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Almost-Magic Stars a Magic Pentagram (5-Pointed Star), We Now Know, Must Have 5 Lines Summing to an Equal Value

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Almost-Magic Stars a Magic Pentagram (5-Pointed Star), We Now Know, Must Have 5 Lines Summing to an Equal Value MAGIC SQUARE LEXICON: ILLUSTRATED 192 224 97 1 33 65 256 160 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 67 254 158 226 130 63 95 232 136 57 89 94 62 131 227 127 31 162 194 121 25 168 200 195 163 30 126 4 100 221 189 9 105 216 184 10 106 215 183 11 107 214 182 188 220 101 5 157 253 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 M AGIC SQUAR E LEX ICON : Illustrated 1 1 4 8 512 4 18 11 20 12 1024 16 1 1 24 21 1 9 10 2 128 256 32 2048 7 3 13 23 19 1 1 4096 64 1 64 4096 16 17 25 5 2 14 28 81 1 1 1 14 6 15 8 22 2 32 52 69 2048 256 128 2 57 26 40 36 77 10 1 1 1 65 7 51 16 1024 22 39 62 512 8 473 18 32 6 47 70 44 58 21 48 71 4 59 45 19 74 67 16 3 33 53 1 27 41 55 8 29 49 79 66 15 10 15 37 63 23 27 78 11 34 9 2 61 24 38 14 23 25 12 35 76 8 26 20 50 64 9 22 12 3 13 43 60 31 75 17 7 21 72 5 46 11 16 5 4 42 56 25 24 17 80 13 30 20 18 1 54 68 6 19 H. D. Heinz & J. R. Hendricks Magic Square Lexicon: Illustrated By Harvey D. Heinz & John R. Hendricks Published November 2000 By HDH 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 ....................................................................................... 181 Magic Square Bibliography..............................................A1-1 to A1-15 John Hendricks Bibliography .............................................A2-1 to A2-3 13 22 10 1 23 9 4 17 14 7 12 20 12b 24 3 2 16 21 11 6 15 8 5 18 19 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 don’t 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 - Franklin’s 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 - Bergholt’s 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
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