A Magic Cube With 6n3 Cells

Abstract

An ordinary magic of the nth order is here defined as an arrangement of n3 different positive integers at the vertices of an n X n X n (three dimensional) lattice work in such a manner that the sum of the numbers in every row, every column, every file, and in each of the four main diagonals* shall be the same number S. When n is greater than three, it is also required that the sum along the diagonals of each horizontal layer shall have the value S. The number S will be called the characteristic of the magic cube. It may be noted in passing that earlier writers on this topic have been content with a definition of a magic which imposed no restriction on the sums along the diagonals of the horizontal layers. It seems to the author that the definition above is more appropriate and more interesting. As an indication of the possibilities of further development along the lines which will be suggested by this paper, it may be said that the writer has recently constructed ordinary magic cubes of orders six and ten which conform to the definition in the first paragraph. It is the purpose of this paper to explain the construction of a magic cube which will now be defined. Consider a as made up of n3 (of equal dimensions) arranged in layers of n2 cubelets each. On each face of every one of these cublets let there be written one of the integers from 1 to 6n3 inclusive, without repetition. Let it be required that the numbers written on the top faces of the cubelets form an ordinary magic cube. Likewise those on the bottom faces, those on the north faces, those on the south faces, those on the east faces, and those on the west faces. (In the four cases last mentioned, it is understood that the n X n arrays whose diagonals are required to have the sum S lie in vertical planes, instead of the horizontal planes as specified in the first paragraph.) It is also required that all of these ordinary magic cubes have the same characteristic. Such an arrangement of the first 6n3 positive integers will be called a generalized magic cube. A generalized magic can be obtained from an ordinary magic by multiplying all of the numbers by 6, and then judiciously adding to or subtracting from each of them a suitably chosen number from the set 0, 1, 2, 3, 4, 5, so as to obtain the proper combination of six magic cubes with no duplication of numbers. Such a generalized magic with n =4 was constructed by the author in 1931. In the present paper there is presented a generalized magic with n = 4 which has the additional property that the sum of the integers on each of the cubelets is equal to 1155. The manner in which this result was obtained will now be explained. First, a magic square was constructed by use of the first sixteen positive