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--- defintion_of_subsets [2023/03/03 11:52] – [The representative line of a magic square] mino
+++ defintion_of_subsets [2024/09/07 11:58] (current) – external edit 127.0.0.1
@@ Line 3: / Line 3: @@
 We introduce some terminologies here to divide the whole set of magic squares into small subsets.
-==== A magic set and its binary representation ====
+==== A magic series and its binary representation ====
-  * We define **a magic set of order //n//** as a set of //n// distinct integers in the range //[1..n<sup>2</sup>]// whose sum is equal to the magic sum //( ( n<sup>2</sup> + 1 ) * n ) / 2//.
+  * **A magic series of order //n//** is a set of //n// distinct integers in the range //[1..n<sup>2</sup>]// whose sum is equal to the magic sum //( ( n<sup>2</sup> + 1 ) * n ) / 2//.
     * Examples:
-      * { 2, 9, 4 } is a magic set of order 3.
+      * { 2, 9, 4 } is a magic series of order 3.
-      * { 10, 7, 14, 3 } is a magic set of order 4.
+      * { 10, 7, 14, 3 } is a magic series of order 4.
   * Any set of distinct positive integers { a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub> ...  } can be represented as an integer whose value is equal to 2<sup>a<sub>1</sub>-1</sup> + 2<sup>a<sub>2</sub>-1</sup> + 2<sup>a<sub>3</sub>-1</sup> + ... . We call it the binary representation of a distinct integer set.
     * Examples:
@@ Line 16: / Line 16: @@
     * We can define order on sets of distinct integers in accord with the order of their binary representation.
     * Example:
-      * A magic set { 5, 16, 2, 11 } is larger than { 12, 1, 15, 6 } because their binary representations are 1000 0100 0001 0010 ( = 0x8412 ) and 0100 1000 0010 0001 ( = 0x4821 ), respectively, and 0x8412 > 0x4821.
+      * A magic series { 5, 16, 2, 11 } is greater than { 12, 1, 15, 6 } because their binary representations are 1000 0100 0001 0010 ( = 0x8412 ) and 0100 1000 0010 0001 ( = 0x4821 ), respectively, and 0x8412 > 0x4821.
-==== Complement of a magic set ====
+==== Complement of a magic series ====
-If you replace each element //a// of a magic set by // n<sup>2</sup> + 1 - a //, the result is also a magic set. We call the resulting set the **compliment** of the original magic set.
+If you replace each element //x// of a magic series by // n<sup>2</sup> + 1 - x //, the result is also a magic series. We call the resulting set the **complement** of the original magic series.
   * Examples in the case of order 4:
-    * { 12, 1, 15, 6 } is the compliment of { 5, 16, 2, 11 }.
+    * { 12, 1, 15, 6 } is the complement of { 5, 16, 2, 11 }.
-    * { 7, 10, 3, 14 } is the compliment of itself.
+    * { 7, 10, 3, 14 } is the complement of itself.
-In binary representations, the compliment of a magic set is obtained by the bit reverse manipulation.
+In binary representations, the complement of a magic series is obtained by the bit reverse manipulation.
   * Example in the case of order 4:
-    * The compliment of 1000 0100 0001 0010 is 0100 1000 0010 0001.
+    * The complement of 1000 0100 0001 0010 is 0100 1000 0010 0001.
-==== The representative line of a magic square ====
+==== The representative magic series of a magic square ====
-Every row and column of a magic square always forms a magic set. We define the ** representative line ** of a magic square as **the largest magic set which forms a row, a column, compliment of a row, or compliment of a column** and classify magic squares by their representative lines. This classification is invariant under rotations, reflections, M-transformations, and the compliment transformation. Note that diagonal or counter-diagonal magic set is not considered a representative line.
+Every row and column of a magic square is always a magic series. We define the ** representative magic series ** of a magic square as **the largest magic series which forms a row, a column, the complement of a row, or the complement of a column**. Note that diagonal magic series are not considered a representative magic series.
 Example:
+The representative magic series of a magic square
+12  1  5
+  3 14 10
+13  4  8
+  6 15 11
+is { 16, 13, 3, 2 } = 0x9006, which forms the complement of the third column.
+**We classify magic squares by their representative magic series. This classification is invariant under rotations, reflections, [[https://oeis.org/A266237|M-transformations]], and the complement transformation.
+**