in each half-rank or half-file, and 514 in each quarter-rank or quarter-file. Furthermore all complementary pairs are balanced about the center. The alternative arrangement shown in Fig. 3 makes each of the small squares perfect in itself, with every rank, file and corner diagonal footing up 514 and complementary pairs balanced about the 3968 130 131 3965 3964 134 135 3961 193 3903 3902 196 197 3899 3898 200 137 3959 3958 140 141 3955 3954 144 3896 202 203 3893 3892 206 207 3889 145 3951 3950 148 149 3947 3946 152 3888 210 211 3885 3884 214 215 3881 3944 154 155 3941 3940 158 159 3937 217 3879 3878 220 221 3875 3874 224 160 3938 3939 157 156 3942 3943 153 3873 223 222 3876 3877 219 218 3880 3945 151 150 3948 3949 147 146 3952 216 3882 3883 213 212 3886 3887 209 3953 143 142 3956 3957 139 138 3960 208 3890 3891 205 204 3894 3895 201 136 3962 3963 133 132 3966 3967 129 3897 199 198 3900 3901 195 194 3904 161 3935 3934 164 165 3931 3930 168 3872 226 227 3869 3868 230 231 3865 3928 170 171 3925 3924 174 175 3921 233 3863 3862 236 237 3859 3858 240 3920 178 179 3917 3916 182 183 3913 241 3855 3854 244 245 3851 3850 248 185 3911 3910 188 189 3907 3906 192 3848 250 251 3845 3844 254 255 3841 3905 191 190 3908 3909 187 186 3912 256 3842 3843 253 252 3846 3847 249 184 3914 3915 181 180 3918 3919 177 3849 247 246 3852 3853 243 242 3856 176 3922 3923 173 172 3926 3927 169 3857 239 238 3860 3861 235 234 3864 3929 167 166 3932 3933 163 162 3936 232 3866 3867 229 228 3870 3871 225 center. As in the other arrangement the squares in each vertical or horizontal row combine to make cubes whose "straight" dimensions all have the right summation. In addition the new form has the two plane diagonals of each original square (eight for each cube), but sacrifices the four cubic diagonals in each cube. In lieu of these we find a complete set of "bent diagonals” (“Franklin") like those described for the magic cube of six in The Monist for July, 1909. If the four squares in either diagonal of the large figure be piled up it will be found that neither cubic diagonal nor vertical column is correct, but that all diagonals of vertical squares facing toward front or back are. Taken as a plane figure the whole group makes up a magic square of 16 with the summation 2056 in every rank, file or corner diagonal, half that summation in half of each of those dimensions, and one-fourth of it in each quarter dimension. 2. Let n=6, then S-3891.-With the natural series I... 1296 squares were constructed which combined to produce the six magic cubes of six indicated by the Roman numerals in Fig. 4. These have all the characteristics of the 6-cube described in The Monist of July last-108 "straight" rows, 12 plane diagonals and 24 "bent" diagonals in each cube, with the addition of 32 vertical-square diagonals if the squares are piled in a certain order. A seventh cube with the same features is made by combining the squares in the lowest horizontal row-i. e., the bottom squares of the numbered cubes. The feature of the cubic bent diagonals is found on combining any three of the small squares, no matter in what order they are taken. In view of the recent discussion of this cube it seems unnecessary to give any further account of it now. The whole figure, made up as it is of thirty-six magic squares, is itself a magic square of 36 with the proper summation (23346) for every rank, file and corner diagonal, and the corresponding fractional part of that for each half, third or sixth of those dimensions. Any square group of four, nine, sixteen or twenty-five of the small squares will be magic in all its dimensions. 3. Let n=8, then S-16388.-The numbers I...4096 may be arranged in several different ways. If the diagrams in Mr. Andrews's book be adopted we have a group of eight cubes in which rank, file, column and cubic diagonal are correct (and in which the halves of these dimensions have the half summation), but all plane diagonals are irregular. If the plan be adopted of constructing the small squares of complementary couplets, as in the 6-cube, the plane diagonals are equalized at the cost of certain other features. I have used therefore a plan which combines to some extent the advantages of both the others. It will be noticed that each of the small squares in Fig. 5 is perfect in that it has the summation 16388 for rank, file and corner diagonal (also for broken diagonals if each of the separated parts contain two, four or six-not an odd number of cells), and in balancing complementary couplets. When the eight squares are piled one upon the other a cube results in which rank, file, column, the plane diagonals of each horizontal square, the four ordinary cubic diagonals and 32 cubic bent diagonals all have S=16388. What is still more remarkable, the half of each of the "straight" dimensions and of each cubic diagonal has half that sum. Indeed this cube of eight can be sliced into eight cubes of 4 in each of which every rank, file, column and cubic diagonal has the footing 8194; and each of these 4-cubes can be subdivided into eight tiny 2-cubes in each of which the eight numbers foot up 16388. So much for the features of the single cube here presented. As a matter of fact only the one cube has actually been written out. The plan of its construction, however, is so simple and the relations of numbers so uniform in the powers of 8 that it was easy to investigate the properties of the whole 8+ scheme without having the squares actually before me. I give here the initial number of each of the eight squares in each of the eight cubes, leaving it for some one possessed of more leisure to write them all out and verify my statements as to the intercubical features. It should be remembered that in each square the number diagonally opposite the one here given is its complement, i. e., the number which added to it will give the sum 4097. Each of the sixty-four numbers given above will be at the upper left-hand corner of a square and its complement at the lower right-hand corner. The footings given are for these initial numbers, but the arrangement of numbers in the squares is such that the footing will be the same for every one of the sixty-four columns in each cube. If the numbers in each horizontal line of the table above be added they will be found to have the same sum: consequently the squares headed by them must make a cube as nearly perfect as the example given in Fig. 5, which is cube I of the table above. But the sum of half the numbers in each line is half of 16388, and hence each of the eight cubes formed by taking the squares in the horizontal rows is capable of subdivision into 4-cubes and 2-cubes, like our original cube. We thus have sixteen cubes, each with the characteristics described for the one presented in Fig. 5. If we pile the squares lying in the diagonal of our great square (starting with 1, 289, etc., or 2304, 2528, etc.) we find that its columns and cubic diagonals are not correct; but all the diagonals of its vertical squares are so, and even here the remarkable feature of the half-dimension persists. Of course there is nothing to prevent one's going still further and examining constructions involving the fifth or even higher powers, but the utility of such research may well be doubted. The purpose of this article is to suggest in sketch rather than to discuss exhaustively an interesting field of study for some one who may have time to develop it. WABASH COLLEGE. H. M. KINGERY. |