<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=content-type content=text/html;charset=us-ascii>
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV>Seqfans, <BR> Do the following sets of
numbers belong in the OEIS? </DIV>
<DIV>There are too many magic squares in the universe to include, but
perhaps </DIV>
<DIV>this finite set has some historical significance that makes them
worthy? </DIV>
<DIV> </DIV>
<DIV>First, I copy Benjamin Franklin's 8x8 magic square: <BR>52 61 4
13 20 29 36 45 <BR>14 3 62 51 46 35 30 19 <BR>53 60 5 12 21 28 37 44
<BR>11 6 59 54 43 38 27 22 <BR>55 58 7 10 23 26 39 42 <BR>9 8
57 56 41 40 25 24 <BR>50 63 2 15 18 31 34 47 <BR>16 1 64 49 48 33 32
17 </DIV>
<DIV>where the the column and row sums equal 260. </DIV>
<DIV>Should this be in the OEIS? </DIV>
<DIV> </DIV>
<DIV>At the bottom of this email, I copy Benjamin Franklin's 16x16 magic
square.</DIV>
<DIV> <BR>Related question - is it trivial or is it false in
general that </DIV>
<DIV>magic squares obey the following "rule"? <BR><EM>The matrix powers
of magic squares form number squares <BR>having the same sum for each of
the columns and rows.</EM></DIV>
<DIV>(Something tells me that this is not in true in general ... </DIV>
<DIV>and depends on how the magic square is constructed.) </DIV>
<DIV> </DIV>
<DIV>Example: the matrix square of Ben Franklin's 8x8 magic
square is: </DIV>
<DIV>[7794, 7378, 9522, 9106, 8946, 8530, 8370, 7954;<BR>9266, 9746, 7154, 7634,
7858, 8338, 8562, 9042;<BR>7954, 7602, 9298, 8946, 8850, 8498, 8402,
8050;<BR>8786, 9074, 7826, 8114, 8146, 8434, 8466, 8754;<BR>8274, 8050, 8850,
8626, 8658, 8434, 8466, 8242;<BR>8466, 8626, 8274, 8434, 8338, 8498, 8402,
8562;<BR>7474, 6930, 9970, 9426, 9138, 8594, 8306, 7762;<BR>9586, 10194, 6706,
7314, 7666, 8274, 8626, 9234]</DIV>
<DIV>where the the column and row sums equal 67600 = 260^2. </DIV>
<DIV> </DIV>
<DIV>
<DIV>For these magic squares, all matrix powers produce
<STRONG>non-unique elements </STRONG></DIV>
<DIV>in a number square having the same sum for each of the columns
and rows. </DIV>
<DIV> </DIV></DIV>
<DIV>I wonder how many, say, 3X3 magic squares have <EM>matrix
squares</EM> </DIV>
<DIV>consisting of <STRONG>unique elements</STRONG> and
have the same column and row sums? </DIV>
<DIV> </DIV>
<DIV>Just curious ... not very mathematically deep or serious ... </DIV>
<DIV>
Paul <BR>----------------------------------</DIV>
<DIV> </DIV>
<DIV>Benjamin Franklin's 16x16 magic square:</DIV>
<DIV> </DIV>
<DIV>200 217 232 249 8 25 40 57 72 89
104 121 136 153 168 185<BR> 58 39 26 7 250 231 218
199 186 167 154 135 122 103 90 71<BR>198 219 230 251
6 27 38 59 70 91 102 123 134 155 166 187<BR>
60 37 28 5 252 229 220 197 188 165 156 133 124 101
92 69<BR>201 216 233 248 9 24 41 56
73 88 105 120 137 152 169 184<BR> 55 42 23 10 247
234 215 202 183 170 151 138 119 106 87 74<BR>203 214 235 246
11 22 43 54 75 86 107 118 139 150 171
182<BR> 53 44 21 12 245 236 213 204 181 172 149 140 117
108 85 76<BR>205 212 237 244 13 20 45
52 77 84 109 116 141 148 173 180<BR> 51 46
19 14 243 238 211 206 179 174 147 142 115 110 83 78<BR>207 210
239 242 15 18 47 50 79 82 111 114 143 146
175 178<BR> 49 48 17 16 241 240 209 208 177 176 145 144
113 112 81 80<BR>196 221 228 253 4 29
36 61 68 93 100 125 132 157 164 189<BR> 62
35 30 3 254 227 222 195 190 163 158 131 126 99
94 67<BR>194 223 226 255 2 31 34 63
66 95 98 127 130 159 162 191<BR> 64 33
32 1 256 225 224 193 192 161 160 129 128 97 96
65<BR><BR>Source:<BR><A
href="http://www.mathpages.com/home/kmath155.htm">http://www.mathpages.com/home/kmath155.htm</A><BR><BR>Ps.
the 16x16 magic square is slightly editted (from BF's original) by the
website owner.<BR></DIV></BODY></HTML>