# Unusual Magic Squares

### A Pandiagonal Torus

This pattern generates 50 order-5 pandiagonal magic squares.

### Magic Circles

Two diagrams show characteristics of order-4.

### 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.

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.

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