/*
 *  File:    Order5CDoc.txt
 *  Author:  S Harry White
 *  Created: 2009-07-19
 */


                        Order 5 Concentric Magic Squares
                        --------------------------------

The numbers from 1 to 25 are partitioned into two sets:

  . one set of  9 values for the center order-3 square
  . one set of 16 values for the order-5 border

The center value is always 13 and the remaining values are
placed in each set in complementary pairs.
Thus, there are 12!/(8!4!) = 495 partitions.

A center magic square can be made for 26 of the partitions.
These partitions and the number of borders and squares are:

center  1, 2, 3,12,13,14,23,24,25
border  4, 5, 6, 7, 8, 9,10,11,15,16,17,18,19,20,21,22
borders 1008, squares 8064

center  1, 3, 5,11,13,15,21,23,25
border  2, 4, 6, 7, 8, 9,10,12,14,16,17,18,19,20,22,24
borders 1116, squares 8928

center  1, 4, 7,10,13,16,19,22,25
border  2, 3, 5, 6, 8, 9,11,12,14,15,17,18,20,21,23,24
borders 648, squares 5184

center  1, 6, 8,11,13,15,18,20,25
border  2, 3, 4, 5, 7, 9,10,12,14,16,17,19,21,22,23,24
borders 756, squares 6048

center  2, 3, 4,12,13,14,22,23,24
border  1, 5, 6, 7, 8, 9,10,11,15,16,17,18,19,20,21,25
borders 1080, squares 8640

center  2, 4, 6,11,13,15,20,22,24
border  1, 3, 5, 7, 8, 9,10,12,14,16,17,18,19,21,23,25
borders 1080, squares 8640

center  2, 5, 8,10,13,16,18,21,24
border  1, 3, 4, 6, 7, 9,11,12,14,15,17,19,20,22,23,25
borders 936, squares 7488

center  2, 6, 9,10,13,16,17,20,24
border  1, 3, 4, 5, 7, 8,11,12,14,15,18,19,21,22,23,25
borders 1044, squares 8352

center  2, 7, 8,12,13,14,18,19,24
border  1, 3, 4, 5, 6, 9,10,11,15,16,17,20,21,22,23,25
borders 864, squares 6912

center  3, 4, 5,12,13,14,21,22,23
border  1, 2, 6, 7, 8, 9,10,11,15,16,17,18,19,20,24,25
borders 828, squares 6624

center  3, 5, 7,11,13,15,19,21,23
border  1, 2, 4, 6, 8, 9,10,12,14,16,17,18,20,22,24,25
borders 828, squares 6624

center  3, 6, 9,10,13,16,17,20,23
border  1, 2, 4, 5, 7, 8,11,12,14,15,18,19,21,22,24,25
borders 468, squares 3744

center  3, 7, 9,11,13,15,17,19,23
border  1, 2, 4, 5, 6, 8,10,12,14,16,18,20,21,22,24,25
borders 972, squares 7776

center  4, 5, 6,12,13,14,20,21,22
border  1, 2, 3, 7, 8, 9,10,11,15,16,17,18,19,23,24,25
borders 1080, squares 8640

center  4, 6, 8,11,13,15,18,20,22
border  1, 2, 3, 5, 7, 9,10,12,14,16,17,19,21,23,24,25
borders 828, squares 6624

center  4, 8, 9,12,13,14,17,18,22
border  1, 2, 3, 5, 6, 7,10,11,15,16,19,20,21,23,24,25
borders 1080, squares 8640

center  5, 6, 7,12,13,14,19,20,21
border  1, 2, 3, 4, 8, 9,10,11,15,16,17,18,22,23,24,25
borders 576, squares 4608

center  5, 7, 9,11,13,15,17,19,21
border  1, 2, 3, 4, 6, 8,10,12,14,16,18,20,22,23,24,25
borders 756, squares 6048

center  5, 8,10,11,13,15,16,18,21
border  1, 2, 3, 4, 6, 7, 9,12,14,17,19,20,22,23,24,25
borders 720, squares 5760

center  6, 7, 8,12,13,14,18,19,20
border  1, 2, 3, 4, 5, 9,10,11,15,16,17,21,22,23,24,25
borders 684, squares 5472

center  6, 8,10,11,13,15,16,18,20
border  1, 2, 3, 4, 5, 7, 9,12,14,17,19,21,22,23,24,25
borders 684, squares 5472

center  6, 9,10,12,13,14,16,17,20
border  1, 2, 3, 4, 5, 7, 8,11,15,18,19,21,22,23,24,25
borders 1008, squares 8064

center  7, 8, 9,12,13,14,17,18,19
border  1, 2, 3, 4, 5, 6,10,11,15,16,20,21,22,23,24,25
borders 684, squares 5472

center  8, 9,10,12,13,14,16,17,18
border  1, 2, 3, 4, 5, 6, 7,11,15,19,20,21,22,23,24,25
borders 1044, squares 8352

center  8,10,11,12,13,14,15,16,18
border  1, 2, 3, 4, 5, 6, 7, 9,17,19,20,21,22,23,24,25
borders 648, squares 5184

center  9,10,11,12,13,14,15,16,17
border  1, 2, 3, 4, 5, 6, 7, 8,18,19,20,21,22,23,24,25
borders 360, squares 2880

The total number of concentric squares is 174,240.

The last partition makes the 2880 consecutively concentric squares.