By Martin I.M. Andrew

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**Example text**

All that remains to convert this cube to a normal eighth-order perfect magic cube is to convert it to base 10 and add the number I to the number in each cell. This method is much more general than it appears to be at first. Great liberties may be taken in the construction of the original natural-order cube. Let us consider the "units" digits first. It is not necessary to write the numbers in their natural 52 Magic Cubes: New Recreations order. Write the "units" digits 0 to 7, inclusive, in any orthogonal perpendicular to the left side.

F. A. P. Barnard, "Theory of Magic Squares and of Magic Cubes," in The Memoirs of the NationalAcademy of Science 4 (1888):209, 244-248. Magic Cubes: New Recreations 38 parallel to the front of the eighth-order magic cube), we shall construct from them Figure 5-5 (the left side of the cube), Figure 5-6 (the top of the cube), and Figure 5-7 (the two diagonal sections perpendicular to the left side of the cube, which show the space diagonals). Examination of these figures, which are to the base 8 (in every orthogonal, every main diagonal in the orthogonal sections, and every space diagonal, the sum of the "hundreds" and "tens" digits is 28 and the sum of the "units" digits is 36, as required), will show that they prove conclusively what Barnard claimed, namely, that the cube was a perfect eighth-order magic cube-and it was published over 100 years ago.

For the vertical columns, it is obvious that, in the first square filled, we have a series of terms of which the first is unity and the common difference is N. Of the terms of this series we have the first one-fourth and the last onefourth of the number in the first square on the left, the middle two-fourths being in the second on the right. By the properties of arithmetical series it follows that the sum of the antecedents on one side is equal to the sum of those on the other. Hence, when their consequents are written in, the sums of the entire columns will be equal, and, as before, will be severally expressible by N(N 3 + 1)/2.