### Explanation most perfect (Franklin pan)magic 8x8 square

The most perfect (Franklin pan)magic 8x8 square consist of 4 proportional 4x4 panmagic squares.

4x4 panmagic square                                                4x4 sub-square of 8x8

 1 8 13 12 1 54 12 63 15 10 3 6 16 59 5 50 4 5 16 9 53 2 64 11 14 11 2 7 60 15 49 6

In both squares the sum of two numbers of a colour always totals to the lowest plus the highest number of the magic square (1+16=17 respectively 1+64=65). With each time two colours you can get all eight (pan)diagonals (see 4x4 magic square, explanation).

Look below at the patterns of the panmagic 4x4 square and the most perfect 8x8 square.

 1 8 13 12 1 8 13 12 15 10 3 6 15 10 3 6 4 5 16 9 4 5 16 9 14 11 2 7 14 11 2 7

9 + 25 = 34                                                           16 + 18 = 34

 1 54 12 63 3 56 10 61 1 54 12 63 3 56 10 61 16 59 5 50 14 57 7 52 16 59 5 50 14 57 7 52 53 2 64 11 55 4 62 9 53 2 64 11 55 4 62 9 60 15 49 6 58 13 51 8 60 15 49 6 58 13 51 8 17 38 28 47 19 40 26 45 17 38 28 47 19 40 26 45 32 43 21 34 30 41 23 36 32 43 21 34 30 41 23 36 37 18 48 27 39 20 46 25 37 18 48 27 39 20 46 25 44 31 33 22 42 29 35 24 44 31 33 22 42 29 35 24

55 + 75 + 59 + 71 = 130 + 130 = 260                  17 + 113 + 49 + 91 = 130 + 130 = 260

Because of the structure the sum of the numbers of each 1/2 row/column/(pan)diagonal and of each 2x2 sub-square is allways (half of the magic sum: 1/2 x 260 =) 130.

There are the 3 following swap possibilities:

[1th] You can swap row 1&3 and/or row 2&4 and/or row 5&7 and/or row 6&8 and/or column 1&3 and/or column 2&4 and/or column 5&7 and/or column 6&8.

[2nd] You can swap the upper half with the down half and/or the right half with the left half.

[3rd] You can swap row 1&2 and row 3&4 and row 5&6 and row 7&8 and/or column 1&2 and column 3&4 and column 5&6 and column 7&8.

If you combine the 3 swap possibilities you can get each number out of 1 up to 64 in the top left corner. Try it!!!

From Willem Barink we learn that a small part (1/64) of the most perfect (8x8) magic squares has an extra magic feature. See the following most perfect magic 8x8 square:

 1 32 43 54 9 24 35 62 1 32 43 54 9 24 35 62 60 37 18 15 52 45 26 7 60 37 18 15 52 45 26 7 22 11 64 33 30 3 56 41 22 11 64 33 30 3 56 41 47 50 5 28 39 58 13 20 47 50 5 28 39 58 13 20 17 16 59 38 25 8 51 46 17 16 59 38 25 8 51 46 44 53 2 31 36 61 10 23 44 53 2 31 36 61 10 23 6 27 48 49 14 19 40 57 6 27 48 49 14 19 40 57 63 34 21 12 55 42 29 4 63 34 21 12 55 42 29 4

33 + 97 = 130                                                61 + 69 = 130

The extra feature is that in each row and each column (not only ) adding the numbers from position (1 up to 4 and 5 up to 8, but also from) 3 up to 6 gives the magic sum of 130.

Most perfect on this website is different from most perfect on other websites. In my opinion the 8x8 Franklin panmagic square is most perfect. But according to Kathleen Ollerenshaw, the famous mathematician, the 8x8 complete magic square is most perfect. See below how you can transform a 8x8 Franklin panmagic square into a 8x8 complete magic square (by swapping rows and columns systematically).

 Franklin panmagic 8x8 (= according to me most perfect) square 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 1 32 38 59 5 28 34 63 260 260 130 130 46 51 9 24 42 55 13 20 260 260 130 130 27 6 64 33 31 2 60 37 260 260 130 130 56 41 19 14 52 45 23 10 260 260 130 130 11 22 48 49 15 18 44 53 260 260 130 130 40 57 3 30 36 61 7 26 260 260 130 130 17 16 54 43 21 12 50 47 260 260 130 130 62 35 25 8 58 39 29 4 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 1 32 38 59 5 28 34 63 46 51 9 24 42 55 13 20 11 22 48 49 15 18 44 53 40 57 3 30 36 61 7 26 27 6 64 33 31 2 60 37 56 41 19 14 52 45 23 10 17 16 54 43 21 12 50 47 62 35 25 8 58 39 29 4 Complete (= K. Ollerenshaw's most perfect) magic 8x8 square 98 162 98 162 98 162 98 162 162 98 162 98 162 98 162 98 128 128 66 194 1 32 5 28 38 59 34 63 260 260 194 66 46 51 42 55 9 24 13 20 260 260 66 194 11 22 15 18 48 49 44 53 260 260 194 66 40 57 36 61 3 30 7 26 260 260 66 194 27 6 31 2 64 33 60 37 260 260 194 66 56 41 52 45 19 14 23 10 260 260 66 194 17 16 21 12 54 43 50 47 260 260 194 66 62 35 58 39 25 8 29 4 132 132 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130

Addition of the numbers in half rows/columns/diagonals of the complete magic square don't give half of the magic sum. That is why I think the Franklin panmagic 8x8 square is most perfect and the complete magic square is not.

You can transform all most perfect magic squares which are a multiple of 4 from 8x8 to infinite (= 8x8, 12x12, 16x16, 20x20, ...) into complete magic squares. In all downloads of most perfect squares on this website you find the transformation from most perfect into complete.

most perfect 8x8 magic square.xls