## Tuesday, October 14, 2014

### A Special Order 13 Magic Square

A Special Order 13 Magic Square:
This is an order 13 magic square with the normal “magic square properties”, but it also has some special additions.
 157 13 23 147 109 31 111 138 36 66 102 100 72 145 25 17 153 61 139 59 32 134 104 68 98 70 16 154 144 26 57 56 30 112 136 99 105 60 110 22 148 156 14 113 114 140 58 34 65 71 133 37 97 73 94 76 151 18 21 89 146 135 35 29 141 79 91 78 92 27 82 150 155 11 63 107 33 137 74 96 75 95 143 159 15 20 88 115 55 101 69 90 80 93 77 19 24 81 149 152 54 116 103 67 164 6 3 167 85 142 158 12 28 64 106 108 62 7 163 168 86 1 132 44 39 125 50 48 118 124 162 8 84 2 169 38 126 131 45 120 122 52 46 5 83 161 10 166 129 43 40 128 123 117 49 51 87 165 9 160 4 41 127 130 42 47 53 121 119
The square has 169 spaces for number, and contains all of the numbers from 1 to 169 (13 squared) one time each.  All of the rows have the same sum (1105), all of the columns have the same sum (1105), and both of the main diagonals have the same sum.
What makes it special?  It also contains 12 smaller magic squares (well, one of them is only semi-magic because only one of its diagonals adds up to the magic constant for that square) ranging in size from 3 x 3 to 11 x 11.  So this magic square is really a total of 13 magic squares.
A 3 x3 semi-magic square with 3 in the upper left corner.  All rows and columns, but only 1 main diagonal have a magic constant of 255.
Four 4 x 4 magic squares with 157, 97, 132 or 50 in the upper left corner of each square.  All rows, columns and the two main diagonals sum to 340.
Two 5 x 5 magic squares with 164 or 151 in the upper left corner.  All rows, columns and the two main diagonals sum to 425.
Two 7 x 7 magic squares with 94 or 57 in the upper left corner.  All rows, columns, and the two main diagonals sum to 595.
Two 9 x 9 magic squares with 109 and 97 in the upper left corner.  Their magic constant is 765.
An 11 x 11 magic square with 23 in the upper left corner.  Its magic constant is 935.
And finally, a 13 x 13 magic square with 157 in the upper left corner (the entire array).  Its magic constant is 1105.
My daughter was born on the 13th of the month, turned 13 (became a teenager) on a Friday the 13th (with 12 friends attending a sleepover, all 13 dressed in black).  The last time I warned her about a Friday the 13th coming up (so that she could avoid hospital duty that nigh) she said "Why?  That's when all the fun stuff happens."  What can I say - she has a thing for the number 13.
David

#### 1 comment:

1. Hi,

Question is how many 13x13 tables exist? Any information is greatly appreciated.

Thank you,

RFR