What's New? by Kanji Setsuda
These days I have been studying about various "Euler Squares" of order 3, 4, 5, 7, 8, 9 and Cubes of order 3 and 4 analyzing directly on their decomposed layers, and I have found several interesting things about them. At last I invented how to reconstruct those objects by "New Euler's Method" using Positional Number System of Base N.
1. At first I could write such a new style of program for "CES(Complete Euler Squares)" dictating their definitions literally, applying "Greco-Latinian Method" even to all pandiagonals, that I could build all solutions of the following 4 types of order 5 and 7 by this style of programming:
(1) Simultaneous 'CES' of order 5: 16 Solutions
(2) Pandiagonal 'CES' of order 5: 3600 Solutions
Please read this article of mine.
(3) Simultaneous 'CES' of order 7: 3456 Solutions
(4) Pandiagonal 'CES' of order 7: 38102400 Solutions
Please read this article of mine.
2. While I was making some structural analyses on them, I could invent such a new, simple but powerful method of reconstruction of those 4 types of 'CES' above as I would call "Composition by Knight's Tour":
(1) Make their layer units from one of the primary diagonals by the unique "Knight's Tour" movements.
(2) Combine them to compose a whole solution.
(3) Test your compositions under the Definition-3 of CES and some list-forming inequality conditions.
And I could rebuild all the solutions above. This new method ran so quick that it took only a few minutes even to count up the Pandiagonal types to the last.
Please read this article of mine.
3. I also found that we cannot compose any CES of order 4, 6, 8, 9 and 10 by the Positional Number System of Base 4, 6, 8, 9 and 10. It is shown by our failures in making their own valid layer units for those CES with our Knight's Tour Composition.
4. I invented how to compose CES of order 4 and 8 directly with their definitions and by Binary Number System.
(1) Make their own Layer Units only with '0' and '1' according to your definitions and careful design.
(2) Choose 4/6 units to combine and make them act as the layers of binary decompositions to compose each solution.
(3) Check whether your combination is appropriate or not, by consulting with your reference table to know how similar any two units are.
(4) Calculate and Compose each solution and test it whether it is against the definition-3 of CES or not.
(5) Before listing out your answers, select the standard solutions by the list-forming inequality conditions.
Please read this article and that one of mine.
By this way I could make various types of Pan-diagonal magic squares of order 4 and 8. I would like to call this way as "New Euler's Method". I have succeeded in counting up every 'Composite and Pan-diagonal' magic square of order 8 through.
I also verified the existence of Simultaneous MS88: Self-complementary and Pan-diagonal (Grand Harvey Heinz discovered).
Finally I have known about the existence of 'Three-type Simultaneous MS88: Self-complementary, Pan-diagonal and Composite'. I found 5760 solutions of them.
Please read this article of mine.
.... ** Diagram for Various Sets of Solutions **
.---------------------------------.
| [Self-Complementary MS88] |
.---+---------------------------------+-[Pan-diagonal MS88]-.
| | | |
| .-+-[Complete Euler Squares 8x8]----+---------------. |
| | | | | |
| | | .---------------+-[Composite--. | |
| | | [Simultaneous | [Three-type |& Pandiagonal| | |
| | | S-C.& P-D. | Simultaneous | MS88] | | |
| | | MS88] | MS88] 5760 | | | |
| | '-----------------+---------------' | | |
| | | | | |
| | .-[Complete MS88]-+----------. | | |
| | | |[Composite| [Composite | | |
| | | |& Complete| & Pandiagonal | | |
| | | | MS88] | MS88] | | |
| | | | 368640 | 119064960 | | |
| | | '----------+---[119439360]----' | |
| | | | | |
| '-+-------------------[Complete Euler Squares 8x8]--' |
| | | |
| '------[Complete MS88]-------' |
| |
'-----------------------------------------------------------'
5. I found that 'Composite and Pandiagonal' magic squares of order 8 are not equivalent to 'Complete Euler Squares'. The concept of CES proves to be far broader and more potential than what I have ever had in my mind. It is powerful enough to make every type of panmagic squares of order 8.
6. I examined this "New Euler's Method" by making various types of 3-dimensional magic cubes of order 4 using Binary Number System.
I have succeeded in remaking all the objects we know very well. I am so happy that I could really count up CES type of Complete MC444 through. The count of solutions with N1=1 type is 133856760.
Please read this article of mine.
I could also certify and verify the existence of Simultaneous MC444 both Self-Complementary and Pan-triagonal.
Please read this article.
7. I also examined the New Euler's Method by making various types of magic squares of order 9 using Positional Number System of Base 3.
I could build Simultaneous Magic Squares of order 9 both S-C and P-D including Multiple 3x3 mini-squares within. I found 22272 solutions of Complete Euler Squares of that type.
I could also find the most rare solution set of 48 Multiple 3x3 type of Self-Complementary MS99 developed from the 4-Dimensional Extra Cubic Magic Forms of order 3.
Please read this article of mine.
** Magic Square of Order 9 Composed by "New Euler's Method" **
2/ 1/ 3/ 4/ 1#
021210102 012201120 021102210 021210102 1 72 50 78 29 16 44 22 57
102021210 201120012 210021102 210102021 54 5 64 11 79 33 58 39 26
210102021 120012201 102210021 102021210 68 46 9 34 15 74 21 62 40
210102021 201120012 210021102 102021210 80 31 12 37 27 59 6 65 52
021210102 120012201 102210021 021210102 13 75 35 63 41 19 47 7 69
102021210 012201120 021102210 210102021 30 17 76 23 55 45 70 51 2
102021210 120012201 102210021 210102021 42 20 61 8 67 48 73 36 14
210102021 012201120 021102210 102021210 56 43 24 49 3 71 18 77 28
021210102 201120012 210021102 021210102 25 60 38 66 53 4 32 10 81
8. I also examined the New Euler's Method by making various types of magic squares of order 3, 5 and 7 using Positional Number System of Base 3, 5 and 7. I found we can make even "Non-Complete Type" of Euler Squares, without any definitions about pan-diagonals or primary diagonals of them.
I verified our old "Knight's Tour Compositions" for order 5 and 7 by our new "New Euler's Method" using PNS of Base 5 and 7.
Please read this article of mine.
9. I wonder what is essentially meant by our New Euler's Method.
The composition/decomposition by PNS of Base N combines the Layer Units and the combinations of them with the complete set of object solutions, and puts the one-to-one correspondence between them.
If you only have the layer units and number list of used units, you can always reconstruct the whole set of object solutions you have wanted by our New Euler's Method.
I tried to invent some tiny simple applications of this method.
You can read my discussion and test report in this article.
... On November 17, 2003 by Kanji Setsuda: jag12001@nifty.com