===== Definition of subsets =====

We introduce some terminologies here to divide the whole set of magic squares into small subsets.

==== A magic series and its binary representation ====
  * **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 series of order 3.
      * { 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:
      * { 2, 9, 4 } is represented as 1 0000 1001<sub>2</sub> = 0x10b.
      * { 10, 7, 14, 3 } is represented as 0010 0010 0100 0100<sub>2</sub> = 0x2244.

==== Order on distinct integer sets ====
    * We can define order on sets of distinct integers in accord with the order of their binary representation.
    * Example:
      * 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 series ====

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 complement of { 5, 16, 2, 11 }.
    * { 7, 10, 3, 14 } is the complement of itself.

In binary representations, the complement of a magic series is obtained by the bit reverse manipulation.

  * Example in the case of order 4:
    * The complement of 1000 0100 0001 0010 is 0100 1000 0010 0001.

==== The representative magic series of a magic square ====

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

    16 12  1  5
     7  3 14 10
     9 13  4  8
     2  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.
**