/* * File: Order7CDoc.txt * Author: S Harry White * Created: 2010-02-26 */ Order 7 Concentric Magic Squares -------------------------------- The numbers from 0 to 48 are partitioned into two sets: . one set of 25 values for the center order-5 square . one set of 24 values for the order-7 border Each of the order-5 sets is further 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 24 and the remaining values are placed in each set in complementary pairs. Thus, there are 24!/(12!12!) = 2,704,156 partitions of all the numbers, and 12!/(4!8!) = 495 partitions of each order-5 set. Concentric magic squares can be made for 2,669,946 of the partitions. The first and last 10 of these partitions and the number of order-5 border groups, order-7 border groups, and squares are shown below. There are 8 rotations of each center order-3 square and of each order-5 border. There are 3!3! = 36 permutations of each order-5 border group, and 5!5! = 14,400 permutations of each order-7 border group. There is zero or one order-3 center square for each order-5 partition. When there is an order-3 square, the number of order-7 squares is: 8 x 8 x 36 x 14400 x (order-5 border groups) x (order-7 border groups) center 0 1 2 3 4 5 6 7 8 9 14 22 24 26 34 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 15 16 17 18 19 20 21 23 25 27 28 29 30 31 32 33 35 36 37 38 order-5 border groups 4, order-7 border groups 4968, squares 659,305,267,200 center 0 1 2 3 4 5 6 7 8 9 14 23 24 25 34 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 15 16 17 18 19 20 21 22 26 27 28 29 30 31 32 33 35 36 37 38 order-5 border groups 2, order-7 border groups 4158, squares 275,904,921,600 center 0 1 2 3 4 5 6 7 8 9 15 21 24 27 33 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 16 17 18 19 20 22 23 25 26 28 29 30 31 32 34 35 36 37 38 order-5 border groups 4, order-7 border groups 4660, squares 618,430,464,000 center 0 1 2 3 4 5 6 7 8 9 15 22 24 26 33 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 16 17 18 19 20 21 23 25 27 28 29 30 31 32 34 35 36 37 38 order-5 border groups 8, order-7 border groups 4102, squares 1,088,756,121,600 center 0 1 2 3 4 5 6 7 8 9 15 23 24 25 33 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 16 17 18 19 20 21 22 26 27 28 29 30 31 32 34 35 36 37 38 order-5 border groups 4, order-7 border groups 4460, squares 591,888,384,000 center 0 1 2 3 4 5 6 7 8 9 16 20 24 28 32 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 15 17 18 19 21 22 23 25 26 27 29 30 31 33 34 35 36 37 38 order-5 border groups 4, order-7 border groups 4850, squares 643,645,440,000 center 0 1 2 3 4 5 6 7 8 9 16 21 24 27 32 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 15 17 18 19 20 22 23 25 26 28 29 30 31 33 34 35 36 37 38 order-5 border groups 2, order-7 border groups 3847, squares 255,268,454,400 center 0 1 2 3 4 5 6 7 8 9 16 22 24 26 32 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 15 17 18 19 20 21 23 25 27 28 29 30 31 33 34 35 36 37 38 order-5 border groups 2, order-7 border groups 4668, squares 309,746,073,600 center 0 1 2 3 4 5 6 7 8 9 16 23 24 25 32 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 15 17 18 19 20 21 22 26 27 28 29 30 31 33 34 35 36 37 38 order-5 border groups 4, order-7 border groups 3948, squares 523,940,659,200 center 0 1 2 3 4 5 6 7 8 9 17 20 24 28 31 39 40 41 42 43 44 45 46 47 48 border 10 11 12 13 14 15 16 18 19 21 22 23 25 26 27 29 30 32 33 34 35 36 37 38 order-5 border groups 4, order-7 border groups 4102, squares 544,378,060,800 . . . . . . . . . . center 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 29 30 31 32 33 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 20 28 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 296, order-7 border groups 977, squares 9,594,696,499,200 center 11 12 13 14 15 16 17 18 20 21 22 23 24 25 26 27 28 30 31 32 33 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 19 29 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 325, order-7 border groups 822, squares 8,863,395,840,000 center 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 18 30 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 273, order-7 border groups 788, squares 7,137,298,022,400 center 11 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 17 31 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 361, order-7 border groups 663, squares 7,940,826,316,800 center 11 12 13 14 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 33 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 16 32 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 327, order-7 border groups 616, squares 6,683,030,323,200 center 11 12 13 14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 15 33 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 311, order-7 border groups 499, squares 5,148,798,566,400 center 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 36 37 border 0 1 2 3 4 5 6 7 8 9 10 14 34 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 361, order-7 border groups 406, squares 4,862,708,121,600 center 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 border 0 1 2 3 4 5 6 7 8 9 10 13 35 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 435, order-7 border groups 252, squares 3,636,928,512,000 center 11 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 37 border 0 1 2 3 4 5 6 7 8 9 10 12 36 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 411, order-7 border groups 263, squares 3,586,266,316,800 center 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 border 0 1 2 3 4 5 6 7 8 9 10 11 37 38 39 40 41 42 43 44 45 46 47 48 order-5 border groups 605, order-7 border groups 185, squares 3,713,402,880,000 The total number of concentric squares is 3,835,791,613,181,952,000. The last partition of the order-5 set of the last partition makes the 61,378,560,000 consecutively concentric squares.