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12 7635
52 269
59 4519
48 471
16
623922
61 38
74
50 964
14 8128
57 4026
24
33
498
1178 346337 23
5642 25
541 681380 30
1873 32
5844 21476 70
66
65 7 51
36 77 10
29 79 1527 41 55
67 3 53
72 5 46177531
2043 60
H. D. Heinz & J. R. Hendricks
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MAGIC SQUARE
LEXICON:
MAGIC SQUARE LEXICON:
ILLUSTRATED
192 224 97 1
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96 64 129 225
193 161 32 128
2 98 223 191 8 104 217 185 190 222 99 3
159 255 66 34 153 249 72 40 35 67 254 158
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157 253 68 36 152 248 73 41 151 247 74 42 150 246 75 43 37 69 252 156
228 132 61 93 233 137 56 88 234 138 55 87 235 139 54 86 92 60 133 229
125 29 164 196 120 24 169 201 119 23 170 202 118 22 171 203 197 165 28 124
6 102 219 187 12 108 213 181 13 109 212 180 14 110 211 179 15 111 210 178 16 112 209 177 186 218 103 7
155 251 70 38 149 245 76 44 148 244 77 45 147 243 78 46 146 242 79 47 145 241 80 48 39 71 250 154
230 134 59 91 236 140 53 85 237 141 52 84 238 142 51 83 239 143 50 82 240 144 49 81 90 58 135 231
123 27 166 198 117 21 172 204 116 20 173 205 115 19 174 206 114 18 175 207 113 17 176 208 199 167 26 122
All rows, columns, and 14 main diagonals sum correctly in proportion to length 16 4x4 magic squares.
H. D. Heinz & J. R. Hendricks
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MAGIC SQUARE
LEX ICON :Illustrated
12 7635
52 269
59 4519
48 471
16
623922
61 38
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50 964
14 8128
57 4026
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5642 25
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5844 21476 70
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29 79 1527 41 55
67 3 53
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Magic Square Lexicon: Illustrated By
Harvey D. Heinz &
John R. Hendricks
Published November 2000 By HDH
Second print run July 2005
Copyright 2000 by Harvey D. Heinz
Published in small quantities by HDH
as demand indicates
ISBN 0-9687985-0-0
Binding courtesy of Pacific Bindery Services Ltd.
For Erna & Celia Two ladies with patience and forbearance,
while their men are playing with numbers.
To commemorate the year 2000
Prime magic square A
67 241 577 571
547 769 127 13
223 139 421 673
619 307 331 199
Plus prime magic square B
1933 1759 1423 1429
1453 1231 1873 1987
1777 1861 1579 1327
1381 1693 1669 1801
Equals magic square C
2000 2000 2000 2000
2000 2000 2000 2000
2000 2000 2000 2000
2000 2000 2000 2000
Designed by John E. Everett (July, 2000) By permission, Carlos Rivera, http://www.primepuzzles.net/
Contents
Tables and Illustrations iii
Preface 1 xi
Preface 2 xiii
Magic Square Lexicon 1 to 174
References 175
The Authors 1 81
Magic Square Bibliography A1-1 to A1-15
John Hendricks Bibliography A2-1 to A2-3
12b
1
23
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1819
20
21
22 23
24
Tables and Illustrations
1 - Two types of almost magic stars ..................................................... 2
2 - The Alphamagic square of order-3 .................................................. 3
3 - An order-4 and an order-5 anti-magic square.................................. 4
4 - Number of anti-magic squares......................................................... 4
5 - Two order-5 anti-magic stars........................................................... 5
6 - The 8 aspects of a magic square. ..................................................... 6
7 - One of four order-3 magic cubes, all associated. ............................ 7
8 - Three associated magic squares....................................................... 8
9 - An order-9 associated, pandiagonal, 32 ply magic square............. 9
10 - One of 4 basic magic cubes .........................................................11
11 - An order-4 Basic magic square ...................................................12
12 - A basic order-6 magic star...........................................................13
13 - Basic tesseract MT# 9 shown in the standard position................14
14 - A disguised version of MT# 9 (previous figure...........................15
15 - The 58 basic tesseracts of order-3 in indexed order ....................16
16 - An order-8 bent-diagonal magic square ......................................17
17 - An order-9 bimagic square with unusual features .......................19
18 - An order-5 and an order-6 bordered square.................................21
19 -The continuous nature of a pandiagonal magic square.................21
20 -Two order-3 magic squares and their complements .....................23
21 - An order-6 magic square and its complement .............................24
22 - An order-5 pandiagonal and its complement. pair pattern...........24
23 - An order-4 complete cube of binary digits ..................................25
24 - 12 order-3 magic squares form an order-12 composition ...........27
25 - Two order-4 magic squares formed from the above....................27
iii
Tables and Illustrations
26 - 2 order-5 magic square that dont obey the rule for Bordered....28
27 - A hypercube of order-3 (a cube) showing the coordinates. .........30
28 - Crosmagic Quadrant pattern for order-9 and order-13 ................32
29 - Diamagic Quadrant pattern for order-9 and order-13. .................34
30 - Diametrically equidistant pairs in even and odd orders...............35
31 - A magic square in ternary, decimal 0-8 and decimal 1 to 9.........36
32 - Digital-root magic squares with digital roots of 3, 6 and 9. ........36
33 - Divide magic square from a multiply magic square. ...................37
34 - An order-4 domino magic square. ...............................................37
35 - An order-7 magic square using a complete set of dominoes. ......38
36 - Six doubly-even magic squares in one. ......................................39
37 - The 12 Dudeney groups. .............................................................40
38 - An order-7 magic square with embedded orders 3 and 4. ..........41
39 - Table Summary of magic squares count...................................42
40 - Table Summary of magic stars count. ......................................43
41 - Index # 6 of the 36 essentially different pandiagonals. ...............44
42 - An order-4, doubly-even, within a singly-even order-6. .............45
43 - An order-3 magic square with 2 expansion bands. ......................46
44 - Franklins order-8 magic square..................................................49
45 - Some Franklin order-8 patterns. ..................................................50
46 - Two magic Generalized parts. .....................................................52
47 - Exponential geometric magic square. .........................................53
48 - Ratio geometric magic square. ...................................................53
49 - Order-4 Latin, Greek and Graeco-Latin squares. ........................54
50 - A bipartite anti-magic graph........................................................54
iv
Tables and Illustrations
51 - A bipartite super-magic graph. ....................................................55
52 - An order-4 supermagic graph and magic square .........................55
53 - An order-5 star and two isomorphic tree-planting graphs. ..........56
54 - Orders 3 and 4 Heterosquare. ......................................................57
55 - Horizontal and vertical steps in an order-3 magic square............58
56 - This non-normal order-4 magic square does not use integers. ....59
57 - The first 3 order-4 magic squares. ...............................................60
58 - Star A shows the name of the cells, Star B is solution # 1. .........60
59 - The first six order-6 magic stars in tabular form. ........................61
60 - An order-5 with inlaid diamond and even corner numbers. ........62
61 - Comparison of order-4 Inlaid and Bordered magic squares. .......62
62 - The primary intermediate square for order-5...............................63
63 - Pandiagonal order-5 and solution set...........................................64
64 - An order-9 iso-like magic star. ....................................................66
65 - The order-9 diamagic square for the above star. .........................67
66 - An order-8 magic star isomorphic to an order-5 square. .............68
67 - Four aspects of the IXOHOXI magic square...............................69
68 - An order-8 with two half-board re-entrant Knight Tours. ...........71
69 - The two re-entrant knight tour paths for the above square. .........72
70 - Variations of a numerical Latin square........................................73
71 - Bergholts general form for order-4. ...........................................74
72 - A solution set and the resulting magic square. ............................75
73 - An order-7 Lozenge magic square. .............................................76
74 - Lringmagic Quadrant pattern for order-9 and order-13...............76
75 - An order-6 multiply magic square is 22 ply and 32 ply. .......77
v
Tables and Illustrations
76 - Two magic circles........................................................................78
77 - An order-3 basic normal magic cube...........................................79
78 - Number of lower hyperplanes within a given hypercube. ...........80
79 - An order-4 semi-pandiagonal and its magic line diagram. ..........81
80 - A 3 x 9 magic rectangle with correct diagonals...........................82
81 - An order-5 pandiagonal square with special numbers. ................83
82 - An order-9 pandiagonal magic square that is also 32-ply............83
83 - An order-12 pattern b normal magic star.....................................84
84 - An order-11 pattern b normal magic star.....................................85
85 - A summary of some magic star facts...........................................86
86 - Order-10 magic stars, Type S and Type T...................................87
87 - The 3-D order-3 magic star. ........................................................89
88 - Comparison of magic squares cubes and tesseracts. ...................90
89 - A magic tesseract shown with Hendricks projection...................91
90 - The above magic tesseract in tabular form. .................................92
91 - Order-3 hypercube comparison. ..................................................93
92 - Magic triangular regions, orders 4 and 6. ....................................94
93 - Transforming a 2-d magic star to 3-D cube and octahedron. ......95
94 - An order-4 magic square mapped to a tetrahedron.....................95
95 - An order-8 most-perfect magic square. .......................................97
96 - An order-4 multiplication magic square and its reverse. .............98
97 - # of segments in n-agonals for dimensions 2, 3, and 4 ................99
98 - Table - Hypercube has both a general and a specific meaning. 100
99 - Four different types of number squares .................................... 101
100 - This bordered magic square consists of two odd orders ......... 102
vi
Tables and Illustrations
101 - Two order-3 number squares and magic squares....................104
102 - An add magic square with an inlaid multiply magic square105
103 - A pentagram of five magic diamonds.....................................106
104 - This ornamental magic star consists of two interlocked stars.107
105 - An order-3 cube showing coordinates and line paths. ............108
106 - L. S. Frierson overlapping square, orders 3, 5, 7 and 9. .........109
107 - A. W. Johnson, Jr. bordered palindromic magic square. ........110
108 - An essentially different pandiagonal and a derivative. ...........112
109 - An order-7 pan-magic star. ....................................................113
110 - The order-7 pandiagonal square used for the above star. .......114
111 - A pan-3-agonal magic cube of order 4. ..................................116
112 - Order-5 pandiagonal magic square showing parity pattern. ...117
113 - An order-4 square partitioned into cells. ................................117
114 - Collisons order-14 patchwork magic square. .......................118
115 - An Order-7 with inlaid order-5 pandiagonal magic square. ...120
116 - Values of the corner cells of the order-16 perfect tesseract...121
117 - Order-5 and order-11 perfect prime squares...........................123
118 - Two of 18 order-4 perimeter-magic triangles.........................124
119 - A perimeter-magic order-5 pentagon and order-3 septagon. 124
120 - Three of the 5 anti-magic Octahedrons. .................................125
121 - Plusmagic Quadrant pattern for order-9 and order-13............126
122 - Minimum starting prime for consecutive primes squares.......127
123 - A. W. Johnson, Jr.s bordered, order-8 prime number square.127
124 - Two order-5 prime stars, minimal and consecutive primes. ...128
125 - Pythagorean magic squares, all order-4, Sc2 = sa2 + Sb2. ....130
vii
Tables and Illustrations
126 - Pythagorean magic squares, orders 3, 4, 5; Sc = sa + Sb.......130
127 - The eight opposite corner pairs of a magic tesseract. .............131
128 - Eight order-13 quadrant magic patterns ................................132.
129 - Even order crosmagic and lringmagic patterns, order 8, 12 ...133.
130 - This order-13 quadrant square is 14 times quadrant magic ....134.
131 - Some patterns of the order-13 quadrant magic square............135.
132 - A regular order-4 magic square from a Graeco-Latin square .137.
133 - Three representations of an order-4 magic square..................138.
134 - Continuous and separate patterns for Order-8 magic stars .....139.
135 - First four solutions, orders 8A and 8B....................................140.
136 - Two traditional tesseract projections. .....................................141
137 - The modern Hendricks projection.........................................141
138 - Order-4 reverse magic square pair..........................................142
139 - A principal reversible square with 2 of its 16 variations. .......143
140 - The 3 principal reversible squares of order-4. ........................144
141 - Number of Most-perfect magic squares .................................145
142 - The Sagrada magic square sums to 33....................................147
143 - The Sator word magic square. ................................................148
144 - An order-7 associated, and thus self-similar, magic square. ..149
145 - An order-4 self-similar magic square that is not associated. ..149
146 - An order-7 semi-magic square of squares. .............................150
147 - Two semi-pandiagonal magic squares. ...................................151
148 - An order-9 serrated magic square...........................................152
149 - Patterns for the above serrated magic square..........................153
150 - 2 pandiagonal squares contained in the above square. ...........153
viii
Tables and Illustrations
151 - An order-6 simple, normal, singly-even magic square. ..........155
152 - An algebraic pattern for an order-6 pandiagonal. ...................156
153 - The order-6 pandiagonal in base 7 and base-10 .....................157
154 - Even and odd number placement in a square and cube. .........158
155 - Species # 1 of the order-3 magic tesseract..............................159
156 - Square of primes make a semi-magic square..........................160
157 - The smallest orthomagic arrangement of distinct squares. .....161
158 - Sringmagic quadrant pattern for order-9 and order-13. ..........161
159 - Order-4, standard position, and two disguises. .......................162
160 - An order-6 magic star and a disguised version of it. .............163
161 - Subtraction magic square. ......................................................164
162 - Hypercubes number of correct summations. .......................165
163 - Symmetrical cells in even and odd order magic squares. .......166
164 - Two Kravitz Talisman squares. ..............................................167
165 - Transforming an associated magic square to a pandiagonal...168
166 - # of parallel segmented triagonals for orders 3 to 10..............170
167 -Collisons order-5 pandiagonal upside-down magic square. ...171
168 - The smallest possible consecutive primes order-3..................172
169 - The smallest possible consecutive primes type-2 order-3 ......172
170 - An order-8A weakly magic star..............................................173
171 - An unorthodox use of wrap-around........................................174
ix
7 47 11 40 6 49 15
48 31 18 13 12 45 8
4 26 33 30 41 14 27
16 29 22 25 28 21 34
23 36 9 20 17 24 46
42 5 38 37 32 19 2
35 1 44 10 39 3 43
PREFACE 1.
With the increasing popularity of the World Wide Web has come an explosive increase in published material on magic squares and cubes.
As I look at this material, I can appreciate how it is expanding our knowledge of this fascinating subject. However, frequently an author comes up with a new idea (or what he thinks is a new idea) and defines it using a term that has been in other use, in some cases, for hundreds of years.
On the other hand, because the subject is growing so fast, it is important that new words and phrases be defined and publicized as quickly as possible. For these reasons, in the winter of 1999 I decided to research this subject and publish a glossary on my Web page.
While this book is an attempt to standardize definitions, unfortunately not all magic square hobbyists will have a copy of this book at hand. Therefore, I suggest that when using a term not too well known, an attempt be made to clarify its meaning.
After posting the result to my site, I also printed it as a booklet for my personal reference. Because John Hendricks has been a good source of information for me, I sent him a copy of the booklet as a courtesy gesture. He then suggested I publish an expanded version of this in book form. I was immediately interested, and when he graciously accepted the request to serve as co-author, I decided, with his knowledge and experience to support me, I could do it.
What definitions have been included in this book is arbitrary. We have tried to include the more popular terms by drawing on a wide range of resources. Inevitably, with a book of this nature, personal preferences enter the picture. I am sure that every person reading this book will say to himself at some point why did he bother putting that item in, or why that illustration, or what about.
In any case I have worked on the assumption that a picture is worth a thousand words, and so have kept the descriptive text to a minimum. I have tried, when picking the illustrations, to find items of additional interest besides just referring to the particular term being defined. Hopefully, this will encourage the use of the book for browsing as well as for reference.
xi
Where I felt it would be appropriate, I have included a source reference. Where a definition appears or is used in a variety of sources, no mention is made of the source unless one particular location is especially informative. Where a term or definition is primarily (or solely) the work of one author, his work is cited as the source.
To add to the usefulness of this reference, I have in many cases, included relevant facts or tables of comparisons.
In the definitions text, bold type indicates a term that has its own definition.
This book uses m to indicate order (of magic squares, cubes, etc) and n to indicate dimension. This is the terminology used by Hendricks in his writings where so much of the work involves dimensions greater then 2. For magic stars, because all work is in two dimensions, the traditional n will continue to be used for the order.
Unless I specifically indicate otherwise, all references to magic squares mean normal (pure) magic squares composed of the natural numbers from 1 to m2. Likewise for cubes, tesseracts, etc. Normal magic stars use the numbers from 1 to 2n.
A special thanks to John Hendricks for the support and encouragement he has given me on this project.
Writing and publishing this book is a first venture for me. My hope is that it will prove to be an informative and a worthy reference on this fascinating subject.
Harvey D. Heinz
July 2005
It is now time to print a second run of this book. I have corrected the mistakes I am aware of that appeared in the first run.
The only other changes are minor variations in wording, or slight elaborations where space permitted.
H.D.H
xii
PREFACE 2.
The analogy with squares and cubes is not complete, for rows of numbers can be arranged side-by-side to represent a visible square, squares can be piled one upon another to make a visible cube, but cubes cannot be so combined in drawing as to picture to the eye their higher relations.
Magic Squares and Cubes, by W.S. Andrews, Dover Publication.
Faced with that, I proceeded anyway. Many professors, even today, teach the wrong model of the tesseract The problem has to do with partitioning the tesseract into cells, so that numbers can be assigned to various cells & coordinate positions.
In 1950, I sketched the first magic tesseract.. Nobody would look at it. Andrews had said it was impossible. I did not have a chance to look into it again for about five years and was on Gimli Airforce Station during a cold winter with not much else to do. So, I managed to make a 5- and 6-dimensional magic hypercube of order 3. I reasoned that if the establishment would not look at the magic tesseract, then they might look at the higher dimensional hypercubes. However, it was not until I was in Montreal before a mathematician from Seattle, home for Christmas, heard about me and wished to see my magic hypercubes. As he looked over it all, he said, This stuff has got to be published.
He phoned a friend at McGill University. The next thing I knew, I got a reprint order form for my article The Five- and Six-Dimensional Magic Hypercubes of Order 3 which was published in the Canadian Mathematical Bulletin, May 1962..
There were many hurdles to overcome with terminology and symbolism. A simple concept such as a row of numbers is not so simple in six-dimensional space. One runs out of names row, column, pillar, post, file, rank,then what? So, the customary practice for higher dimensional spaces is to number the coordinate axes x1, x2, xixn . Thus, I coined 1-row, 2-row, 3-row, ,,,, n-row.. There were both dimension and order to be taken into account now, so I used n for dimension and m for order. This was to be n-dimensional magic hypercubes of order m.
xiii
The old guard , adept in cubes, had just finished coining long diagonal and space diagonal and here I had to put a halt to that because for 4, 5, 6, 7, 8 dimensions you could not very well have short diagonal, long diagonal and longer than that diagonal. I noticed that when one traversed the square that 2 coordinates always changed on a diagonal as you moved along it. Three coordinates changed for the space diagonals of the cube, while only two coordinates changed for the facial diagonals. Therefore, it was clearly in order to talk in terms of 2-agonal, 3-agonal, 4-agonal, , n-agonal depending upon how many coordinates change as you move along one of them. This means that triagonal, quadragonal, etc. were born and were a most logical solution to the problem..
One of the greatest challenges of all, was the concept of a perfect cube. As a boy, I learned that the four-space diagonals of a cube were required as well as all rows, columns and pillars to sum a constant magic sum. It was accepted that facial diagonals alone would be the requirement for a perfect cube. Eventually, Benson and Jacoby made a magic cube that had all broken triagonals and all broken diagonals summing the magic sum in every cross-section of the cube. It was both pandiagonal and pantriagonal. Thus, it was perfect.
Not until I made the perfect tesseract of order 16 and the 5-dimensional perfect magic hypercube of order 32 did I realize that perfect means all planar cross-sections are pandiagonal magic squares and all hypercubes have everything summing the magic sum
Planck had shown that the order of the hypercube had to be 2n or more before one could have pandiagonal squares with every cross-section. However, not until I actually made one did the point become clear. So the definition of perfect is upgraded. Through every cell on the hypercube there are (3n-1)/2 different routes that must sum the magic sum.
Over the years, it has been my pleasure to participate in the development of mathematics and to offer what I can on the subject.
John R. Hendricks
xiv
Algorithm 1
AAAA Algorithm
A step-by step procedure for solving a problem by hand or by using a computer.
Algebraic pattern
A generalized magic square, cube, tesseract, hypercube, or border, etc. using algebraic digits for the numbers. A pattern is used extensively for making inlays. See Solution Set.
Almost-magic Stars
A magic pentagram (5-pointed star), we now know, must have 5 lines summing to an equal value. However, such a figure cannot be constructed using consecutive integers. Charles Trigg calls a pentagram with only 4 lines with equal sums but constructed with the consecutive numbers from 1 to 10, an almost-magic pentagram.
C. W. Trigg, J. Recreational Mathematics, 29:1, 1998, pp.8-11, Almost Magic
Pentagams
Marin Trenkler (Safarik University, Slovakia) has independently coined the phrase almost-magic, but generalizes it for all orders of stars. His definition: If there are numbers 1, 2, , 2n located in a star Sn (or Tn) so that the sum on m 2 lines is 4n + 2, on the others 4n + 1 and 4n + 3, we call it an almost-magic star. See Magic stars type T for information on Sn and Tn.
Marin Trenkler, Magicke Hviezdy (Magic stars), Obsory Matematiky, Fyziky a
Informatiky, 51(1998).
2 Magic Square Lexicon: Illustrated
.. Almost-magic Stars
NOTE that by Trenklers definition, the order-5 almost-magic star has only 3 lines summing correctly. Triggs order-5 (the only order he defines) requires 4 lines summing the same.
Neither author has defined almost-magic for higher order stars.
NOTE2: This book will retain the customary n as the order for magic stars but use m to indicate the order of magic squares, cubes, etc, leaving n free to indicate dimension.
1 Two types of almost magic stars.
This Trigg Almost-magic order-5 star has 4 lines which sum to 24, and 1 line to 14. This Trenkler Almost-magic order-5 star has 3 lines which sum to 22, 1 line to 21 and 1 line to 23.
H.D.Heinz, http://www.geocities.com/~harveyh/trenkler.htm
C. W. Trigg, J. Recreational Mathematics, 29:1, 1998, pp.8-11, Almost Magic Pentagams
1
4
9
105 3 6
2
8
7
1
4
910
5
2 8
S5A
7
6
3
Trigg Trenkler
Alphamagic square 3
Alphamagic square
2 - The Alphamagic square of order-3.
Spell out the numbers in the first magic square. Count the letters in each number word and make a second magic square with these integers. Lee Sallows discovered this magic square oddity in 1986.
Lee Sallows, Abacus 4, 1986, pp28-45 & 1987 pp20-29
Anti-magic graphs
See Graphs anti-magic
Anti-magic squares
An array of consecutive numbers, from 1 to m2, where the rows, columns and two main diagonals sum to a set of 2(m + 1) consecutive integers.
Anti-magic squares are a sub-set of heterosquares. Joseph S. Madachy, Mathemaics On Vacation, pp 101-110. (Also JRM 15:4, p.302)
5 22 18
28 15 2
12 8 25
4 9 8
11 7 3
6 5 10
Five twenty- two eighteen
Twenty- eight f if t een two
Twelve eight twenty- f ive
4 Magic Square Lexicon: Illustrated
.. Anti-magic squares
65
34 5 8 20 9 22 64
2 15 5 13 35 19 23 13 10 2 67
16 3 7 12 38 21 6 3 15 25 70
9 8 14 1 32 11 18 7 24 1 61
6 4 11 10 31 12 14 17 4 16 63
33 30 37 36 29 68 69 60 62 66 71
3 - An order-4 and an order-5 anti-magic square.
Note that in each case, the sums of the lines form a consecutive series.
In 1999, John Cormie, a graduate mathematics student at the University of Winnipeg, did a research project on this subject. He developed several methods of constructing these squares for both odd and even orders.
Order
(m)
Magic
squares
Anti-
magic
squares
1 1 0
2 0 0
3 1 0
4 880 299,710
5 275,305,224 ?
4 - Number of anti-magic squares. Cormie & Lineks anti-magic square page is at
http://www.uwinnipeg.ca/~vlinek/jcormie/index.html
Anti-magic stars 5
Anti-magic stars
A normal magic star diagram, but instead of each line of 4 numbers summing to a constant, each line has a different sum. If the sums consist of consecutive numbers, the star is anti-magic; if the sums are not consecutive, the star is a heterostar.
The illustration shows two of the 2208 possible order-5 anti-magic stars. Note that there can be no normal magic stars of order-5, that is those using the integers 1 to 10. The smallest series possible is 1 to 12 with no 7 or 11.
5 - Two order-5 anti-magic stars. C. Trigg, J. Recreational Mathematics, 10:3, 1977, pp 169-173, Anti-magic
pentagrams.
Arithmetic magic squares
Sometimes used to refer to squares that have a magic sum, especially to differentiate from geometric magic squares (Andrews).
Arrays
An array is an orderly arrangement of a set of cardinal numbers, algebraic symbols, or other elements into rows, columns, files, or any other lines.
1
72 4
5
8
10
3
9
6
1
7
24
5
8 10
39
6
6 Magic Square Lexicon: Illustrated
.. Arrays
NOTE. For these purposes, the arrays used for magic squares, cubes and hypercubes would be narrowed down to square and rectangular ones like matrices and their cubic and higher dimensional equivalents. An array may also be a variable in a computer program. For these purposes, it would be the storage location for the magic square, cube, etc.
Aspect
An apparently different but in reality only a disguised version of the magic square, cube, tesseract, star, etc. It is obtained by rotations and/or reflections of the basic figure. Once one has a hypercube of any dimension, through mirror images and rotations one can view the hypercube in many ways. There are: A = (2n) n! ways of viewing a hypercube of dimension n.
Dimension (n) Name Aspects 2 square 8 3 cube 48 4 tesseract 384 5 hypercube 3840
In counting the number of any given type of hypercube, one can count all, including the aspects (the long count); or only the basic ones.
2 7 6 4 9 2 8 3 4 6 1 8
9 5 1 3 5 7 1 5 9 7 5 3
4 3 8 8 1 6 6 7 2 2 9 4
Original Rotate 90 Rotate 180 Rotate 270
4 3 8 8 1 6 6 7 2 2 9 4
9 5 1 3 5 7 1 5 9 7 5 3
2 7 6 4 9 2 8 3 4 6 1 8
Vertical reflections of squares immediately above
6 - The 8 aspects of a magic square.
.. Aspect 7
.. Aspect
Where n is the order (number of points) of a magic star there are 2n aspects for each star. NOTE that with rectilinear magic arrays, the number of aspects is determined by the dimension. With magic stars (which are normally only 2 dimensions) the number of aspects is determined by the order.
See Isomorphisms.
Associated magic cubes, tesseracts, etc.
Just as the one order-3 magic square is associated, so to are the 4 order-3 magic cubes and the 58 order-3 magic tesseracts. In fact, all order-3 magic hypercubes are associated.
All associated magic objects can be converted to another aspect by complementing each number (the self-similar feature). This figure is another aspect of the cube shown in basic magic cube.
7 - One of four order-3 magic cubes, all of which are associated.
Notice that the two numbers on each side of the center number sum to 28 which is 33 + 1. Go to Tesseract to see an order-3 associated 4-dimensional hypercube.
J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9, p 59.
17
6
241
15 19 8
26 10
23 3 16
7 14 21
12 25 5
18 22 2
20 9 13
4 11 27
8 Magic Square Lexicon: Illustrated
Associated magic square
A magic square where all pairs of cells diametrically equidistant from the center of the square equal the sum of the first and last terms of the
series, or m2 + 1 for a pure magic square. These number pairs are said to be complementary. This type of magic square is often referred to as a symmetrical magic square.
The center cell of odd order associated magic squares is always equal to the middle number of the series. Therefore the sum of each pair is equal to 2 times the center cell. In an order-5 magic square, the sum of the 2 symmetrical pairs plus the center cell is equal to the constant, and any two symmetrical pairs plus the center cell sum to the constant. i.e. the two pairs do not have to be symmetrical to each other.
In an even order magic square the sum of any m/2 symmetrical pairs will equal the constant (the sum of the 2 members of a symmetrical pair is equal to the sum of the first and last terms of the series).
8 - Three associated magic squares.
The order-3 associated magic square with each pair symmetrical summing to 32 + 1. The order-4 associated magic square with each pair symmetrical summing to 42 + 1 The order-5 associated magic square with each pair symmetrical summing to 52 + 1 .
As with any magic square, each associated magic square has 8 aspects due to rotations and reflections. any associated magic square can be converted to another aspect by complementing each number (the self-similar feature).
1 15 24 8 17
16 9 5 4 23 7 16 5 14
2 7 6 3 6 10 15 20 4 13 22 6
9 5 1 2 7 11 14 12 21 10 19 3
4 3 8 13 12 8 1 9 18 2 11 25
.. Associated magic square 9
.. Associated magic square
There are NO singly-even Associated pure magic squares. The one order-3 magic square is associative. There are 48 order-4 associative magic squares.
Order-5 is the smallest order having associated magic squares that are also pandiagonal.
1 42 80 64 24 35 46 60 17
50 61 12 5 43 75 68 25 30
72 20 31 54 56 13 9 38 76
8 37 78 71 19 33 53 55 15
48 59 16 3 41 79 66 23 34
67 27 29 49 63 11 4 45 74
6 44 73 69 26 28 51 62 10
52 57 14 7 39 77 70 21 32
65 22 36 47 58 18 2 40 81
9 An order-9 associated, pandiagonal, 32-ply magic square.
Associated magic squares are occasionally referred to as regular.
All associated magic squares are semi-pandiagonal but not all semi-pandiagonal magic squares are associated.
Note that while an associated magic square is also referred to as symmetrical, it should properly be called center symmetrical. There are magic squares (rare) that are symmetrical across a line.
W. S. Andrews, Magic squares & Cubes, 1917, p.266
Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0
Auxiliary square
See Intermediate square.
10 Magic Square Lexicon: Illustrated
BBBB Base
Also called the radix. The number of distinct single-digit numbers, including zero in a counting system.
When the radix. exceeds ten, then more symbols then the familiar 0, 1, 2, 3,...,9 are required. Sometimes Greek symbols are used. More common is a, b, c, etc.
It is often convenient to use a number base equal to the order of the magic figure. The number of digits making up the number in each cell are then equal to the dimension of the magic figure. When the magic square (or other figure) is completely designed, the numbers are then converted to base 10 (decimal) and 1 is added to each to make the series range from 1 to mn. where m = the order and n = the dimension.
Basic magic cube
There are 4 basic magic cubes of order-3. All four are associated (as is the single basic magic square). The squares in the three center planes of these four cubes is magic. Each of the four may be disguised to make 48 other (apparently) different magic cubes by means of rotations and reflections. These variations are NOT normally considered as new cubes by the magic square researcher for the purposes of enumeration. They may become important to use in determining degree of rarity by a statistician. See Relative frequency.
Which of the 48 aspects is considered to be the basic cube? Normally that is of no importance. However, if listing all the magic cube solutions of a given order, it is necessary to have a standard position. The basic cube is determined in this case by three conditions.
The bottom left corner is the smallest corner.
The value of the second cell of the bottom row is smaller then the first cell of the second row.
The value of the first cell of the second row is smaller then the second cell of the first pillar.
.. Basic magic cube 11
.. Basic magic cube
This definition is modeled after Frnicles (1693) definition for the basic magic square, but with one important difference. Frnicle considered the starting point for the magic square to be the top left corner. We have set the starting point for the cube (and higher dimensions) to be the bottom left corner. This is consistent with the modern coordinate system in geometry.
While this term will not get the same frequency of use that the equivalent term for magic squares does, it is presented here in the interest of completeness.
10 - One of 4 basic magic cubes.
This diagram is the same magic cube illustrated in Fig. 7. However, this one is normalized to the basic position. The other is a disguised version of this. Point of interest. There are 58 basic magic tesseracts of order-3. Each may be disguised to make 384 other (apparently) different magic tesseracts by means of rotations and reflections. There is only 1 basic magic square of order-3.
See Aspects and Basic magic square.
14
17 6
24
1 15
19
8
26
10
7
21
12
25
5
18
22
2
20
9
13
4
11
27
16
3
23
12 Magic Square Lexicon: Illustrated
Basic magic square
There is 1 basic magic square of order-3 and 880 of order-4, each with 7 variations due to rotations and reflections. These variations are called aspects or disguised versions.
In fact, any magic square may be disguised to make 7 other (apparently) different magic squares by means of rotations and reflections. These variations are NOT considered new magic squares for purposes of enumeration.
Any of the eight variations may be considered the basic one except for enumerating and listing them. Normally, which one you consider the basic one has no importance. However, for purposes of listing and counting, a standard must be defined. Refer to Aspect, Index and Standard Position for a more in-depth discussion of this subject. Basic magic squares are also known as Fundamental magic squares.
4 9 14 7
15 6 1 12
5 16 11 2
10 3 8 13
11 An order-4 Basic magic square.
This order-4 is basic because
The cell in the top left corner has the lowest value of any corner cell.
The cell to the right of this corner cell has a lower value then the first cell of row two.
It is now possible to put this magic square in an ordered list where it appears as # 695 of 880.
Bensen & Jacoby, New Recreations with Magic Squares, Dover, 1976, 0-486-23236-0, p. 123
Basic magic star 13
Basic magic star
All normal magic stars have n lines of 4 numbers that total to the magic sum. A magic star may be disguised to make 2n-1 apparently different magic stars where n is the order (number of points) of the magic star.
These variations are NOT considered new magic stars for purposes of enumeration. This is also referred to as a fundamental magic star. Any of these 2n variations may be considered the fundamental one. However, see Standard position, magic star and Index.
Figure 12 is a basic magic star because:
The point cell with the smallest value is at the top.
The value of the top right valley cell is lower then the top left one.
This star is number 31 in the indexed list of 80 order-6 basic magic stars.
Note: One of the authors (Heinz) has found all basic solutions for magic stars of order 5 to 11 (and some for higher orders).
12 - A basic order-6 magic star. H.D.Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm
10
1
8
7
952
12
11 4
6
3
14 Magic Square Lexicon: Illustrated
Basic magic tesseract
13 - Basic tesseract MT# 9 shown in the standard position.
One of the authors (Hendricks) has found and listed all 58 basic magic tesseracts of order-3. He lists and displays them in his book All Third-Order magic Tesseracts using the following indexing method:
Identify the lowest of the 16 corner numbers.
Take the adjacent number to this corner in each of the four lines. Rearrange these four numbers (if necessary) in ascending order and write them after the corner number.
In figure 13, the lowest corner number is 12 and the four numbers adjacent to it are 52, 61,62, and 76. Taking them in order; row, column, pillar and file they are already in ascending order, and, because the lowest corner is in the bottom left position we realize this tesseract is in the standard position. This definition is consistent with that of the Basic magic cube.
1276
35
522
69
5945
1948
471
16
6239
22
6138
2450
964
1481
2857
4026
7433
498
1178
3463
3723
5642
2554
168
1380
30
1873
3258
4421
476
70
66
65751
367710
297915
274155
67353
72546
1775
31
204360
row
column
pillar
file
.. Basic magic tesseract 15
.. Basic magic tesseract
The following table is from page 3 of All Third Order Magic Tesseracts. Column C shows the lowest corner number. Although 7. 8, and 9 sometimes serve as corners, they never serve as minimum corners.
In the table, Column # is for Magic Tesseract # (the order that Hendricks found the tesseract).
Column S is for species (based on arrangement of even and odd numbers).
Simply sorting the four numbers adjacent to the lowest corner insures that the tesseracts appear in index order (which is the order listed here).
For order-3 magic tesseracts, there are three species. For each of the 58 basic tesseracts, there are 384 aspects or disguised versions.
See Basic magic cube, Basic magic square, Index, Magic tesseract, Species and Standard position.
14 A disguised version of MT# 9 (previous figure).
12
76
35
52
2
69
59
45
19
48
4
71
16
62
39
22
61
38
24
50
9
64
14
81
28
57
40
26
74
33
49
8
11
78
34
63
37
23
56
42
25
54
1
68
13
80
30
18
73
32
58
44
21
47
6
70
66
65
7
51
36
77
10
29
79
15
27
41
55
67
3
53
72
5
46
17
75
31
20
43
60
16 Magic Square Lexicon: Illustrated
.. Basic magic tesseract
C Adjacent axis Numbers
# S C Adjacent axis Numbers
# S
1 45, 51, 53, 54 38 3 4 54, 66, 75, 80 53 3
1 45, 69, 71, 72 46 3 4 54, 72, 74, 75 55 3
1 51, 69, 77, 78 54 3 5 43, 46, 48, 54 51 3
1 53, 71, 77, 80 34 3 5 43, 64, 66, 72 43 3
1 54, 72, 78, 80 1 1 5 45, 46, 48, 52 45 3
2 43, 51, 52, 54 12 2 5 45, 64, 66, 70 39 3
2 43, 69, 70, 72 11 2 5 48, 72, 73, 79 19 2
2 45, 49. 52, 54 26 2 5 52, 72, 73, 75 21 2
2 46, 67, 70, 72 16 2 5 54, 66, 73, 79 20 2
2 49, 69, 76, 78 15 2 5 54, 70, 73, 75 6 2
2 51, 67, 76, 78 25 2 6 43, 46, 47, 52 28 2
2 52, 70, 78, 81 35 3 6 43, 64, 65, 70 29 2
2 52, 72, 76, 81 36 3 6 52, 70, 73, 74 10 3
2 52, 72, 78, 79 31 3 10 51, 59, 60, 78 30 2
2 54, 70, 76, 81 37 3 10 53, 59, 62, 80 27 2
2 54, 70, 78, 79 32 3 10 54, 60, 62, 81 49 3
2 54, 72, 76, 79 33 3 10 54, 60, 63, 80 47 3
3 43, 49, 52, 53 44 3 10 54, 62, 63, 78 48 3
3 43, 67, 70, 71 50 3 11 49, 58, 60, 78 42 3
3 52, 70, 76, 80 4 1 11 51, 58, 60, 76 40 3
3 53, 71, 77, 80 3 3 11 52, 61, 63, 78 5 2
4 45, 47, 48, 54 13 2 11 54, 60, 61, 79 7 2
4 45, 65, 66, 72 23 2 11 54, 61, 63, 76 17 2
4 47, 71, 74, 80 24 2 12 49, 58, 59, 76 22 2
4 48, 66, 80, 81 56 3 12 52, 61, 62, 76 9 3
4 48, 72, 74, 81 58 3 13 53, 56, 62, 74 41 3
4 48, 72, 75, 80 52 3 13 54, 57, 62, 75 2 2
4 53, 65, 74, 80 14 2 13 54, 57, 63, 74 18 2
4 54, 66, 74, 81 57 3 14 54, 57, 61, 73 8 3
15 - The 58 basic tesseracts of order-3 in indexed order (C = corner #, # = order discovered, S = species)
J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999,
0-9684700-2-5
Bent diagonals 17
Bent diagonals
Diagonals that proceed only to the center of the magic square and then change direction by 90 degrees. For example, with an order-8 magic square, starting from the top left corner, one bent diagonal would consist of the first 4 cells down to the right, then the next 4 cells would go up to the right, ending in the top right corner. Bent diagonals are the prominent feature of Franklin magic squares (which are actually only semi-magic because the main diagonals do not sum correctly). Most bent-diagonal magic squares (and all order-4) have the bent-diagonals starting and ending only in the corners. However, some (including the order-8 example shown here) may use wrap-around but must be symmetric around either the horizontal or the vertical axis of the magic square.
For example: In the following magic square, line;
1 + 55 + 64 +10 + 47 + 25 + 18 + 40 is correct.
58 + 9 + 7 +52 + 21 + 34 + 48 + 31 is correct.
4 + 54+ 57 +15 + 42 + 32 + 19 + 37 is correct.
40 + 58 + 9 + 7 +10 + 24 + 39 + 41 is incorrect ( because it is not centered horizontally).
1 16 57 56 17 32 41 40
58 55 2 15 42 39 18 31
8 9 64 49 24 25 48 33
63 50 7 10 47 34 23 26
5 12 61 52 21 28 45 36
62 51 6 11 46 35 22 27
4 13 60 53 20 29 44 37
59 54 3 14 43 38 19 30
16 - An order-8 bent-diagonal magic square.
This remarkable bent-diagonal pandiagonal magic square has many combinations of 8 numbers that sum correctly to 260.
18 Magic Square Lexicon: Illustrated
.. Bent diagonals
The Following are all magic:
8 Rows and 8 columns
16 Diagonals and broken diagonal pairs
8 Bent diagonals in each of 4 directions = 32 total
Any 2 x 4 rectangle (including wrap-around)
Any 2 x 2 square = 130 (including wrap-around)
Corners of any 3 x 3, 4 x 4, 6 x 6 or 8 x 8 square = 130 (including wrap-around) This magic square is from David H. Ahl Computers in Mathematics: A Sourcebook
of Ideas. Creative Computer Press,1979, 0-916688-16-X, P. 117
Bimagic cube
A magic cube that is still magic when all integers contained within it are squared.
Hendricks announced the discovery of the worlds first bimagic cube on June 9, 2000. It is order 25 so consists of the first 253 natural numbers. The magic sum in each row, column, pillar, and the four main triagonals is 195,325. When each of the 15,625 numbers is squared, the magic sum is 2,034,700,525.
The numbers at the eight corners are; 3426, 14669, 6663, 14200, 9997, 5590,12584, and 4491.
J. R. Hendricks, A Bimagic Cube Order 25, self-published 1999, 0-9684700-6-8 and
& H. Danielsson, Printout of A Bimagic Cube Order 25, 2000
Bimagic square
If a certain magic square is still magic when each integer is raised to the second power, it is called bimagic. If (in addition to being bimagic) the integers in the square can be raised to the third power and the resulting square is still magic, the square is then called a trimagic square. These squares are also referred to as doublemagic and triplemagic. To date the smallest bimagic square seems to be order 8, and the smallest trimagic square order 32.
Benson & Jacoby, New Recreations in Magic Squares, Dover, 1976, 0-486-23236-0,
pp 78-92
.. Bimagic square 19
.. Bimagic square
1 23 18 33 52 38 62 75 67
48 40 35 77 72 55 25 11 6
65 60 79 13 8 21 45 28 50
43 29 51 66 58 80 14 9 19
63 73 68 2 24 16 31 53 39
26 12 4 46 41 36 78 70 56
76 71 57 27 10 5 47 42 34
15 7 20 44 30 49 64 59 81
32 54 37 61 74 69 3 22 17
17 - An order-9 bimagic square with unusual features.
This special order-9 bimagic square was designed by John Hendricks in 1999. Each row, column, both diagonals and the 9 numbers in each 3 x 3 square sum to 369. If each of the 81 numbers are squared, the above combinations all sum to 20,049. Different versions of this bimagic square along with theory of construction appear in Bimagic Squares.
Aale de Winkel reports, based on John Hendricks digital equations, that there are 43,008 order-9 bimagic squares.
J. R. Hendricks, Bimagic Squares: Order 9 self-published 1999, 0-9684700-6-8
e-mail of May 14, 2000
Bordered magic square
It is possible to form a magic square (of any odd or even order) and then put a border of cells around it so that you get a new magic square of order m + 2 (and in fact keep doing this indefinitely). Each element of the inside magic square (order-3 or 4) must be increased by 2m + 2, with the remaining numbers (low and high) being placed in the border.
Or to put it differently, there must be (m2 -1)/2 lowest numbers
and their complements (the highest numbers) in the border where m
2 is the order of the square the border surrounds. This applies to each border.
20 Magic Square Lexicon: Illustrated
.. Bordered magic square
The outside border is called the first border and the borders are numbered from the outside in.
When a border (or borders) is removed from a Bordered magic square, the square is still magic (although no longer normal). Any (or all) borders may be rotated and /or reflected and the square will still be magic. The Bordered Magic Square is similar but not identical to Concentric and Inlaid magic squares.
Orders 5 and 6 are the two smallest orders for which you can have a bordered magic square.
Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0, pp 26-33 W. S. Andrews, Magic squares & Cubes, 1917
There are 2880 basic order-5 bordered magic squares (not counting the 7 disguised versions of each). There are 328,458,240 different basic order-6 bordered magic squares. See Enumeration for more on this and order-6.
J. R. Hendricks, Magic Square Course, 1991, unpublished, pp 85-98
M. Kraitchik, Mathematical Recreations., Dover Publ. , 1942, 53-9354, pp 166-170
All bordered magic squares show a consistent relationship between the sum of the numbers in each border and the value of the center cell (or in the case of even order the sum of the center 4 cells. Notice that here we number the borders from the inside out.
For the order-3 square below: value of center cell = 13 sum of border 1 = 1 x 8 x 13 = 104 sum of border 2 = 2 x 8 x 13 = 208 next border if there was one would be 3 x 8 x center cell.
For the order-4 square below: value of center 4 cells = 74 sum of border 1 = 3 x 74 = 222 sum of border 2 = 5 x 74 = 370 next border if there was one would be 7 x sum of center 4 cells.
This feature also applies even if the number series is not consecutive, such as prime number magic squares
.. Bordered magic square 21
.. Bordered magic square
1 34 33 32 9 2
3 4 18 21 19 29 11 18 20 25 8
25 12 11 16 1 30 22 23 13 16 7
6 17 13 9 20 6 17 12 26 19 31
24 10 15 14 2 10 24 21 15 14 27
7 22 8 5 23 35 3 4 5 28 36
18 - An order-5 and an order-6 bordered square.
Broken diagonal pair
Two short diagonals that are parallel to but on opposite sides of a main diagonal and together contain the same number of cells as are contained in each row, column and main diagonal (i.e. the order). These are sometimes referred to as pan-diagonals, and are the prominent feature of Pandiagonal magic squares.
J. L. Fults, Magic Squares, 1974
10 19 3 12 21 10 19 3 12 21 10 19 3 2 11 25 9 18 2 11 25 9 18 2 11 25
24 8 17 1 15 24 8 17 1 15 24 8 17
16 5 14 23 7 16 5 14 23 7 16 5 14
13 22 6 20 4 13 22 6 20 4 13 22 6
10 19 3 12 21 10 19 3 12 21 10 19 3
2 11 25 9 18 2 11 25 9 18 2 11 25
24 8 17 1 15 24 8 17 1 15 24 8 17 16 5 14 23 7 16 5 14 23 7 16 5 14 13 22 6 20 4 13 22 6 20 4 13 22 6
19 -The continuous nature of a pandiagonal magic square.
Notice how the two parts of the broken diagonal 24, 5, 6, 12, 18 of the center pandiagonal magic square may be considered joined to make a complete line of m (in this case 5) numbers.
See Modular space where the broken diagonals become continuous. See Pandiagonal, Pantriagonal, etc., for more on n-dimensional.
22 Magic Square Lexicon: Illustrated
CCCC Cell
The basic element of a magic square, magic cube, magic star, etc. Each cell contains one number, usually an integer. However, it can hold a symbol or the coordinates of its location.
There are m2 cells in a magic square of order m, m3 cells in a magic cube, m4 cells in a magic tesseract, 2n cells in a normal magic star, etc. (Note the use of n for order of the magic star.)
RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition,
p.194
Column
Each vertical sequence of numbers. There are m columns of height m in an order-m magic square.
See Orthogonals for a cube illustrating all the lines.
Complementary numbers
In a normal magic square, the first and last numbers in the series are complementary numbers. Their sum forms the next number in the series (m2 + 1). All other pairs of numbers which also sum to m2 + 1 are also complementary.
If the numbers are not consecutive (the magic square is not normal), the complement pair total is the sum of the first and the last number. Sets of two complementary numbers are sometimes called complementary pairs. Associated magic squares have the complementary pair numbers symmetrical around the center of the magic square.
Please see Associated and Complementary magic squares.
Following are complementary magic squares. Because they are associated, the middle number in the series is its own complement.
.. Complementary numbers 23
.. Complementary numbers
8 1 6 1669 199 1249
3 5 7 619 1039 1459
4 9 2 829 1879 409
A B
2 9 4 409 1879 829
7 5 3 1459 1039 619
6 1 8 1249 199 1669
20 -Two order-3 magic squares and their complementary magic squares.
Each number in the bottom magic square is the complement of the number in the top magic square.
Each pair sums to 10 which is 1+ 9 (the first and last numbers of the series). Also, because the series consist of 1 to m2 (this is a normal magic square), the sum is m2 + 1.
Each pair in this prime number magic square sums to 2078 which is 199 + 1879 (the first and last numbers of the series).
Complementary magic squares
A well know method of transforming one magic square into another of the same order, is to simply complement each number. If the magic square is associated, the resulting square is self-similar. That is, it is the same as the original but rotated 90. If the complement pairs are symmetrical across either the horizontal or vertical axis, the resulting complementary magic square is also self-similar but reflected horizontally or vertically respectively. Robert Sery refers to this process as Complementary Pair Interchange (CPI).
Complimenting works even if the numbers are not consecutive. See Complimentary numbers, figure 20B (above) and figure 21.
24 Magic Square Lexicon: Illustrated
.. Complementary magic squares
1 35 36 7 29 42 49 15 14 43 21 8
47 17 12 45 19 10 3 33 38 5 31 40
6 30 41 2 34 37 44 20 9 48 16 13
43 21 8 49 15 14 7 29 42 1 35 36
5 31 40 3 33 38 45 19 10 47 17 12
48 16 13 44 20 9 2 34 37 6 30 41
21 - An order-6 magic square and its complementary.
An order-6 pandiagonal magic square using 36 of the numbers from1 to 49. (It is impossible to form an order-6 pandiagonal magic square using consecutive numbers.) It is transformed to another order-6 pandiagonal by subtracting each number from 50 (the sum of the first and last numbers).
Complementary pair patterns
The two numbers that together sum to the next number in the series are a complement pair. Join these two numbers with a line. The resulting pattern may be used as a method of classifying the magic squares of a given order. See Dudeney groups for more on this.
There is only 1 pattern for order-3 and 12 patterns for order-4. No one has yet figured out how many patterns there are for order-5 or higher.
22 - An order-5 pandiagonal magic square and its complementary pair pattern.
1 7 19 25 13
20 23 11 2 9
12 4 10 18 21
8 16 22 14 5
24 15 3 6 17
Complete projection cubes 25
Complete projection cubes
On the sci.math newsgroup Dec. 2, 1996 K. S. Brown asked the following:
Do there exist 4x4x4 cubes of binary digits such that the projection onto each face of the cube gives the decimal digits from 0 to 15 (binary 0000 to 1111)?
To state it differently; can each of the four rows of each of the four horizontal planes of an order-4 cube be filled with binary digits such that when read in either direction, the decimal integers 0 to 15 are obtained? And to make the cube complete, can this be done so the result is valid for each of the other two orthogonal sets of four planes?
The answer is yes! Dan Cass found such a cube (shown here), and posted it on Dec. 10, 1996
23 - An order-4 complete cube of binary digits.
K. S. Brown shows on his web site at http://www.seanet.com/~ksbrown/kmath353.htm that this solution is unique.
1 0
1 01
0
1
0
10
10
10
101 01010
10 1
01
0 10
10 10
10 10
1
0
10
101 0
10 10
10 1
0 101
0
10
10
10
26 Magic Square Lexicon: Illustrated
Components
A magic square, or cube, may be broken down into parts which are called components. Some authors use the method of components to build their magic squares. One of these methods which is most meaningful is to show a magic square broken into two squares where the various digits are separated, as shown below:
4 9 2 1 2 0 0 2 1
3 5 7 + 0 1 2 + 2 1 0 + 1
8 1 6 2 0 1 1 0 2
Magic Square Decimal System
3s digit ternary number system
Units digit ternary # system
where the first square is a magic square in the decimal number system. The second square is a Latin square in the ternary number system of the 3s digit. The third square, is a Latin square in the ternary number system for the units digit and is a rotation of the second square, The number one at the end balances the equation.
Composition magic square
It is simple to construct magic squares of order mn (m times n) where m and n are themselves orders of magic squares. For a normal magic square of this type, the series used is from 1 to (mn)2. An order 9 composite magic square would consist of 9 order 3 magic squares themselves arranged as an order 3 magic square and using the series from 1 to 81. An order 12 composite magic square could be made from nine order 4 magic squares by arranging the order 4 squares themselves as an order-3 square, (or sixteen order 3 magic squares arranged as an order 4 magic square). In either case, the series used would be from 1 to 144.
The example order-12 composition magic square was constructed out of 16 order-3 magic squares. They are arranged as per the numbers in the order-4 pattern. Numbers used are consecutive from 1 to 144. The magic sums of these order-3 squares in turn form another order-4 magic square.
Composition magic square 27
Composition magic square
8 1 6 125 118 123 62 55 60 107 100 105
3 5 7 120 122 124 57 59 61 102 104 106
4 9 2 121 126 119 58 63 56 103 108 101
134 127 132 35 28 33 80 73 78 53 46 51
129 131 133 30 32 34 75 77 79 48 50 52
130 135 128 31 36 29 76 81 74 49 54 47
89 82 87 44 37 42 143 136 141 26 19 24
84 86 88 39 41 43 138 140 142 21 23 25
85 90 83 40 45 38 139 144 137 22 27 20
71 64 69 98 91 96 17 10 15 116 109 114
66 68 70 93 95 97 12 14 16 111 113 115
67 72 65 94 99 92 13 18 11 112 117 110
24 - Twelve order-3 magic squares form an order-12 composition magic square with a magic sum of 870.
1 14 7 12 15 366 177 312
15 4 9 6 393 96 231 150
10 5 16 3 258 123 420 69
8 11 2 13 204 285 42 339
A. B.
25 - Two order-4 magic squares, from the order-12 composition magic square.
The order-4 pandiagonal magic square used as a pattern to place the order-3 squares.
The magic sum of each of the order-3 squares form an order 4 pandiagonal magic square with the magic sum 870.
28 Magic Square Lexicon: Illustrated
Concentric magic square
Traditionally this has been another name for Bordered magic squares. It has also been used for Inlaid magic squares.
But Collison found several order-5 magic squares (2 are shown below) by computer search. They depend upon being able to carry over to the next column excess from the units column, which is not normally taken into account in constructing bordered or inlaid magic squares.
24 22 1 12 6 6 24 15 12 8
8 19 5 15 18 21 22 1 16 5
3 9 13 17 23 3 7 13 19 23
10 11 21 7 16 17 10 25 4 9
20 4 25 14 2 18 2 11 14 20
26 - Two order-5 magic square that dont obey the rule for Bordered or Inlaid magic squares.
See Bordered and Inlaid magic squares. J. R. Hendricks, The Magic Square Course, self-published 1991, p 88
Congruence, Congruence equation.
See Modular Arithmetic.
Constant (S)
The sum produced by each row, column, and main diagonal (and possibly other arrangements). This value is also called the magic sum.
The constant (S) of a normal magic square is (m3+m)/2
If the magic square consists of consecutive numbers, but not starting at 1, the constant is (m3+m)/2+m(a-1) where a equals the starting number and m is the order. If the magic square consists of numbers with a fixed increment, then S = am + b(m/2)(m2-1) where a = starting number and b = increment. See Series.
.. Constant (S) 29
.. Constant (S)
For a normal magic square, S = m(m2+1)/2.
For a normal magic cube, S = m(m3+1)/2.
For a tesseract S = m(m4+1)/2.
In general; for a n-dimensional hypercube S = m(mn+1)/2.
For a normal magic star, when n is the order, S = 4n + 2.
NOTE:Hendricks always uses m to indicate the order and reserves n to indicate the dimension of the magic object. See Magic sum and Summations for more information and comparison tables.
Continuous magic square
See Pandiagonal magic square.
Coordinates
A set of numbers that determine the location of a point (cell) in a space of a given dimension..
A coordinate system is normally not required for most work in magic squares. But, for 3-dimensions, or higher, a coordinate system is essential. Customarily, (x, y, z) are the coordinates for 3-dimensional space and (w, x, y, z) for 4-dimensional space.
Coordinates have been handled by Hendricks in a slightly different manner. For dimensions less then ten, only one digit is required per dimension, so the brackets and commas are not required, thus permitting a more concise and space saving notation.
For 2-dimensional space, the x-axis is in its customary position left-to-right. The y-axis is also in its usual position but is reversed. This is because of the way Frnicle defined the Basic magic square.
The origin is considered as being at the top-left, rather than the bottom left of the square.
Rows are parallel to the x-axis and columns are parallel to the y-axis. Pillars are parallel to the z-axis,
30 Magic Square Lexicon: Illustrated
.. Coordinates
For three dimensions and higher a customary left-to-right x-axis; a front-to-back y-axis; and, a bottom to top z-axis is used. Then, the cube is ready to be presented in its usual presentation from the top layer down to the bottom layer. When working in 5- and 6-dimensional space and higher, it becomes more expedient to use numbered axes and the coordinates become:
(x1, x2, x3,,xi,.., xn)
for an n-dimensional magic hypercube. All xi must lie between 1 and m inclusive which is the order of the hypercube. There really is no origin, or coordinate axes in Modular space. So, we simply define them as passing through the coordinate (1,1,1,,1). A row would be parallel to the x1 axis, a column parallel to the x2 axis, a pillar parallel to the x3 axis , a file parallel to the x4 axis and we run out of names. Hence, we number the various kinds of rows according to which axis they are parallel to and say that an i-row lies parallel to the xi axis.
See modular space and orthogonal. See Journal of Recreational Mathematics, Vol.6, No. 3, 1973 pp.193-201. Magic
Tesseracts and n-Dimensional Magic Hypercubes.
Coordinates could also be considered as the indices (subscripts) of a variable array in a computer program used to store a magic square, cube, etc., being generated or displayed.
27 - A hypercube of order-3 showing the coordinates.
333
232
313
323
133
213
223
233
311
321
331
322
332
212
222
312
131
123
211
231
221
111
112
121
122
132
113
Coordinate iteration 31
Coordinate iteration
Coordinate iteration is a systematic process of moving at unit intervals from 1 coordinate location to the next coordinate location along a line in modular space.
Moving along any orthogonal line requires changing only one coordinate digit, but moving along an n-agonal requires changing all n coordinate digits. See orthogonal for an illustration.
Coordinates could also be considered as the indices (subscripts) of a variable array in a computer program used to store a magic square, cube, etc., being generated or displayed. In this case, iterating one subscript at a time would permit storing (or retrieving) the value of the cells, as you move along the line.
See Pathfinder.
Corners
The corners are those cells where the lines that form the edges of the hypercube meet. They have coordinates which are either 1 or m, where m is the order of the hypercube. See Coordinates and Magic tesseract for illustrations.
There are 2n corners in a hypercube of dimension n.
Counting
How many magic squares, cubes, tesseracts, etc. are there?
There is a long count and a short count. Seasoned researchers in magic squares and cubes feel there is a duplication involved when you count rotations and reflections of a known square. Statisticians wishing to study the probability of a magic square, require to know them all.
The number of variations, called aspects, due to rotations and reflections varies with the dimension of the object. For a magic square (dimension 2) there are 8 aspects. So, for example, the researcher says there are 880 order-4 magic squares and the statistician claims there are 7040.Close attention must be paid to which number is being referred to. Normally, the count of magic squares considers the basic squares only.
32 Magic Square Lexicon: Illustrated
.. Counting
Then there are unique magic squares that can be transformed to a range of magic squares. For example, order-5 has 3600 basic pandiagonal magic squares. They are derived from 36 essentially different squares that form 100 squares each by simple transformations. Furthermore, these 36 squares in turn can be formed from one square, using more complicated component substitution methods.
Bensen & Jacoby, New Recreations with Magic Squares, Dover, 1976, 0-486-23236-0, p. 125
For magic stars, the same consideration applies. However, here the number of aspects changes with the order of the star and is equal to 2n.
When talking about the number of magic objects, say magic squares, normally what is meant is the number of basic magic squares. However, keep the above considerations in mind and determine what is meant by the context.
Crosmagic
An array of m cells in the shape of an X that appears in each quadrant of an order-m quadrant magic square.
See Quadrant magic patterns and Quadrant magic square.
28 - Crosmagic Quadrant pattern for order-9 and order-13. H.D.Heinz, http://www.geocities.com/~harveyh/quadrant.htm
Cyclical magic squares
This is another, though seldom used, term for regular pandiagonal magic squares. See Regular and Irregular
Cyclical permutations 33
Cyclical permutations
A pandiagonal magic square may be converted to another by simply moving one row or column to the opposite side of the square. For example, an order-5 pandiagonal magic square may be converted to 24 other pandiagonal magic squares. Any of the 25 numbers in the square may be brought to the top left corner (or any other position) by this method.
In 3-dimensional space, there can be cyclical permutations of a plane face of a cube to the other side of the cube. Pantriagonal magic cubes remain magic when this is done. In 4-dimensional space, entire cubes may be permuted. The Panquadragonal magic tesseract has this feature.
See also Transformations and Transposition.
J. R. Hendricks, American Mathematical Monthly, Vol. 75, No.4, p.384.
1
2
3
4
5 6
7
8 9
10
11
12
13
34 Magic Square Lexicon: Illustrated
DDDD Degree (of a magic square.)
The power, or exponent to which the numbers must be raised, in order to achieve a magic square. The term is used in Bimagic and Trimagic squares.
Diabolic magic square
See Pandiagonal magic square.
Diagonal
Occasionally called a 2-agonal. See n-agonal. Also see Broken, Leading, Long, Main, Right, Opposite Short, Short.
Diagonal Latin square
A Latin square with the extra condition that both the diagonals also contain one of each symbol. See Latin square.
Diamagic
An array of m cells in the shape of a diamond that appears in each quadrant of an order-m quadrant magic square. For order-5 diamagic and crosmagic are the same.
29 - Diamagic Quadrant pattern for order-9 and order-13 H.D.Heinz, http://www.geocities.com/~harveyh/quadrant.htm
Diametrically equidistant 35
Diametrically equidistant
A pair of cells the same distance from but on opposite sides of the center of the magic square. Other terms meaning the same thing are skew related and symmetrical cells. The two members of a complementary pair in an associated (symmetrical) magic square are diametrically equidistant.
X Y Z
Z Y
Y Z Z Y
X X
30 - Diametrically equidistant pairs X, Y and Z shown in an even and an odd order array.
Digital equations
One uses modular arithmetic in finding the various digits that comprise a number at a specific location in a magic square, or cube. If the digits of a number can be expressed as a function of their coordinate location, then the equation(s) describing the relationship can be called the digital equations. They are sometimes referred to as congruence equations or modular equations. For example: If at coordinate location (1, 3) we wish to find the number and it is known that:
D2 x + y (mod 3)
AndD1 2x + y + 1 (mod 3
then the two digits D2 and D1 can be found.
D2 1 + 3 4 1 (mod 3)
AndD1 2 + 3 + 1 6 0 (mod 3)
So the number 10 is located at (1, 3).
36 Magic Square Lexicon: Illustrated
.. Digital equations
10 is in the ternary number system because the modulus is 3. If the coordinate system is shown by:
(1,3), (2,3), (3,3) (1,2), (2,2), (3,2) (1,1), (2,1), (3,1)
and the numbers are all calculated as assigned to their respective locations, then one achieves the magic square below in the ternary number system which is then converted to decimal and finally 1 is added to each number.
10 22 01 3 8 1 4 9 2
02 11 20 2 4 6 3 5 7
21 00 12 7 0 5 8 1 6
31 - The magic square in ternary, decimal 0-8 and dec. 1 to 9. J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998, 0-9684700-0-9
Digital-root magic squares
A digital root magic square is a number square consisting of sequential integers starting from 1 to m2 and with each line sum equal to the same digit when reduced to its digital root. This type of magic square was investigated by C. W. Trigg in 1984. He found that there are 27 basic squares of this type for order-3, nine each of digital root 3, 6, and 9.
1 3 8 3 4 8 5 1 3
2 4 6 1 5 9 4 6 8
9 5 7 2 6 7 9 2 7
32 - Digital-root magic squares with digital roots of 3, 6 and 9. C. W. Trigg, J. Recreational Mathematics, 17:2, 1978-79, pp.112-118, Nine-digit Digit-root Magic Squares.
Disguised magic square 37
Disguised magic square
See Aspect and Basic magic square.
Division magic square
Construct an order-3 multiply magic square, then interchange diagonal opposite corners. Now, by multiplying the outside numbers of each line, and dividing by the middle number, the constant is obtained. See Geometric magic square for information and illustrations on multiply magic squares.
12 1 18 3 1 2
9 6 4 9 6 4
2 36 3 18 36 12
A B
33 - Divide magic square. A. multiply magic square, B. resulting divide magic square. RB = 5
Domino magic square
It is possible to arrange in the form of magic squares, any set of objects that contain number representations. Playing cards and dominos are two types that are often used. This order-4 requires duplicate dominoes and duplicate numbers but has four different numbers on each line.
34 An order-4 Domino magic square.
38 Magic Square Lexicon: Illustrated
.. Domino magic square
Here is an order-7 magic square that uses using a complete set of dominoes from double-0 to double-6.
35 - An order-7 magic square using a complete set of dominoes. RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition,
p.214.
Double magic square
See bimagic magic square.
Doubly-even
The order (side) of the magic square is evenly divisible by 4. i.e. 4, 8, 12, etc. It is probably the easiest to construct.
The order-8 normal pandiagonal magic square shown here contains an order-4 pandiagonal (not normal) magic square in each quadrant and also an order-4 semi-pandiagonal magic square in the center.
.. Doubly-even 39
.. Doubly-even
1 58 15 56 17 42 31 40
16 55 2 57 32 39 18 41
50 9 64 7 34 25 48 23
63 8 49 10 47 24 33 26
3 60 13 54 19 44 29 38
14 53 4 59 30 37 20 43
52 11 62 5 36 27 46 21
61 6 51 12 45 22 35 28
36 - Six doubly-even magic squares in one. J. R. Hendricks, The Magic Square Course, self-published 1991, p 205
Dudeney group patterns
When each pair of complementary numbers in a magic square are joined by a line, the resulting combination of lines forms a distinct pattern which may be called a complementary pair pattern.
H. E.Dudeney introduced this set of 12 patterns to classify the 880 order 4 magic squares. There are 48 group I, which are all pandiagonal. The 48 group III are associated All of groups II, III, IV and V are semi-pandiagonal, as are 96 of the 304 group VI. The other 448 order-4 magic squares are all simple.
Patterns I to III are fully symmetrical around the center point of the square. However, be aware that patterns IV to X also appear
rotated 90for some of the basic magic squares. Pattern XI also
appears rotated 180 and 270 while pattern 12 appears rotated
90 and 180.
40 Magic Square Lexicon: Illustrated
.. Dudeney group patterns
37 - The 12 Dudeney groups.
H.E.Dudeney, Amusements in Mathematics, 1917, p 120
Jim Moran Magic Squares, 1981, 0-394-74798-4 (lots of material) Bensen & Jacoby, New Recreations with Magic Squares, 1976, 0-486-23236-0
I II III IV
V VI VII VIII
IX X XI XII
Embedded magic square 41
E Embedded magic square
38 - An order-7 magic square with embedded orders 3 and 4.
A magic square (or other magic object) that has another magic square contained in it.
The order-7 is a simple, normal magic square. The order-4 (square cells) is simple and the order-3 (round cells) is associated. Neither of these two squares are normal. This square was designed by David Collison.
PS. There is a bonus serrated magic diamond contained in this square! Can you see it? Rotate the square 45 to the right. The first 2 lines (of 6) of the diamond formation are 49, 1 and 7, 22,23, 48. Lines of two sum 50, lines of 3 = 75, of 4 = 100 and lines of 6 = 150.
J. R. Hendricks, The Magic Square Course, self-published 1991, pp 46-47
9 1 37 48 38 26 16
49 10 23 47 4 18 24
15 22 36 11 29 42 20
7 33 44 25 43 17 6
35 46 14 2 21 27 30
19 32 8 3 28 40 45
41 31 13 39 12 5 34
42 Magic Square Lexicon: Illustrated
Enumeration magic squares
This summary is concerned only with the basic number of each order. To determine the long count,. multiply the number shown here by the aspects. See Relative frequency.
Order Square Cube Tesser-
act
5-D 6-D
3 1 4 58 2992 543328
4 880 ? ? ? ?
5 275305224 ? ? ? ?
Aspects 8 48 384 3840 46080
39 - Table Summary of magic squares count.
For Bordered magic squares of order-6 there are 81 different borders for the order-4 nucleus. There are 4! (factorial) different ways of permuting non-corner elements along a row edge. The same with a column edge. And there are 8 ways of rotating and reflecting the borders. So there are 81 x 4! x 4! x 8 borders and 880 basic order-4 nucleus. So there are 328,458,240 different order-6 basic bordered magic squares. For order-5, the total is much smaller because there are only 10 different borders, the variations are (3!)2 and there is only one nucleus order-3 magic square, making only 2880 basic bordered magic squares.
J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999, 0-9684700-2-5, The Magic Square Course, self-published 1991, pp. 85-98.
Enumeration magic stars
This summary is concerned only with the basic number of each order. The basic count will always be an even number because all magic stars appear as a complement pair. To determine the long count. Multiply the number shown here by the aspects.
For magic stars, the order is the same as the number of points. Unlike magic squares, cubes, etc., regardless of the order of a normal magic star, there is always four numbers in a line. As the order increases, the variety of patterns increases. I have identified these patterns arbitrarily as A, B, C and D, which covers up to order-12.
.. Enumeration magic stars 43
.. Enumeration magic stars
The total counts arrived at was obtained by exhaustive computer search. The validity of Heinzs algorithm was confirmed by the counts for orders 6, 7 and 8 matching those obtained by other researchers in 1965. Also, all solutions found were successfully matched in complementary pairs.
Order Aspects A B C D
6 12 80 -- -- --
7 14 72 72 -- --
8 16 112 112 -- --
9 18 3014 1676 1676 --
10 20 115552 10882 10882 --
11 22 53528 75940 75940 53528
12 24 >500000 826112 >500000 >500000
40 - Table Summary of magic stars count.
There are 12 basic magic stars of order-5 using integers from 1 to 12 with no 7 and 11. There are no normal magic stars of order-5. There are 168 Trigg type almost-magic stars (order-5) in 14 groups of 12. (Trigg). JRM 29(1), 1998 There are 2208 anti-magic stars order-5 (Trigg). JRM 10(3), 1977
Martin Gardner, Mathematical Carnival, Alfred A. Knoff, 1975, 0-394-49406-7 H.D.Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm
Essentially different
There are 36 basic essentially different order-5 pandiagonal magic squares each of which have 99 variations by permutations of the rows, columns and diagonals. Rotations and reflections are not included in this count, so the total number of order-5 pandiagonal magic squares is 3600 time 8 rotations and reflections.. A magic square is essentially different from any other when it cannot be generated from another essentially different magic square by:
any combination of the 1-3-5-2-4 transformation,
the interchange of rows/columns with diagonals, and/or cyclical permutations.
44 Magic Square Lexicon: Illustrated
.. Essentially different
Which of the set of 100 in each case is the essentially different one, is determined by
The number in the top left-hand corner is 1,
The number in the cell next to the 1 in the top row is less then any other number in the top row, in the left hand column or in the diagonal containing the 1, and
The number in the left-hand column of the second row is less then the number in the left-hand column of the last row.
There are 3 essentially different pandiagonal magic squares of order-4 each of which produces 16 variations..
There are 129,600 essentially different pandiagonal magic squares of order-7 each of which produces 294 variations (ther
