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{{no footnotes, date=December 2014 A pantriagonal magic cube is a
magic cube In mathematics, a magic cube is the dimension, 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an ''n'' × ''n'' × ''n'' pattern such that the sums of the numbers on each row, ...
where all 4''m''2 pantriagonals sum correctly. There are 4 one-segment pantriagonals, 12(''m'' − 1) two-segment pantriagonals, and 4(''m'' − 2)(''m'' − 1) three-segment pantriagonals. This class of magic cubes may contain some simple magic squares and/or
pandiagonal magic square A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the squa ...
s, but not enough to satisfy any other classifications. The magic constant for magic cubes is ''S'' = ''m''(''m''3 + 1)/2. A proper pantriagonal magic cube has 7''m''2 lines summing correctly. It contains ''no'' magic squares. The smallest pantriagonal magic cube has order 4. A pantriagonal magic cube is the 3-dimensional equivalent of the pandiagonal magic square – instead of the ability to move a ''line'' from one edge to the opposite edge of the square with it remaining magic, you can move a ''plane'' from one edge to the other.


See also

*
Magic cube classes Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics. This new system is more precise in defining magic cubes. But possibly of more importance, it is consistent for all orders and all dimensions of ...
*
triagonal A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...


References

* Heinz, H.D. and Hendricks, J. R., Magic Square Lexicon: Illustrated. Self-published, 2000, 0-9687985-0-0. * Hendricks, John R., The Pan-4-agonal Magic Tesseract, The American Mathematical Monthly, Vol. 75, No. 4, April 1968, p. 384. * Hendricks, John R., The Pan-3-agonal Magic Cube, Journal of Recreational Mathematics, 5:1, 1972, pp51-52. * Hendricks, John R., The Pan-3-agonal Magic Cube of Order-5, JRM, 5:3, 1972, pp 205-206. * Hendricks, John R., Pan-n-agonals in Hypercubes, JRM, 7:2, 1974, pp 95-96. * Hendricks, John R., The Pan-3-agonal Magic Cube of Order-4, JRM, 13:4, 1980-81, pp 274-281. * Hendricks, John R., Creating Pan-3-agonal Magic Cubes of Odd Order, JRM, 19:4, 1987, pp 280-285. * Hendricks, J.R., ''Inlaid Magic Squares and Cubes'' 2nd Edition, 2000, 0-9684700-3-3. *
Clifford A. Pickover Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Researc ...
(2002). ''The Zen of Magic Squares, Circles and Stars''. Princeton Univ. Press. 0-691-07041-5 page 178.


External links

*http://www.magichypercubes.com/Encyclopedia/ Aale de Winkel: Magic Encyclopedia *http://members.shaw.ca/hdhcubes/cube_perfect.htm Harvey Heinz: Perfect Magic Hypercubes Magic squares