The number of magic squares of order 6

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

• We define a magic set of order n as a set of n distinct integers in the range [1..n²] whose sum is equal to the magic sum ( ( n² + 1 ) * n ) / 2.
  • Examples:
    • { 2, 9, 4 } is a magic set of order 3.
    • { 10, 7, 14, 3 } is a magic set of order 4.
• Any set of distinct positive integers { a₁, a₂, a₃ … } can be represented as an integer whose value is equal to 2^a₁-1 + 2^a₂-1 + 2^a₃-1 + … . We call it the binary representation of a distinct integer set.
  • Examples:
    • { 2, 9, 4 } is represented as 1 0000 1001₂ = 0x10b.
    • { 10, 7, 14, 3 } is represented as 0010 0010 0100 0100₂ = 0x2244.

• 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.

If you replace each element a of a magic set by n² + 1 - a , the result is also a magic set. We call the resulting set the compliment of the original magic set.

• Examples in the case of order 4:
  • { 12, 1, 15, 6 } is the compliment of { 5, 16, 2, 11 }.
  • { 7, 10, 3, 14 } is the compliment of itself.

In binary representations, the compliment of a magic set 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.

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, and the compliment transformation. Note that diagonal or counter-diagonal magic set is not considered the representative line.

Example: