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1 | (3) |
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4 | (2) |
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§ 3 Banal transformations of ordinary magic squares |
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6 | (2) |
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8 | (5) |
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§ 5 Main sources considered |
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13 | (3) |
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§ 6 Squares transmitted to the Latin West |
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16 | (5) |
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Chapter II Ordinary magic squares |
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21 | (1) |
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21 | (3) |
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§ 2 Method of diagonal placing |
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24 | (8) |
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24 | (1) |
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2 Discovery of this method |
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25 | (3) |
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28 | (2) |
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30 | (1) |
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5 Modifying the square's aspect |
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31 | (1) |
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§ 3 A method brought from India |
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32 | (2) |
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34 | (2) |
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§ 5 Use of the knight's move |
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36 | (3) |
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§ 6 Principles of these methods |
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39 | (5) |
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B Squares of evenly-even orders |
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44 | (1) |
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§ 7 The square of order 4 |
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44 | (1) |
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45 | (2) |
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§ 9 Exchange of subsquares |
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47 | (1) |
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48 | (2) |
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50 | (1) |
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§ 12 Principles of these methods |
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51 | (6) |
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§ 13 Filling according to parity |
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57 | (1) |
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58 | (2) |
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§ 15 Crossing the quadrants |
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60 | (3) |
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§ 16 Descent by the knight's move |
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63 | (6) |
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63 | (2) |
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2 Squares of higher orders |
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65 | (4) |
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§ 17 Filling pairs of horizontal rows |
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69 | (2) |
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§ 18 Four knight's routes |
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71 | (2) |
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73 | (1) |
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§ 20 Filling by knight's and bishop's moves |
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74 | (3) |
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§ 21 Filling according to parity |
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77 | (2) |
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§ 22 Filling the subsquares of order 4 |
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79 | (9) |
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85 | (3) |
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C Squares of evenly-odd orders |
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88 | (1) |
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§ 23 Exchanges in the natural square |
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88 | (3) |
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91 | (2) |
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§ 25 Principles of these methods |
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93 | (2) |
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95 | (7) |
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§ 27 Construction of a border |
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102 | (2) |
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§ 28 Method of the central square |
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104 | (3) |
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Chapter III Composite magic squares |
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A Equal subsquares displaying different sums |
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107 | (1) |
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§ 1 Composition using squares of orders larger than 2 |
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107 | (9) |
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§ 2 Composition using squares of order 2 |
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116 | (4) |
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B Equal subsquares displaying equal sums |
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120 | (1) |
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120 | (7) |
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C Division into unequal parts |
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127 | (1) |
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127 | (4) |
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131 | (10) |
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Chapter IV Bordered magic squares |
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§1 Preliminary observations |
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141 | (36) |
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142 | (1) |
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§ 2 Empirical construction |
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142 | (4) |
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§ 3 Grouping the numbers by parity |
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146 | (2) |
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§ 4 Placing together consecutive numbers |
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148 | (1) |
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149 | (1) |
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150 | (1) |
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§ 7 Mathematical basis of these general methods |
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151 | (7) |
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B Squares of evenly-even orders |
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158 | (1) |
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§ 8 Equalization by means of the first numbers |
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159 | (1) |
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160 | (1) |
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161 | (2) |
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§ 11 General principles of placing for even orders |
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163 | (4) |
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C Squares of evenly-odd orders |
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167 | (1) |
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§ 12 Equalization by means of the first numbers |
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167 | (1) |
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168 | (2) |
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170 | (1) |
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§ 15 Principles of these methods |
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171 | (6) |
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Chapter V Bordered squares with separation by parity |
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§ 1 The main square and its parts |
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177 | (2) |
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§ 2 Filling the inner square |
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179 | (1) |
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§ 3 Filling the remainder of the square by trial and error |
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180 | (2) |
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A Methodical filling of the oblique square |
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182 | (1) |
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§ 4 Completing the placing of odd numbers |
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183 | (1) |
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B Methodical placing of the even numbers |
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184 | (1) |
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§ 5 Situation after filling the oblique square |
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185 | (3) |
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1 Determining the number of cells remaining empty |
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185 | (1) |
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2 Determining the sum required |
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186 | (2) |
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§ 6 Rules for placing the even numbers |
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188 | (5) |
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1 First main equalization rule |
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190 | (1) |
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2 Second main equalization rule |
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191 | (1) |
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192 | (1) |
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§ 7 Case of the order n = 4t + 1 (with t < 2) |
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193 | (8) |
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193 | (2) |
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195 | (6) |
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§ 8 Case of the order n = 4t + 3 (with t < I) |
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201 | (7) |
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202 | (1) |
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202 | (6) |
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C Particular case of the order 5 |
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208 | (3) |
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Chapter VI Magic squares with non-consecutive numbers |
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Particular case: The given numbers form arithmetical progressions |
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1 The numbers form a single progression |
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211 | (1) |
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2 The numbers form n progressions |
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212 | (1) |
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3 Magic square with a set sum |
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213 | (1) |
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214 | (1) |
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General case: Squares with arbitrary given numbers |
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215 | (1) |
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215 | (4) |
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215 | (1) |
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2 The given numbers are in the first row |
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216 | (1) |
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3 The middle number is in the median lower cell |
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217 | (1) |
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4 The given numbers are in the diagonal |
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218 | (1) |
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5 The given numbers are in the middle row |
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218 | (1) |
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219 | (5) |
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1 The given numbers are in the first row |
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219 | (1) |
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2 The given numbers are in the second row |
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220 | (1) |
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3 The given numbers are in the diagonal |
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220 | (1) |
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4 The given numbers are in opposite lateral rows |
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221 | (1) |
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5 The given numbers are in the first two rows |
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222 | (2) |
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B Squares of evenly-even orders |
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224 | (1) |
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224 | (15) |
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224 | (2) |
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2 The given numbers are in the upper row |
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226 | (3) |
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3 The given numbers are in the second row |
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229 | (2) |
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4 The given numbers are in the end cells of the first row and the median of the second |
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231 | (1) |
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5 The given numbers are in the end cells of the first row and the median of the third |
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232 | (1) |
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6 The given numbers are in the diagonal |
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232 | (1) |
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7 The given numbers are in the corner cells |
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233 | (1) |
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8 Writing in the sum as a whole |
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234 | (1) |
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234 | (1) |
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b Filling according to `substance' |
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235 | (4) |
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239 | (9) |
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1 The given numbers are in the first row |
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239 | (1) |
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a Particular case: method of division |
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239 | (1) |
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239 | (1) |
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c Filling the second half |
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240 | (2) |
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2 The given numbers are in the first two rows, within the quadrants' diagonals |
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242 | (1) |
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3 The given numbers are in the first and third rows, within the diagonals |
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243 | (1) |
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4 The given numbers are in the first and fourth rows, within the diagonals |
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244 | (1) |
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5 Writing in the global sum |
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245 | (3) |
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C Squares of evenly-odd orders |
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248 | (1) |
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248 | (3) |
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1 The given numbers are in the first row |
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248 | (1) |
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2 The given numbers are equally distributed in the lateral rows |
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249 | (2) |
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§ 6 Squares of higher evenly-odd orders |
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251 | (2) |
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Chapter VII Other magic figures |
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253 | (12) |
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253 | (1) |
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253 | (1) |
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b Case of the order n -- 9 |
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254 | (1) |
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c Case of the order n -- 3 ·e; z, with z ≠ 3 |
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255 | (1) |
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d Case of the square order n ≠ z2, with z ≠ 3 |
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256 | (1) |
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e Case of the composite order n = t middote;, with t, z odd ≠ 3 |
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257 | (6) |
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263 | (2) |
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§ 2 Squares with one empty cell |
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265 | (5) |
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1 Squares of odd orders with central cell empty |
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265 | (1) |
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2 Case ot the square or order 3 |
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266 | (2) |
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3 Squares of evenly-even orders |
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268 | (2) |
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§ 3 Squares with divided cells |
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270 | (1) |
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271 | (2) |
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273 | (1) |
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274 | (3) |
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274 | (2) |
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276 | (1) |
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277 | (3) |
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277 | (1) |
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2 Both sides are evenly even |
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278 | (1) |
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3 One side evenly even and the other evenly odd |
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278 | (1) |
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4 Both sides are evenly odd |
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279 | (1) |
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280 | (1) |
| Appendices |
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281 | (22) |
| Bibliography |
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303 | (4) |
| Index |
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307 | |
Lisainfo e-raamatute kohta