euler Type" of Simultaneous MS77: Both Self-complementary and Pan-diagonal; by the 7-th Increment Number System and New Euler's Method(.pdf)[Newest]
#12. What does New Euler's Method Mean? What else can we Apply it for?(.pdf)[Retyped]
#13. Make 'Complete Euler Squares' of Order 11 for the Simultaneous Type: Self-complementary and Pan-diagonal(.pdf)[New]
#14. Make 'Complete Euler Squares' of Order 13 for the Simultaneous Type: Self-complementary and Pan-diagonal(.pdf)[New]
#15. Make 'Complete Euler Cubes' of Order 5
#15-1. Make 'Complete Euler Cubes' of Order 5 for the Self-complementary Type(.pdf)[New]
#15-2. Make 'Complete Euler Cubes' of Order 5 for the Pan-triagonal Type(.pdf)[New]
#15-3. Make 'Complete Euler Cubes' of Order 5 for the Simultaneous Type: Both Self-Complementary and Pan-triagonal(.pdf)[New]
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]
#3. 'Composite & Pan-triagonal' Magic Cubes of Order 4
... Reconstructed by Prototype Cubes and 'DAMT' Again (.pdf)[Newest]
Chapter 6. Fundamental Studies of Higher-Dimensional Extra-Cubic
... Magic Objects and their Decompositions
#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) [Revised]
#3-3. Reconstruction of Standard 120 of 'Composite & Complete' MC444 from
... the Fundamental 15 Solutions by the new Transformation System(.pdf) [New]
#3-4. The Most Fundamental Solution and Transformation System of
... 'C&C' MC444 and MS88(.pdf) [Revised]
#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) [New]
#2. Composition of Four-Dimensional Extra-Cubic Objects of Order 2
... and their Developed Forms(.pdf) [New]
#3. Composition of Six-Dimensional Extra-Cubic Objects of Order 2
... and their Developed Forms(.pdf) [New]
#4. Composition of 'Three-Type Simultaneous' Magic Squares 8^2
... by Down-Converting from the Extra-Cubic Object 2^6(.pdf) [New]
#5. New Method of Composing Two Types of Magic Squares 4^2
... Using All View-Forms of Extra-Cubic Object 2^4(.pdf) [New]
#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) [New]
#6-1. Another Experiment of Composing 'C&C' MC444
... by the Same Method using Developed ECO2^6(.pdf) [Newest]
#6-2. New Method of Making 6720 'C&P' Magic Cubes 4^3
... with All View-Forms of ECO2^6 and Original Transfromation System(.pdf) [New]
#6-3. New Method of Making Standard 'C&C' Magic Squares 8^2
... with All View-Forms of ECO2^6 and Additional Transfromation System(.pdf) [New]
#7. New Method of Composing 'Multiple' Type of Magic Squares 9x9
... Including 9 Little Squares 3x3 within(.pdf) [New]
#8. Internal Relationship among the Family of 'C&C' Magic Cubes 444
... and Two Types of Magic Squares 88(.pdf) [New]
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 Tour 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).