Magic Square Wartune

The constant values $ M $ of the sums of the magic squares have a minimum value (for non-zero integer positive values). $$ M = n (n ^ 2 + 1) / 2 $$ For a size 3x3, the minimum constant is 15, for 4x4 it is 34, for 5x5 it is 65, 6x6 it is 111, then 175, 260. The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which the Luo Shu principle is clearly evident. The order 5 square is a bordered magic square, with central 3×3 square formed according to Luo Shu principle. The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a pan-diagonal magic square.It is also an instance of most perfect magic square.Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. Aug 20, 2017 If this paragraph is too short for you to learn the basics you should watch some Wartune Youtube videos or read some DolyGames blog posts or whatever. ##### 2) Pattern Break: Why, How and when? The.'Pattern'. is basically the order of kills which you should know as somebody who ran Magic Square before.

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 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, andanother 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 theyare rotations or reflections.

1	9	12	20	23	..........	2	11	25	9	18	1	25	19	13	7
17	25	3	6	14	10	19	3	12	21	14	8	2	21	20
8	11	19	22	5	13	22	6	20	4	22	16	15	9	3
24	2	10	13	16	16	5	14	23	7	10	4	23	17	11
15 A.	18	21	4	7	24 B.	8	17	1	15	18 C.	12	6	5	24

A 3-D model

Wartune Magic Square Image

Magic Circles

A. B.

These two diagrams, between them, illustrate some relationships in this order-4 magicsquare.
1 6 12 15 A. B.
11 16 2 51 + 15 + 4 + 14 -- biggestcircle 1 + 15 + 10 + 8 -- 1 of 4 big circles
8 3 13 10 1 + 12 + 13 + 8 -- 1 of 4medium circles11 + 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.

Pandiagonal 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.

Because this is also an Associative magic square, 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 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 Number Heterosquares

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 Order-3 heterosquare on the left consists of 9 prime numbers. The 3 rows, 3 columnsand the 2 main diagonals all sum to different prime numbers. The sum of all 9 cells isalso a prime number.
Is this the square with the smallest possible total with eighteenunique primes (including the totals)?

Magic Square Wartune Deck

The Square on the right has identical features, but in addition consists of consecutiveprimes.
Is this the square with the smallest possible total with nineconsecutive primes?

These squares designed by Carlos Rivera, Sept. 98. See his Web page on Prime Puzzles& Problems at
http://www.sci.net.mx/~crivera/

Double HH

178

162

194

210

255

239

143

114

159

158

179

163

195

211

254

238

115

142

100

180

164

196

212

253

237

141

116

157

155

182

166

198

214

251

235

118

139

102

101

181

165

197

213

252

236

140

117

156

153

184

168

200

216

249

233

120

137

104

103

183

167

199

215

250

234

138

119

154

151

186

170

202

218

247

231

122

135

106

105

185

169

201

217

248

232

136

121

152

149

188

172

204

220

245

229

124

133

108

107

187

171

203

219

246

230

134

123

150

148

189

173

205

221

244

228

125

132

109

110

190

174

206

222

243

227

131

126

147

146

191

175

207

223

242

226

127

130

111

112

192

176

208

224

241

225

129

128

145

160

177

161

193

209

256

240

113

144

This is an Order-16 pandiagonal pure magic square so uses the consecutive numbers from1 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 arethe results of that test.

If there are 16 cells in the pattern, they sum to S. If there areonly 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 ANYofthe 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 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

See more on the Franklin page

Shineman's Magic Diamonds

Constructed by E. W. Shineman, Jr. , treasurer, to commemorate his company's75^th (Diamond) Anniversary in 1966. Itcontains 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.
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.

Order-8 with embedded star

This order-8 magic square is composed of four order-4 pure magic squares. Theembedded magic star is index # 16 and is super-magic (the points also sum to the constant34).
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 basicsolution # shown.

Frénicle, assigned these magic square index numbers about 1675, when he published alist 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.

Wartune Magic Square Pattern

The magic star index numbers were designed and assigned by me and afull description appears at Magic Star Definitions.

Magic Square Wartune Online

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-4magic 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 8 x 8 Magic Square

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

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

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?

Millennium Magic Square

Magic Square 3x3

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 20^thor the 21^st century (and the 2^nd or 3^rd millennium). Herewe consider it to be the latter.

Sagrada Familia MagicSquare

How To Play Magic Square Wartune

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. Themagic 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 (e-mail 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, aspanish architect considered as a true genius. He worked on this building from 1882 untilhis death in 1926. Although it is not completed yet, it is the most important and amazingbuilding in Barcelona. It has no roof so far, for instance, but there is a saying inBarcelona: 'The only worthy roof for the Sagrada Familia is the sky'. There issome 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

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A Pandiagonal Torus

Magic Circles

Square with Special Numbers

Prime Heterosquare

Double HH

Shineman's Magic Diamonds

Square with Embedded Star

Square to Star

Franklin's Order-8

A Beastly Magic Square

Millennium Magic Square

Sagrada Familia Magic Square

A Pandiagonal Torus

Wartune Magic Square Image

Magic Circles

Pandiagonal with Special Numbers.

Prime Number Heterosquares

Magic Square Wartune Deck

Double HH

Shineman's Magic Diamonds

Order-8 with embedded star

Wartune Magic Square Pattern

Magic Square Wartune Online

Order-4 square to order-8 star.

Franklin 8 x 8 Magic Square

A Beastly Magic Square

Millennium Magic Square

Magic Square 3x3

Sagrada Familia MagicSquare

How To Play Magic Square Wartune