Unusual Magic Squares

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A Pandiagonal Torus

This pattern generates 50 order-5 pandiagonal magic squares.

Magic Circles

Two diagrams show characteristics of order-4.

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 order-16 with HH embedded.

Shineman's Magic Diamonds

Two magic diamond patterns with special numbers.

Square with Embedded Star

Arto Heino's order-8 contains a magic hexagon.

Square to Star

Heino's order-4 magic square converts to an order-8 star.

Franklin's Order-8

Benjamin Franklin's order-8 semi-magic square.

A Beastly Magic Square

Patrick De Geest's Order-6 magic square sums to 666.

Millennium Magic Square

Shineman's order-16 pandiagonal with inlaid 2000.

Sagrada Familia Magic Square

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 order-5 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 3-D model     

Magic Circles

                         A.                                                            B.

These two circle diagrams, between them, illustrate some relationships in this order-4 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 90th 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 twenty-five 3 x 3 and twenty-five 5 x 5 squares, along with the center square in each case (including wrap-around) 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 wrap-around). An example of a 5 x 5 rhombic; 32 + 7 + 28 + 20 + 3 = 90. It is easier to visualize wrap-around (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 Order-3 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 Order-16 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 2-cell segment bent-diagonals
All vertical 2-cell segment bent-diagonals, R, L starting on ODD rows
All vertical 2-cell segment bent-diagonals, L, R starting on EVEN rows
All horizontal 4-cell segment bent-diagonals starting in column 4, 8, 12 and 16
All vertical 4-cell segment bent-diagonals starting in column 2, 6, 10, 14
NO 8-cell segment (regular) bent-diagonals
All knight-move horizontal 8-cell segment, bent-diagonals
All knight-move vertical 8-cell segment, bent-diagonals

Shineman's Magic Diamonds

Constructed by E. W. Shineman, Jr. , treasurer, to commemorate his company's 75th (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 75th birthday.    
This one contains 11 special numbers.
75       Age on reaching diamond anniversary.
33       (1933) Year graduated from high school.
4-9-15    Date of birth.
1878      Year father was born.
22        Age when graduated from college
86       (1886) Birthyear of Father-in-law & mother-in-law
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 order-8 magic square is composed of four order-4 pure magic squares. The embedded magic star is index # 16 and is super-magic (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 order-4 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 e-mailed me this pattern on Jul. 15/98.

Order-4 Square to Order-8 Star

This diagram shows some relationships between an order-8B magic star and an order-4 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 Order-8

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 (1706-1790).

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 order-16 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 order-6 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 3-digit 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 right-hand 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 (e-mail 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

 

This page was originally posted October 1998
It was last updated October 15, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz