More Magic Squares
© 2000 Paul C. Pasles 
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Here are some obscure but interesting magic squares which are mentioned in my paper "The Lost Squares of Dr. Franklin" (Amer. Math. Monthly, June-July 2001). Space considerations prohibited their inclusion there. Most of these are difficult to find outside of rare book rooms.

 

Michael Stifel (Arithmetica Integra, 1544) wrote two magic squares, of orders 9 & 16 (below). These are fully magic: The rows, columns and 2 main diagonals each sum to the same "magic constant" (369 or 2056).
 

16 81 79 77 75 11 13 15 2
78 28 65 63 61 25 27 18 4
76 62 36 53 51 35 30 20 6
74 60 50 40 45 38 32 22 8
9 23 33 39 41 43 49 59 73
10 24 34 44 37 42 48 58 72
12 26 52 29 31 47 46 56 70
14 64 17 19 21 57 55 54 68
80 1 3 5 7 71 69 67 66

 
256
9
247
246
12
13
243
242
16
17
239
238
20
21
235
2
3
226
213
45
46
210
209
49
50
206
205
53
54
201
32
254
4
33
200
63
193
192
66
67
189
188
70
71
185
58
224
253
252
34
59
178
169
89
90
166
165
93
94
161
80
198
223
5
251
222
60
81
160
101
155
154
104
105
151
98
176
197
35
6
7
221
196
82
99
146
141
117
118
137
112
158
175
61
36
250
8
37
62
174
100
113
136
123
122
133
144
157
83
195
220
249
23
38
73
173
107
114
129
126
127
132
143
150
84
184
219
234
24
218
183
85
108
115
125
130
131
128
142
149
172
74
39
233
232
217
75
86
148
138
124
135
134
121
119
109
171
182
40
25
231
41
76
87
147
145
116
140
139
120
111
110
170
181
216
26
27
42
180
162
159
156
102
103
153
152
106
97
95
77
215
230
28
43
179
177
88
168
167
91
92
164
163
96
79
78
214
229
228
202
199
194
64
65
191
190
68
69
187
186
72
57
55
29
227
225
44
212
211
47
48
208
207
51
52
204
203
56
31
30
255
248
10
11
245
244
14
15
241
240
18
19
237
236
22
1

The latter square was originally published with several typos, corrected here.

Somehow these examples have largely escaped mention in the standard accounts of magic squares. Stifel is mentioned in the magic square writings of Günther and Cammann, but the squares themselves do not appear.

Stifel also comments on multiplicative magic squares! I did not know that these were explored so long ago.

 

Divers ouvrages de mathematique et de physique (Bernard Frénicle de Bessy, et. al., 1693) lists all fully magic squares of order 4. Here are a few of them:
 

1
14
11
8
15
4
5
10
6
9
16
3
12
7
2
13
2
3
13
16
14
15
1
4
7
6
12
9
11
10
8
5
2
7
11
14
8
13
1
12
9
4
16
5
15
10
6
3
4
7
14
9
12
6
1
15
5
11
16
2
13
10
3
8

Although original copies of this book are scarce, the 4 x 4 squares have also appeared in more readily available places:


Here is a magic square from the anonymous 18th century author whose notes were found among the Franklin papers:

1
8
58
63
9
16
50
55
62
59
5
4
54
51
13
12
7
2
64
57
15
10
56
49
60
61
3
6
52
53
11
14
17
 24
 42
47
25
32
34
39
46
43
21
20
38
35
29
28
23
18
48
41
31
26
40
33
44
45
19
22
36
37
27
30

 

Links

Web pages cited in the paper

A construction of Franklin squares, by Neil Abrahams.
Benjamin Franklin, A Documentary History, by J.A. Leo Lemay (forthcoming).
How to make a Franklin square, by Paul C. Pasles.
 

Other relevant links
 

Franklin’s Autobiography: Poor Richard’s Almanac:
Project Gutenberg 1733 1753 1759 (facsimile pages)
earlyamerica.com 1733-59 (selected excerpts only)

 
American Philosophical Society Athenæum of Philadelphia
Library Company of Philadelphia Royal Society
Rosenbach Museum and Library University of Pennsylvania
Historical Society of Pennsylvania Franklin & Marshall College
Material on this page © 2000 Paul C. Pasles. All rights reserved. This page cannot be copied, published or redistributed in any form without the express written consent of the author.