By W. S. Andrews

**Read or Download MAGIC SQUARES AND CUBES, Second Edition, Revised and Enlarged PDF**

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**Additional info for MAGIC SQUARES AND CUBES, Second Edition, Revised and Enlarged**

**Sample text**

This is a general but not a universal relationship between A and B primary squares. 3. The sum of the series I to 6 is 21 and the sum of every column in both A and B 6 X 6 primary squares must also be 2I. MAGIC SQUARES. 4. The sum of every column in a 6 X 6 root square must be 90, and under these conditions it follows that the sum of every column of a 6 X 6 magic square which is formed by the combination of a primary square with a root square must be III (21 90 = III). + 5. With the necessary changes in numbers the above rules hold good for all sizes of A and B primary squares and root squares of this class.

Referring to the 6 X 6 A primary square shown in Fig. 75, it will be noted that the two corner diagonal columns contain the numbers I to 6 111 arithmetical order, starting respectively from the upper and lower left hand corner cells, and that the diagonal columns of the B primary square in Fig. f 6 2 If J 2 6 I (. 3 4 2 6 j J 4- J 6 2 J / 2 ,;- 2 I 2 4 '" J S '2" 4 oS 6 6 I J Fig. 75. " S ~- / {J Fig. 76. from the two upper corner cells. The numbers 111 the two corner diagonal columns are subject to many arrangements which differ from the above but it will be unnecessary to consider them in the present article.

7 d, a, a, d, b, " d 2- / a, b, c, d, or e. J' e, aa, or cc. b, d, cc, or ee. b, d, cc, or ee. e, aa, or cc. c, aa, dd, or ee. a, b, c, d, or e. Fig. 87. Fig. 89 is the A printary square developed from column No. I in Fig. 88, and Fig. 90 is the B primary square made by reversing the direction of all horizontal and perpendicular columns of Fig. 89. Substituting root numbers for the primary numbers in Fig. 90, and 42 MAGIC SQUARES. adding these root numbers to the primary numbers in Fig. 91. The latter will be found identical with the square which may be written out directly from diagrams in Fig.