Contents

This pattern generates 50
order5 pandiagonal magic squares. 

Two diagrams show
characteristics of order4. 

Commemorates my Dad's
90th birthday. 

Rivera's prime # squares
have each line summing different. 

Ed Shineman constructed
this order16 with HH embedded. 

Two magic diamond
patterns with special numbers. 

Arto Heino's order8
contains a magic hexagon. 

Heino's order4 magic
square converts to an order8 star. 

Benjamin Franklin's
order8 semimagic square. 

Patrick De Geest's
Order6 magic square sums to 666. 

Shineman's order16
pandiagonal with inlaid 2000. 

A beautiful Spanish
cathedral's magic square sums to 33. 
A Pandiagonal Torus

This pattern, which is a torus drawn in two
dimensions may be used as an order5 pandiagonal magic square
generator.
Examples:
Start at number 1, and follow the big circles, to generate the
rows of the A. magic square (below).
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.
Another 25 different magic squares can be constructed by forming the
rows and columns with the numbers along the spiral lines. See Magic
square C, below.
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 

Magic Circles
A.
B. 
These two circle diagrams, between them, illustrate
some relationships in this order4 magic square.
(Compare the examples below with this square.)
1 
6 
12 
15 
11 
16 
2 
5 
8 
3 
13 
10 
14 
9 
7 
4 
Thanks for the idea to W.
S. Andrews, Magic Squares and Cubes, Dover, 1960. 
A.
1 + 15 + 4 + 14  biggest circle
1 + 12 + 13 + 8  1 of 4 medium circles
1 + 6 + 16 + 11  1 of 5 small circles 
B.
1 + 15 + 10 + 8  1 of 4 big circles
1 + 2 + 13 + 8  1 of 4 small circles 
Square with Special Numbers
32 
4 
23 
3 
28 
17 
12 
49 
5 
7 
22 
8 
1 
26 
33 
10 
47 
6 
25 
2 
9 
19 
11 
31 
20 

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.
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 (and
large patterns) 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. 
Prime Heterosquare
19 





137 





5 
41 
13 
59 


31 
37 
41 
109 
A. 
17 
3 
47 
67 

B. 
53 
59 
61 
173 

7 
83 
11 
101 


67 
43 
47 
157 
23 
29 
127 
71 
227 

167 
151 
139 
149 
439 
These squares designed by
Carlos Rivera, Sept. 98. See his
Web page on Prime Puzzles & Problems at
http://www.primepuzzles.net/ 
The Order3 heterosquare (A.) 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)?
Square B. has identical features, but in addition consists of
consecutive primes.
Is this the square with the smallest possible
total with nine consecutive primes? 
Double HH
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 ANY of the 256 cells of the magic square.
See more on my Franklin page


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 
Shineman's Magic Diamonds

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. 
Square with Embedded Star

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. 
Order4
Square to Order8 Star

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. 
Franklin's Order8
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 


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) 
A Beastly Magic Square
This order6 magic square is constructed from the first 36
multiples of 6, and has a magic sum of 666.
66 
108 
78 
174 
216 
24 
96 
84 
72 
204 
30 
180 
90 
60 
102 
198 
168 
48 
120 
162 
132 
12 
54 
186 
150 
138 
126 
42 
192 
18 
144 
114 
156 
36 
6 
210 

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?
I received this beastly square from
Patrick De Geest on Dec. 7, 1998. Well done Patrick! 
Millennium Magic Square

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.
The large two, which contains 32 numbers, sums to 4000 (the magic sum x
2).
The double zero shown in the top left cell represents the new year.
NOTE: There is controversy as to whether the year 2000 is part of the
20th or the 21st century (and the 2nd or 3rd millennium). Here we consider
it to be the latter. 
Sagrada Familia Magic Square
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. Alex Cohn (email July
15/01) points out that this square also appears multiple times on
the main facade of the uncompleted 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.
Here is one (of several he provided) that uses the integers 0 to 16,
but without the 4. 
0 
5 
12 
16 
15 
11 
6 
1 
10 
3 
13 
7 
8 
14 
2 
9 

