About Definitions: Kanji Setsuda
I am afraid I might be confusing you a little, about my way of strange naming and original style of definitions for magic objects. Let me explain a little about them here.
I named everything as I liked to. I think names are flowers come out of our own culture. Of course, I allow you put your own names to your flowers freely according to your cultural traditions. I accept them and enjoy their individualities.
I am often asked why I would call "Composite" for "Compact". It is because I like the name "Composite" best of all, since Prof. Mutsumi Suzuki gave it to me long ago.
Mr. Gakuho Abe, our Japanese Old Master, once told me he knew nothing about the English name "Compact" for "Soketsu-jin". It might have been a translator's decision, and they might not have had a good discussion about it before publishing.
I believe the way of definition must be clear, self-apparent and self-indicative of inner structure in itself. It must not be self-contradictory.
Since we want to study our objects mathematically and we now don't want to trace them as a history of human culture, we must define them mathematically with the basic simultaneous equations and basic diagrams, I suppose.
We must differentiate any definition from our 'concept' in our mind and express it concretely in methodological view-points. We must design them carefully in advance, explain our premise, and report of our method and result as honest as possible, I think.
Let me show you some examples of my style of definitions.
I myself define various types of Magic Square of order 8 like these.
(1) Pan-diagonal Magic Square of order 8:
It takes both of "Basic Conditions" and "Pan-diagonal Conditions" listed below. Every row, every column, and every pandiagonal adds up to the same sum, called "Magic Constant".
(2) Self-complementary Magic Square of order 8:
It takes both of "Basic Conditions" and "Self-Complementary Conditions". Any 'Complementary Pair' of 65 is always important in this type, and all pairs must be located symmetrically with respect to the geometric center. We call (M, N) as 'Complementary Pair' of 65, if M+N=65 is true.
(3) Complete Magic Square of order 8:
It takes both of "Basic Conditions" and "Complete Conditions". In this type, all 'Complementary Pairs' of 65 must be located only on pandiagonals just as listed below. Every pandiagonal always consists of 4 complementary pairs and the sum is calculated in the same manner as: 65+65+65+65=260(in Classical Notation)
Yes, as you see, it is a special type of Pandiagonal Magic Squares. Though in the case of order 4 both Pandiagonal type and Complete one have the same set of 48 solutions in common, in the case of order 8 each of them has its own solution set different from the others.
(4) Composite and Complete Magic Square of order 8:
It takes these three sets simultaneously: "Basic Conditions", "Composite Conditions" and "Complete Conditions".
As a result it becomes equivalent to the "Most-Perfect" MS88, that Drs. Ollerenshaw and Brae studied and discovered their beautiful laws about.
In the case of order 4 both 'C&C' magic square and Pandiagonal one have the same set of 48 solutions in common, while in the case of order 8 each of them has its own solution set different from the others.
(5) Composite and Pandiagonal Magic Square of order 8:
It takes the three sets simultaneously: "Basic Conditions", "Pan-diagonal Conditions" and "Composite Conditions'.
It has the larger set of solutions than the 'C&C' type above.
(6) Simultaneous MS88: Self-Complementary and Pandiagonal:
It takes these three sets of conditions: "Basic Conditions", "Pan-Diagonal Conditions" and "Self-Complementary Conditions".
I myself didn't know that such type in even order could exist, but Grand Harvey Heinz once told us about his discovery.
(7) Three-type Simultaneous MS88: Self-Complementary, Pan-Diagonal and Composite:
It takes these four sets: "Basic Conditions", "Self-Complementary Conditions", "Pan-Diagonal Conditions" and "Composite Conditions".
I found 5760 solutions of this type. Read this article.
(8) Complete Euler Square of order 8:
Read this article about this new type.
Design and definitions must be made in the decomposed forms by Binary Number System.
I confess I am afraid I might have mis-named to that. I suppose I must have named it just as "Pandiagonal Euler Square" in stead of 'Complete ...', since this type has nothing to do with "Complete Conditions". It might be confusing.
** Basic Diagram of MS88 with Various Conditions **
** Basic Conditions: C=260(in Classical Notation) **
n1+n2+n3+n4+n5+n6+n7+n8=C ... rw1;
n9+n10+n11+n12+n13+n14+n15+n16=C ... rw2;
n17+n18+n19+n20+n21+n22+n23+n24=C ... rw3;
n25+n26+n27+n28+n29+n30+n31+n32=C ... rw4;
n33+n34+n35+n36+n37+n38+n39+n40=C ... rw5;
n41+n42+n43+n44+n45+n46+n47+n48=C ... rw6;
n49+n50+n51+n52+n53+n54+n55+n56=C ... rw7;
n57+n58+n59+n60+n61+n62+n63+n64=C ... rw8;
n1+n9+n17+n25+n33+n41+n49+n57=C ... cl1;
n2+n10+n18+n26+n34+n42+n50+n58=C ... cl2;
n3+n11+n19+n27+n35+n43+n51+n59=C ... cl3;
n4+n12+n20+n28+n36+n44+n52+n60=C ... cl4;
n5+n13+n21+n29+n37+n45+n53+n61=C ... cl5;
n6+n14+n22+n30+n38+n46+n54+n62=C ... cl6;
n7+n15+n23+n31+n39+n47+n55+n63=C ... cl7;
n8+n16+n24+n32+n40+n48+n56+n64=C ... cl8;
** Basic Diagram of MS88 **
.-----------------------.
5 6 7 8| 1| 2| 3| 4| 5| 6| 7| 8| 1 2 3 4
|--+--+--+--+--+--+--+--|
13 14 15 16| 9|10|11|12|13|14|15|16| 9 10 11 12
|--+--+--+--+--+--+--+--|
21 22 23 24|17|18|19|20|21|22|23|24|17 18 19 20
|--+--+--+--+--+--+--+--|
29 30 31 32|25|26|27|28|29|30|31|32|25 26 27 28
|--+--+--+--+--+--+--+--|
37 38 39 40|33|34|35|36|37|38|39|40|33 34 35 36
|--+--+--+--+--+--+--+--|
45 46 47 48|41|42|43|44|45|46|47|48|41 42 43 44
|--+--+--+--+--+--+--+--|
53 54 55 56|49|50|51|52|53|54|55|56|49 50 51 52
|--+--+--+--+--+--+--+--|
61 62 63 64|57|58|59|60|61|62|63|64|57 58 59 60
'-----------------------'
** Pan-Diagonal Conditions: C=260(in Classical N.) **
n1+n10+n19+n28+n37+n46+n55+n64=C ...pd1;
n2+n11+n20+n29+n38+n47+n56+n57=C ...pd2;
n3+n12+n21+n30+n39+n48+n49+n58=C ...pd3;
n4+n13+n22+n31+n40+n41+n50+n59=C ...pd4;
n5+n14+n23+n32+n33+n42+n51+n60=C ...pd5;
n6+n15+n24+n25+n34+n43+n52+n61=C ...pd6;
n7+n16+n17+n26+n35+n44+n53+n62=C ...pd7;
n8+n9+n18+n27+n36+n45+n54+n63=C ...pd8;
n1+n16+n23+n30+n37+n44+n51+n58=C ...pb1;
n2+n9+n24+n31+n38+n45+n52+n59=C ...pb2;
n3+n10+n17+n32+n39+n46+n53+n60=C ...pb3;
n4+n11+n18+n25+n40+n47+n54+n61=C ...pb4;
n5+n12+n19+n26+n33+n48+n55+n62=C ...pb5;
n6+n13+n20+n27+n34+n41+n56+n63=C ...pb6;
n7+n14+n21+n28+n35+n42+n49+n64=C ...pb7;
n8+n15+n22+n29+n36+n43+n50+n57=C ...pb8;
** Self-Complementary Conditions: CCC=65(in Classical N.) **
n1+n64=CCC; n2+n63=CCC; n3+n62=CCC; n4+n61=CCC;
n5+n60=CCC; n6+n59=CCC; n7+n58=CCC; n8+n57=CCC;
n9+n56=CCC; n10+n55=CCC; n11+n54=CCC; n12+n53=CCC;
n13+n52=CCC; n14+n51=CCC; n15+n50=CCC; n16+n49=CCC;
n17+n48=CCC; n18+n47=CCC; n19+n46=CCC; n20+n45=CCC;
n21+n44=CCC; n22+n43=CCC; n23+n42=CCC; n24+n41=CCC;
n25+n40=CCC; n26+n39=CCC; n27+n38=CCC; n28+n37=CCC;
n29+n36=CCC; n30+n35=CCC; n31+n34=CCC; n32+n33=CCC;
n33+n32=CCC; n34+n31=CCC; ...
** Complete Conditions: CCC=65(in Classical Notation) **
n1+n37=CCC; n2+n38=CCC; n3+n39=CCC; n4+n40=CCC;
n5+n33=CCC; n6+n34=CCC; n7+n35=CCC; n8+n36=CCC;
n9+n45=CCC; n10+n46=CCC; n11+n47=CCC; n12+n48=CCC;
n13+n41=CCC; n14+n42=CCC; n15+n43=CCC; n16+n44=CCC;
n17+n53=CCC; n18+n54=CCC; n19+n55=CCC; n20+n56=CCC;
n21+n49=CCC; n22+n50=CCC; n23+n51=CCC; n24+n52=CCC;
n25+n61=CCC; n26+n62=CCC; n27+n63=CCC; n28+n64=CCC;
n29+n57=CCC; n30+n58=CCC; n31+n59=CCC; n32+n60=CCC;
n33+n5=CCC; n34+n6=CCC; ...
** Composite Conditions: CC=130(in Classical N.) **
n1+n2+n9+n10=CC; n2+n3+n10+n11=CC;
n3+n4+n11+n12=CC; n4+n5+n12+n13=CC;
n5+n6+n13+n14=CC; n6+n7+n14+n15=CC;
n7+n8+n15+n16=CC; n8+n1+n16+n9=CC;
n9+n10+n17+n18=CC; n10+n11+n18+n19=CC;
n11+n12+n19+n20=CC; n12+n13+n20+n21=CC;
n13+n14+n21+n22=CC; n14+n15+n22+n23=CC;
n15+n16+n23+n24=CC; n16+n9+n24+n17=CC;
n17+n18+n25+n26=CC; n18+n19+n26+n27=CC;
n19+n20+n27+n28=CC; n20+n21+n28+n29=CC;
n21+n22+n29+n30=CC; n22+n23+n30+n31=CC;
n23+n24+n31+n32=CC; n24+n17+n32+n25=CC;
n25+n26+n33+n34=CC; n26+n27+n34+n35=CC;
n27+n28+n35+n36=CC; n28+n29+n36+n37=CC;
n29+n30+n37+n38=CC; n30+n31+n38+n39=CC;
n31+n32+n39+n40=CC; n32+n25+n40+n33=CC;
.....
Pay your special attention about "Composite Conditions".
When you make 'Composite' MS88 with both "Basic Conditions" and "Composite Conditions", you cannot make any other than 'Composite and Pandiagonal' MS88. Yes. "Composite Conditions" with these three equations below could surely make 'Composite and Pandiagonal' type:
n1+n2+n3+n4+n5+n6+n7+n8=C ........... rw1;
n1+n9+n17+n25+n33+n41+n49+n57=C ..... cl1;
n1+n10+n19+n28+n37+n46+n55+n64=C .... pd1;
Therefore, the next two types of MS88 have the same solution set in common, and they are supposed to be equivalent:
(1) "Three-type Simultaneous MS88": Self-complementary, Pandiagonal and 'Composite'
(2) "Simultaneous MS88": Self-complementary and 'Composite'
Read this article of mine, please.
(On August 26, 2003, February 24, 2007 by Kanji Setsuda)
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