# Reversible Squares

## Description

Reversible squares contain the numbers 1 to n^{2} and have these features:

- for the square itself and for each contained square and rectangle, one
pair of diagonally opposite corner numbers has the same sum as the other
pair of diagonally opposite corner numbers
- in each row and column, all symmetrically opposite pairs have the same sum;
for odd order squares, the row or column middle number is half that sum

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:

**2**^{n-2}(n/2)!^{2}

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

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

- the top row begins with the numbers 1 and 2
- row numbers are in ascending order left to right
- column numbers are in ascending order top to bottom

## Construction

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.

##### REFERENCES

Heinz, Harvey "Most-perfect Magic Squares"
http://recmath.org/Magic%20Squares/most-perfect.htm

Ollerenshaw, Kathleen and David S. Brée
"Most-perfect Pandiagonal Magic Squares"
http://www.agnesscott.edu/lriddle/women/abstracts/ollerenshaw_mostperfect.htm

Stewart, Ian "Most-Perfect Magic Squares"
http://www.klassikpoez.narod.ru/mk/122-123.pdf