### Inlaid 14x14 magic square (with inlays 4x4, 6x6 and 12x12)

I was inspired by the inlaid squares on the website of Harvey Heinz: www.magic-squares.net/magicsquare.htm#Orders 3, 5, 7, 9 Inlaid and the inlaid squares of John Hendricks: www.magic-squares.net/hendricks.htm

First we construct a 12x12 inlay, which consist of four pan-4x4 in 6x6 magic squares in five steps:

[1] To get the four 4x4 panmagic inlay squares we take a 8x8 most perfect (Franklin pan)magic square, add 40 to each number and split up the 8x8 square in four 4x4 (inlay) squares.

Most perfect 8x8 square + 40      =   four 4x4 inlay squares

 1 54 12 63 3 56 10 61 41 94 52 103 43 96 50 101 16 59 5 50 14 57 7 52 56 99 45 90 54 97 47 92 53 2 64 11 55 4 62 9 93 42 104 51 95 44 102 49 60 15 49 6 58 13 51 8 100 55 89 46 98 53 91 48 17 38 28 47 19 40 26 45 57 78 68 87 59 80 66 85 32 43 21 34 30 41 23 36 72 83 61 74 70 81 63 76 37 18 48 27 39 20 46 25 77 58 88 67 79 60 86 65 44 31 33 22 42 29 35 24 84 71 73 62 82 69 75 64

[2] To construct the four borders we need (4 x 20 =) 80 numbers. Take the numbers 1 up to 40 and 105 up to 144. Translate the numbers 105 up to 144 into -/- 1 up to -/- 40.

[3] Put in each side of the border 3 positive and 3 negative numbers and take care that the sum of the 6 numbers is exactly 0. In the four x four corners you need 16 numbers extra = 8 positive and 8 negative numbers double. The average number is ([the lowest number + the highest number] divided by 2: [1+40]/2 =) 20,5. Take as sum of the 8 double numbers (8 x 20,5 = ) 164. Take as sum of 3 numbers (3 x 20,5 =) 61,5 = 61 (8x) or 62 (8x). I puzzeled and got the following table:

 + 15 20 26 61 16 21 25 62 17 22 23 62 18 19 24 61 164 + 7 28 26 61 5 32 25 62 8 31 23 62 1 36 24 61 -/- 15 9 37 61 16 6 40 62 17 10 35 62 18 4 39 61 -/- 13 14 34 61 3 29 30 62 2 27 33 62 11 12 38 61

[4] Use the table to construct the 4 borders and translate the negative numbers into -/- 1 up to -/- 40 into 105 up to 144).

 15 20 -13 -14 -34 26 16 21 -3 -29 -30 25 17 22 -2 -27 -33 23 18 19 -11 -12 -38 24 28 32 31 36 7 5 8 1 -37 -40 -35 -39 -9 -6 -10 -4 -15 -16 -17 -18 15 20 -13 -14 -34 26 16 21 -3 -29 -30 25 17 22 -2 -27 -33 23 18 19 -11 -12 -38 24 -28 28 -32 32 -31 31 -36 36 -7 7 -5 5 -8 8 -1 1 37 -37 40 -40 35 -35 39 -39 9 -9 6 -6 10 -10 4 -4 -26 -20 13 14 34 -15 -25 -21 3 29 30 -16 -23 -22 2 27 33 -17 -24 -19 11 12 38 -18 15 20 132 131 111 26 16 21 142 116 115 25 17 22 143 118 112 23 18 19 134 133 107 24 117 28 113 32 114 31 109 36 138 7 140 5 137 8 144 1 37 108 40 105 35 110 39 106 9 136 6 139 10 135 4 141 119 125 13 14 34 130 120 124 3 29 30 129 122 123 2 27 33 128 121 126 11 12 38 127

[5] Combine the borders and the 4x4 inlay squares.

12x12 magic square = four 6x6 magic squares with pan-4x4 inlay

 15 20 132 131 111 26 16 21 142 116 115 25 117 41 94 52 103 28 113 43 96 50 101 32 138 56 99 45 90 7 140 54 97 47 92 5 37 93 42 104 51 108 40 95 44 102 49 105 9 100 55 89 46 136 6 98 53 91 48 139 119 125 13 14 34 130 120 124 3 29 30 129 17 22 143 118 112 23 18 19 134 133 107 24 114 57 78 68 87 31 109 59 80 66 85 36 137 72 83 61 74 8 144 70 81 63 76 1 35 77 58 88 67 110 39 79 60 86 65 106 10 84 71 73 62 135 4 82 69 75 64 141 122 123 2 27 33 128 121 126 11 12 38 127

The magic sum of each 4x4 panmagic inlay square is 290.The magic sum of each 6x6 magic square is 435. The magic sum of the 12x12 magic square is 870.

Because the 12x12 magic square consists of four proportional 6x6 magic squares, each 1/2 row/column/diagonal gives 1/2 of the magic sum.

And now we construct the 14x14 inlaid square:

[1] Add 26 to each number of the 12x12 magic square;

[2] Use 52 numbers (= 1 up to 26 and 171 up to 196) to construct the 14x14 border.

The sum of the numbers 1 up to 26 is 351. Add 33 to the sum of 351 and you get 384 = 4x96. To get 33 take the numbers 16 and 17 double. I constructed the following table:

 16 17 1 26 2 25 9 96 16 4 24 5 22 6 19 96 17 3 23 7 21 10 15 96 8 11 12 13 14 18 20 96

Use the table to construct the 14x14 border (the numbers 171 up to 196 have been translated into -/- 1 up to -/- 26):

 16 1 26 2 25 9 -8 -11 -12 -13 -14 -18 -20 17 3 23 7 21 10 15 -4 -24 -5 -22 -6 -19 -16 16 1 26 2 25 9 -8 -11 -12 -13 -14 -18 -20 17 -3 3 -23 23 -7 7 -21 21 -10 10 -15 15 4 -4 24 -24 5 -5 22 -22 6 -6 19 -19 -17 -1 -26 -2 -25 -9 8 11 12 13 14 18 20 -16 16 1 26 2 25 9 189 186 185 184 183 179 177 17 194 3 174 23 190 7 176 21 187 10 182 15 4 193 24 173 5 192 22 175 6 191 19 178 180 196 171 195 172 188 8 11 12 13 14 18 20 181

Combine the 14x14 border and the 12x12 inlay to complete the 14x14 magic inlaid square.

Magic 14x14 inlaid square (with inlays 4x4, 6x6 and 12x12)

 16 1 26 2 25 9 189 186 185 184 183 179 177 17 194 41 46 158 157 137 52 42 47 168 142 141 51 3 174 143 67 120 78 129 54 139 69 122 76 127 58 23 190 164 82 125 71 116 33 166 80 123 73 118 31 7