[SeqFan] Benjamin Franklin's Magic Squares

[zak seidov]

Related (or not?) Q:

Did anyone consider "routes on magic squares":
broken line connecting 1->2->3->...->n^2->1.

Some of Qs:
1) The extremal total routes?
2) Distribution of segment lengths?
2a)All distinct segment lengths?
3) Symmetrical routes?
4) ?

Thanks, Zak

--- "Paul D. Hanna" <pauldhanna at juno.com> wrote:

> Seqfans, 
>      Do the following sets of numbers belong in the
> OEIS? 
> There are too many magic squares in the universe to
> include, but perhaps 
> this finite set has some historical significance
> that makes them worthy? 
>  
> First, I copy Benjamin Franklin's 8x8 magic square: 
> 52 61  4 13 20 29 36 45 
> 14  3 62 51 46 35 30 19 
> 53 60  5 12 21 28 37 44 
> 11  6 59 54 43 38 27 22 
> 55 58  7 10 23 26 39 42 
> 9  8 57 56 41 40 25 24 
> 50 63  2 15 18 31 34 47 
> 16  1 64 49 48 33 32 17 
> where the the column and row sums equal 260. 
> Should this be in the OEIS? 
>  
> At the bottom of this email, I copy Benjamin
> Franklin's 16x16 magic
> square.
>   
> Related question - is it trivial or is it false in
> general that 
> magic squares obey the following "rule"? 
> The matrix powers of magic squares form number
> squares 
> having the same sum for each of the columns and
> rows.
> (Something tells me that this is not in true in
> general ... 
> and depends on how the magic square is constructed.)
>  
>   
> Example:  the matrix square of Ben Franklin's 8x8
> magic square is: 
> [7794, 7378, 9522, 9106, 8946, 8530, 8370, 7954;
> 9266, 9746, 7154, 7634, 7858, 8338, 8562, 9042;
> 7954, 7602, 9298, 8946, 8850, 8498, 8402, 8050;
> 8786, 9074, 7826, 8114, 8146, 8434, 8466, 8754;
> 8274, 8050, 8850, 8626, 8658, 8434, 8466, 8242;
> 8466, 8626, 8274, 8434, 8338, 8498, 8402, 8562;
> 7474, 6930, 9970, 9426, 9138, 8594, 8306, 7762;
> 9586, 10194, 6706, 7314, 7666, 8274, 8626, 9234]
> where the the column and row sums equal 67600 =
> 260^2. 
>   
> For these magic squares, all matrix powers produce
> non-unique elements 
> in a number square having the same sum for each of
> the columns and rows. 
>  
> I wonder how many, say, 3X3 magic squares have
> matrix squares 
> consisting of unique elements and have the same
> column and row sums? 
>  
> Just curious ... not very mathematically deep or
> serious ... 
>       Paul 
> ----------------------------------
> 
> Benjamin Franklin's 16x16 magic square:
> 
> 200 217 232 249   8  25  40  57  72  89 104 121 136
> 153 168 185
>   58  39  26   7 250 231 218 199 186 167 154 135 122
> 103  90  71
> 198 219 230 251   6  27  38  59  70  91 102 123 134
> 155 166 187
>   60  37  28   5 252 229 220 197 188 165 156 133 124
> 101  92  69
> 201 216 233 248   9  24  41  56  73  88 105 120 137
> 152 169 184
>   55  42  23  10 247 234 215 202 183 170 151 138 119
> 106  87  74
> 203 214 235 246  11  22  43  54  75  86 107 118 139
> 150 171 182
>   53  44  21  12 245 236 213 204 181 172 149 140 117
> 108  85  76
> 205 212 237 244  13  20  45  52  77  84 109 116 141
> 148 173 180
>   51  46  19  14 243 238 211 206 179 174 147 142 115
> 110  83  78
> 207 210 239 242  15  18  47  50  79  82 111 114 143
> 146 175 178
>   49  48  17  16 241 240 209 208 177 176 145 144 113
> 112  81  80
> 196 221 228 253   4  29  36  61  68  93 100 125 132
> 157 164 189
>   62  35  30   3 254 227 222 195 190 163 158 131 126
>  99  94  67
> 194 223 226 255   2  31  34  63  66  95  98 127 130
> 159 162 191
>   64  33  32   1 256 225 224 193 192 161 160 129 128
>  97  96  65
> 
> Source:
> http://www.mathpages.com/home/kmath155.htm
> 
> Ps. the 16x16 magic square is slightly editted (from
> BF's original) by
> the website owner.



 
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