How many distinct consecutively concentric magic squares of each order are there?
Computed numbers for order n up to 14 are:


^{*} A border group is a configuration of the border numbers as corners, rest of row, and rest of column. For order 5, the bones numbers of these groups are:
Top Left Top Right Top Row Right Column Corner Corner Without Ends Without Ends     12 10 11 5 8 6 7 9 12 10 11 6 7 5 8 9 11 5 9 7 8 6 10 12 10 6 11 7 8 5 9 12 10 6 9 5 8 7 11 12 9 7 12 6 8 5 10 11 9 7 11 5 8 6 10 12 8 6 12 5 9 7 10 11 8 6 10 5 7 9 11 12 7 5 12 6 8 9 10 11
The corner complementary pairs are at opposite ends of the diagonals. The rest of row complementary pairs are at opposite ends of the columns. The rest of column complementary pairs are at opposite ends of the rows.
The number of distinct consecutively concentric squares of order n is:
number(n) = number(n2) × r × c × p × p
where: c = computed number of border groups p = permutations of 'row or column without ends' = (n2)! r = rotations and reflections = 8, (except = 1 for order 1) For example:
number(3) = 1 × 1 × 1 × 1 × 1 = 1 number(5) = 1 × 8 × 10 × 6 × 6 = 2,880 number(6) = 880 × 8 × 140 × 24 × 24 = 567,705,600 number(7) = 2,880 × 8 × 185 × 120 × 120 = 6.14 × 10^{10}
When (n2) is the smallest center square order, i.e., 1 or 4, number(n2) is the number of distinct magic squares; otherwise, number(n2) is the number of distinct consecutively concentric magic squares.
If the constraint that the nested squares consist of consecutive numbers is removed, there are many more concentric magic squares.
For order 5, the computed number of distinct concentric magic squares, (including the consecutively concentric squares), is 174,240.
For order 6, the number is 736,347,893,760, i.e., 7.36 × 10^{11}. Francis Gaspalou has confirmed this number, (July 2009). Miguel Angel Amela also obtained this number, (April 2016).
For order 7, the number is 3,835,791,613,181,952,000, i.e., 3.84 × 10^{18}.
Amela, Miguel Angel
miguel.amel@gmail.com
General Pico  La Pampa  Argentina