# Nasik magic hypercube

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A Nasik magic hypercube is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to ${\displaystyle S={\frac {m(m^{n}+1)}{2}}}$ where S = the magic constant, m = the order and n = the dimension, of the hypercube.

Or, to put it more concisely, all pan-r-agonals sum correctly for r = 1...n.

The above definition is the same as the Hendricks definition of perfect, but different from the Boyer/Trump definition. See Perfect magic cube

## Definitions

A Nasik magic cube is a magic cube with the added restriction that all 13m2 possible lines sum correctly to the magic constant. This class of magic cube is commonly called perfect (John Hendricks definition.). See Magic cube classes. However, the term perfect is ambiguous because it is also used for other types of magic cubes. Perfect magic cube demonstrates just one example of this.
The term nasik would apply to all dimensions of magic hypercubes in which the number of correctly summing paths (lines) through any cell of the hypercube is P = (3n- 1)/2

A pandiagonal magic square then would be a nasik square because 4 magic line pass through each of the m2cells. This was A.H. Frost’s original definition of nasik.
A nasik magic cube would have 13 magic lines passing through each of its m3 cells. (This cube also contains 9m pandiagonal magic squares of order m.)
A nasik magic tesseract would have 40 lines passing through each of its m4 cells.
And so on.

## History

In 1866 and 1878, Rev. A. H. Frost coined the term Nasik for the type of magic square we commonly call pandiagonal and often call perfect. He then demonstrated the concept with an order-7 cube we now class as pandiagonal, and an order-8 cube we class as pantriagonal.[1][2]
In another 1878 paper he showed another pandiagonal magic cube and a cube where all 13m lines sum correctly[3] i.e. Hendricks perfect.[4] He referred to all of these cubes as nasik as a respect to the great Indian Mathematician D R Kaprekar who hails from Deolali in Nasik District in Maharashtra, India. In 1905 Dr. Planck expanded on the nasik idea in his Theory of Paths Nasik. In the introductory to his paper, he wrote;

Analogy suggest that in the higher dimensions we ought to employ the term nasik as implying the existence of magic summations parallel to any diagonal, and not restrict it to diagonals in sections parallel to the plane faces. The term is used in this wider sense throughout the present paper.

— C. Planck, M.A.,M.R.C.S., The Theory of Paths Nasik, 1905[5]

In 1917, Dr. Planck wrote again on this subject.

It is not difficult to perceive that if we push the Nasik analogy to higher dimensions the number of magic directions through any cell of a k-fold must be ½(3k-1).

— W. S. Andrews, Magic Squares and Cubes, Dover Publ., 1917, page 366[6]

In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes. They specifically mentioned that these cubes contained 13m2 correctly summing lines. They also had 3m pandiagonal magic squares parallel to the faces of the cube, and 6m pandiagonal magic squares parallel to the triagonal planes.[7]

## References

1. ^ Frost, A. H., Invention of Magic Cubes, Quarterly Journal of Mathematics, 7,1866, pp92-102
2. ^ Frost, A. H., On the General Properties of Nasik Squares, QJM, 15, 1878, pp 34-49
3. ^ Frost, A. H. On the General Properties of Nasik Cubes, QJM, 15, 1878, pp 93-123
4. ^ Heinz, H.D., and Hendricks, J.R., Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0 pp 119-122
5. ^ Planck, C., M.A.,M.R.C.S., The Theory of Paths Nasik, 1905, printed for private circulation. Introductory letter to the paper.
6. ^ Andrews, W. S., Magic Squares and Cubes, Dover Publ. 1917. Essay pages 363-375 written by C. Planck
7. ^ Rosser, B. and Walker, R. J., Magic Squares: Published papers and Supplement, 1939. A bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4