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More Magic Squares
© 2000 Paul C. Pasles
Click here for the BRAND NEW Franklin Mathematics Page!
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:
- K. H. De Haas, Frenicle's 880 basic magic squares of 4 x 4 cells (Rotterdam: Van Sijn, 1935).
- Kathleen Ollerenshaw & Herman Bondi, Magic Squares of Order Four. Phil. Trans. Royal Soc. London, Series A, 1982. (Some libraries also have this article in a bound reprint.) The 880 squares are listed in the appendix.
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
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.