[SeqFan] Benjamin Franklin's Magic Squares

Thu Dec 14 12:30:54 CET 2006

Dear seqfans,

Two related (nice) sequences:

%S BF01
8,7,2,1,3,4,5,6,6,5,4,3,1,2,7,8,8,7,2,1,3,4,5,6,6,5,4,3,1,2,7,8,8,7,2,1,3,4,5,6,6,5,4,3,1,2,7,8,8,7,2,1,3,4,5,6,6,5,4,3,1,2,7,8

%N BF01 x-coordinates of numbers 1..64 in BF MS (8x8
case)

%C BF01 There is period of 16 terms inside the  period
a(i)=a(17-i). Also of (great) interest is graph with
points (x_i,y_i),i=1..64. Cf. BF02. Are all this old
hat?

%K BF01 nice, nonn, fini, full

%S BF02
2,3,2,3,3,2,3,2,1,4,1,4,4,1,4,1,8,5,8,5,5,8,5,8,7,6,7,6,6,7,6,7,6,7,6,7,7,6,7,6,5,8,5,8,8,5,8,5,4,1,4,1,1,4,1,4,3,2,3,2,2,3,2,3

%N BF02 y-coordinates of numbers 1..64 in BF MS (8x8
case)

%C BF02 The graph is beautiful and almost symmetric.
Also of (great) interest is graph with points
(x_i,y_i),i=1..64. Cf. BF01. Are all this old hat?

%K BF02 nice, nonn, fini, full

Zak

PS I'll also try to check 16x16 BF MS.

--- zak seidov <zakseidov at yahoo.com> wrote:

> 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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