Editing Pantriagonal magic cube

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A '''pantriagonal magic cube''' is a [[magic cube]] where all 4''m''<sup>2</sup> pantriagonals sum correctly. There are 4 one-segment pantriagonals, 12(''m''&nbsp;&minus;&nbsp;1) two-segment pantriagonals, and 4(''m''&nbsp;−&nbsp;2)(''m''&nbsp;−&nbsp;1) three-segment pantriagonals. This [[magic cube classes|class of magic cubes]] may contain some [[Magic_square#Classification_of_magic_squares|simple magic squares]] and/or [[pandiagonal magic square]]s, but not enough to satisfy any other classifications. 

The magic constant for magic cubes is ''S''&nbsp;= ''m''(''m''<sup>3</sup>&nbsp;+&nbsp;1)/2.

A '''proper pantriagonal magic cube''' has 7''m''<sup>2</sup> 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]]
* [[triagonal]]

==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]] (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

[[Category:Magic squares]]