Editing Magic hypercube (section)

==Nasik magic hypercubes==
A '''''Nasik magic hypercube''''' is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to ''S'' = {{sfrac|''m''(''m''<sup>''n''</sup>+1)|2}}  where ''S'' is 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''. This definition is the same as the Hendricks definition of '''''perfect''''', but different from the Boyer/Trump definition.

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'' = {{sfrac|3<sup>''n''</sup> − 1|2}}.

A [[pandiagonal magic square]] then would be a ''nasik'' square because 4 magic line pass through each of the ''m''<sup>2</sup> cells. This was A.H. Frost’s original definition of nasik. A ''nasik'' magic cube would have 13 magic lines passing through each of its ''m''<sup>3</sup> cells. (This cube also contains 9''m'' pandiagonal magic squares of order ''m''.) A ''nasik'' magic tesseract would have 40 lines passing through each of its ''m''<sup>4</sup> 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''.<ref>Frost, A. H., Invention of Magic Cubes, ''Quarterly Journal of Mathematics'', 7,1866, pp92-102</ref><ref>Frost, A. H., ''On the General Properties of Nasik Squares'', QJM, 15, 1878, pp 34-49</ref> In another 1878 paper he showed another ''pandiagonal'' magic cube and a cube where all 13''m'' lines sum correctly<ref>Frost, A. H. ''On the General Properties of Nasik Cubes'', QJM, 15, 1878, pp 93-123</ref> i.e. Hendricks ''perfect''.<ref>Heinz, H.D., and Hendricks, J.R., ''Magic Square Lexicon: Illustrated'', 2000, 0-9687985-0-0 pp 119-122</ref>
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;
{{Blockquote|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<ref>Planck, C., M.A., M.R.C.S., ''The Theory of Paths Nasik'', 1905, printed for private circulation. Introductory letter to the paper.</ref>}}

In 1917, Dr. Planck wrote again on this subject.
{{Blockquote|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(3<sup>k</sup>-1).| W. S. Andrews, Magic Squares and Cubes, Dover Publ., 1917, page 366<ref>Andrews, W. S., Magic Squares and Cubes, Dover Publ. 1917. Essay pages 363-375 written by C. Planck</ref>}}

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 13''m''<sup>2</sup> correctly summing lines. They also had 3''m'' pandiagonal magic squares parallel to the faces of the cube, and 6''m'' pandiagonal magic squares parallel to the [[space diagonal|space-diagonal]] planes.<ref>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</ref>