to this page of unusual magic squares. New material will be added here as it becomes available and time permits. Enjoy! 
A Pandiagonal Torus 
This pattern generates 50 order5 pandiagonal magic squares. 
Magic Circles 
Two diagrams show characteristics of order4. 
Square with Special Numbers 
Commemorates my Dad's 90th birthday. 
Prime Heterosquare 
Rivera's prime # squares have each line summing different. 
Double HH 
Ed Shineman constructed this order16 with HH embedded. 
Shineman's Magic Diamonds 
Two magic diamond patterns with special numbers. 
Square with Embedded Star 
Arto Heino's order8 contains a magic hexagon. 
Square to Star 
Heino's order4 magic square converts to an order8 star. 
Franklin's Order8 
Benjamin Franklin's order8 semimagic square. 
A Beastly Magic Square 
Patrick De Geest's Order6 magic square sums to 666. 
Millennium Magic Square 
Shineman's order16 pandiagonal with inlaid 2000. 
Sagrada Familia Magic Square 
A beautiful Spanish cathedral's magic square sums to 33. 
This pattern, which is a torus drawn in two dimensions may be used as an order5 pandiagonal magic square generator. Examples: Start at number 2, and follow the big circles, to generate the columns of the B magic square. 25 different pandiagonal magic squares can be formed this way by starting with each of
the 25 numbers on the model.

Actually, four magic squares may be constructed by following the radial lines, and another four by following the spiral lines, in either direction around the torus. However, three of these magic squares are just disguised versions of the fourth one, because they are rotations or reflections.

A 3D model 
A. B.
These two diagrams, between them, illustrate some relationships in this order4 magic
square.
1 6 12 15
A.
B.
11 16 2 5
1 + 15 + 4 + 14  biggest
circle
1 + 15 + 10 + 8  1 of 4 big circles
8 3 13 10
1 + 12 + 13 + 8  1 of 4
medium circles 11 + 2 + 13 + 8 
1 of 4 small circles
14 9 7 4
1 + 6 + 16 + 11  1 of 5 small circles
Thanks for the idea to W. S. Andrews, Magic Squares and Cubes, Dover, 1960.

I designed this pandiagonal magic square to commemorate my Dad's 90^{th}
birthday. The three center numbers in the top row are his birth date, April 23/03. The 5
rows, 5 columns, the 2 main diagonals and the 10 broken diagonal pairs all sum to 90. Because this is also an Associative magic square, the corners of twentyfive 3 x 3 and twentyfive 5 x 5 squares, along with the center square in each case (including wraparound) also all sum to 90. 
There is still more! Corners of 25 2 x 2 rhombics along with the center cell of
each. Example: 17 + 4 + 49 + 8 + 12 = 90. Also 25 3 x 3, 4 x 4, and 5 x 5 rhombics (including wraparound). An example of a 5 x 5 rhombic; 32 + 7 + 28 + 20 + 3 = 90. It is easier to visualize wraparound if you lay out multiple copies of the magic square like a magic carpet. For still more patterns summing to 90, see my Deluxe magic square, although not all those patterns are possible because this is not a pure magic square. 
19 
...................  137 

5 
41 
13 
59 
31 
37 
41 
109 

17 
3 
47 
67 
53 
59 
61 
173 

7 
83 
11 
101 
67 
43 
47 
157 

23 
29 
127 
71 
227 
167 
151 
139 
149 
439 
The Order3 heterosquare on the left consists of 9 prime numbers. The 3 rows, 3 columns
and the 2 main diagonals all sum to different prime numbers. The sum of all 9 cells is
also a prime number.
Is this the square with the smallest possible total with eighteen
unique primes (including the totals)?
The Square on the right has identical features, but in addition consists of consecutive
primes.
Is this the square with the smallest possible total with nine
consecutive primes?
These squares designed by Carlos Rivera, Sept. 98. See his Web page on Prime Puzzles
& Problems at
http://www.sci.net.mx/~crivera/
98 
79 
178 
95 
162 
63 
194 
47 
210 
255 
2 
239 
18 
143 
114 
159 
158 
179 
78 
163 
94 
195 
62 
211 
46 
3 
254 
19 
238 
115 
142 
99 
100 
77 
180 
93 
164 
61 
196 
45 
212 
253 
4 
237 
20 
141 
116 
157 
155 
182 
75 
166 
91 
198 
59 
214 
43 
6 
251 
22 
235 
118 
139 
102 
101 
76 
181 
92 
165 
60 
197 
44 
213 
252 
5 
236 
21 
140 
117 
156 
153 
184 
73 
168 
89 
200 
57 
216 
41 
8 
249 
24 
233 
120 
137 
104 
103 
74 
183 
90 
167 
58 
199 
42 
215 
250 
7 
234 
23 
138 
119 
154 
151 
186 
71 
170 
87 
202 
55 
218 
39 
10 
247 
26 
231 
122 
135 
106 
105 
72 
185 
88 
169 
56 
201 
40 
217 
248 
9 
232 
25 
136 
121 
152 
149 
188 
69 
172 
85 
204 
53 
220 
37 
12 
245 
28 
229 
124 
133 
108 
107 
70 
187 
86 
171 
54 
203 
38 
219 
246 
11 
230 
27 
134 
123 
150 
148 
189 
68 
173 
84 
205 
52 
221 
36 
13 
244 
29 
228 
125 
132 
109 
110 
67 
190 
83 
174 
51 
206 
35 
222 
243 
14 
227 
30 
131 
126 
147 
146 
191 
66 
175 
82 
207 
50 
223 
34 
15 
242 
31 
226 
127 
130 
111 
112 
65 
192 
81 
176 
49 
208 
33 
224 
241 
16 
225 
32 
129 
128 
145 
160 
177 
80 
161 
96 
193 
64 
209 
48 
1 
256 
17 
240 
113 
144 
97 
This is an Order16 pandiagonal pure magic square so uses the consecutive numbers from
1 to 256.
Each of the 16 rows, columns, and diagonals sum to the constant 2056
The E. S. each also sum to 2056 and the H. H. each sum to 2056 x 2.
Constructed in Sept./98 by E.W. Shineman, Jr. for myself. Thanks Ed.
Update: Sept. 14, 2001
After investigating the Franklin 16x16 squares, I did the same tests on this one. Here are
the results of that test.
If there are 16 cells in the pattern, they sum to S. If there are
only 4 cells to a pattern, their sum is S/4, and 8 cell patterns produce S/2.
The word ‘All’ with no qualifier means that the pattern may be started at ANYof
the 256 cells of the magic square.
All rows of 16 cells. All columns of 16 cells. All rows of 8 cells starting on EVEN columns All columns of 8 cells starting on rows 8 & 16 All rows of 4 cells starting on EVEN columns All columns of 4 cells starting on rows 2 & 10 All rows of 2 cells starting on EVEN columns All 16 cell diagonals All 2x2 square arrays Corners of all even squares All 16 cell small patterns (fully symmetrical within a 6x6 or 8x8 square array) All 16 cell midsize patterns (fully symmetrical within a 10 or 12 square array) All 16 cell large patterns (fully symmetrical within a 14 or 16 square array) All horizontal 2cell segment bentdiagonals All vertical 2cell segment bentdiagonals, R, L starting on ODD rows All vertical 2cell segment bentdiagonals, L, R starting on EVEN rows All horizontal 4cell segment bentdiagonals starting in column 4, 8, 12 and 16 All vertical 4cell segment bentdiagonals starting in column 2, 6, 10, 14 NO 8cell segment (regular) bentdiagonals All knightmove horizontal 8cell segment, bentdiagonals All knightmove vertical 8cell segment, bentdiagonals 
See more on the Franklin page
Constructed by E. W. Shineman, Jr. , treasurer, to commemorate his company's 75^{th} (Diamond) Anniversary in 1966. It contains 5 special numbers.
75 The anniversary.
18 & 91 1891 The year the company was founded.
206 Net sales in 1966 (millions of dollars).
244 Net earnings (cents per share).
24 combinations of 4 numbers sum to 1966.
Also constructed by E. W. Shineman, Jr., this in 1990 for his 75^{th} birthday.
This one contains 11 special numbers.
75 Age on reaching diamond anniversary.
33 (1933) Year graduated from high school.
4915 Date of birth.
1878 Year father was born.
22 Age when graduated from college
86 (1886) Birthyear of Fatherinlaw &
motherinlaw
1885 Year mother was born.
63 & 68 (1963 &1968) Years of career milestones
24 combinations of 4 numbers sum to 1990.
This order8 magic square is composed of four order4 pure magic squares. The
embedded magic star is index # 16 and is supermagic (the points also sum to the constant
34).
The index numbers of the magic squares are:
upper left # 390 equivalent upper right # 142 the basic solution
lower left # 724 equivalent lower right # 271 equivalent
The equivalent solutions require rotations and/or reflections in order to match the basic
solution # shown.
Frénicle, assigned these magic square index numbers about 1675, when he published a
list of all 880 basic solutions for the order4 magic square. For more information, see
Benson & Jacoby, New Recreations with Magic Squares, Dover Publ., 1976.
The magic star index numbers were designed and assigned by me and a full description appears at Magic Star Definitions.
Thanks to Arto Juhani Heino who emailed me this pattern on Jul. 15/98.
This diagram shows some relationships between an order8B magic star and an order4
magic square.
Both patterns are basic solutions. The star is index # 57 (Heinz) and the square is index
# 666 (Frénicle).
Thanks to Arto Juhani Heino for this design.

This magic square was constructed by Benjamin Franklin (17061790). It has many interesting properties as illustrated by the following cell patterns. Because the square is continuous, (wraps around), each pattern is repeated 64 times ( 8 in each direction). However, because the main diagonals do not sum correctly (one totals 260  32 & the other 260 + 32), it is not a true magic square. 
Franklin also constructed an order16 magic square with similar properties. It also has the property that any 4 by 4 square sums to the constant, 2056, as well as some other combinations. See my Franklin page for more on all of Ben Franklin squares (and his magic circle) 
This order6 magic square is constructed from the first 36 multiples of 6, and has a magic sum of 666.
I received this beastly square from 
This square contains many hidden 3digit palindromes (which I indicate here in blue). The top left 3 by 3 square is magic with S = 252. The bottom left 3 by 3 square is magic with S = 414. The 3 row of 3 cells in top right corner sum to 414. The 3 row of 3 cells in bottom right corner sum to 252. The corners of the 3 squares working from the outside to the center, each sum to 444. The 6 by 6 border cells sum to 2220 which equals 666 + 888 + 666. The border cells of the central 4 by 4 square sum to 1332 which equals 666 + 666. The top half of the righthand column sums to 252 and the bottom half to 414. The top half of the column next to it sums to 414 and the bottom half to 252. By dividing each number in the magic square by 6, a new magic square is obtained, with S = 111. What other features still await discovery? 
Edward W. Shineman, Jr. designed this magic square to commemorate the
start of the new century (and millenium). It is an order 16 pandiagonal using numbers from –3 to 253 with one number not used. (Can you find the missing number?) Each row, column and
diagonal, including the broken diagonal pairs, sum to 2000. In addition, the three groups
of sixteen numbers (the zeros) each sum to 2000. 
NOTE: There is controversy as to whether the year 2000 is part of the 20^{th} or the 21^{st} century (and the 2^{nd} or 3^{rd} millennium). Here we consider it to be the latter.
The Sagrada Familia cathedral in Barcelona, Spain,
contains the unusual magic square shown in the two pictures below.
Both the number 10 and the number 14 are repeated twice and there is no 12 or 16. The
magic sum is 33.
Does anyone know the significance of this magic square?
Many people have speculated that 33 signifies
Jesus Christ's age at the time of his crucifixion.
These pictures were taken by
Jorge Posada and are dedicated to his girlfriend Maite. Thank you Jorge,
for the pictures and for drawing this item to my attention. The picture below shows the placement in the cathedral but is of unknown origin. Alex Cohn (email July 15/01) points out that this square also appears multiple times on the main facade of the incompleted church. 
The Sagrada Familia cathedral is the most important work of Gaudi, a spanish architect considered as a true genius. He worked on this building from 1882 until his death in 1926. Although it is not completed yet, it is the most important and amazing building in Barcelona. It has no roof so far, for instance, but there is a saying in Barcelona: "The only worthy roof for the Sagrada Familia is the sky". There is some information about the cathedral in: http://www.greatbuildings.com/buildings/Sagrada_Familia.html
Lee Sallows (July12,2001) points out that magic squares
with a magic sum of 33 may be constructed without using duplicate integers. 

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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 06, 2005
Copyright © 1998, 1999, 2000 by Harvey D. Heinz