Reversible Squares


Reversible squares contain the numbers 1 to n2 and have these features:

Of course, these squares are not magic. However, the (pan)diagonals have the magic sum. And, like the associative magic squares, every pair of cells symmetrically opposite from the center is a complementary pair. So, a reversible square remains reversible not only under Transform 1 and Transform 2, but by swapping only the described rows or columns. From each even order reversible square, the total number of resulting squares is:


For odd order, the number is that for order n - 1.

For each of these groups there is one principal reversible square in which:


ReversibleSquares makes principal reversible squares. It uses a simple algorithm to generate all the formats and write the squares in Frénicle standard form in ascending order.

How Many

The numbers of principal reversible squares are given by the OEIS sequence A273013

For any prime number order, there is only one principal reversible square consisting of the sequential numbers.

Orders to 100

For doubly-even orders, there is a one-to-one correspondence between reversible squares and most-perfect magic squares.


Heinz, Harvey "Most-perfect Magic Squares"

Ollerenshaw, Kathleen and David S. Brée "Most-perfect Pandiagonal Magic Squares"

Stewart, Ian "Most-Perfect Magic Squares"