Original text of the Langman's perfect magic cube of order 7
by Harry Langman, "Play Mathematics", Hafner Publishing Company, 1962

Just as we have magic squares, we have magic cubes. We may divide a cube, for example, into 64 smaller cubicles, and place a number in each of these cubicles with the requirement that every four numbers in a straight line shall have the same sum. Or we may have a cubic wire arrangement where a number is attached to every crossing, with the same requirement. In the former illustration, we may imagine a number of beads occupying each cubicle.

We shall content ourselves with giving an illustration of a seven-cube constructed with consecutive integers:

(Look at the Excel file for the Plates II to VII)

Imagine these placed together in the order given so as to form a cube of 343 cells. Every plate parallel to a given face forms a magic square. There are 21 of these. Besides, the rectangular array common to a plane through each pair of opposite edges gives a magic square, thus yielding a total of 27 magic squares. Further, every seven numbers in a line parallel to an edge gives the same line total. There are 49x3 of these. There are, besides, 21x2 diagonals parallel to the faces of the cube. There are four 'solid diagonals', giving a total of 193 alignments with the characteristics sum 1204.

The mnemonic for the construction of this will be clear from the following: As is usual with magic squares, we consider the first column to follow the last column cyclically, and the first row to follow the last in similar fashion. (We might visualize something like a torus, cut and flattened out.) In the same manner we visualize the first plate as succeeding the last. With this convention in mind, following the 1 on Plate IV, the first seven numbers are entered in cyclic order.

(This figure was not in the original Langman's text,
and is added here for a better understanding)

Then we jump two spaces to the right on the next block to begin entering the next seven numbers on that in the same manner. This is continued till seven sets of seven numbers are placed. The next 49 numbers are placed in like manner, beginning just below the last of the previous set. This is continued till all numbers are placed.

Harry Langman

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