A Pantriagonal Magic Cube is a magic cube where all 4m2 pantriagonals sum correctly. There are 4 one-segment, 12(m-1) two-segment, and 4(m-2)(m-1) three-segment pantriagonals. This class of magic cubes may contain some simple magic squares AND/OR pandiagonal magic squares , but not enough to satisfy any other classifications.
The constant for magic cubes is S = m(m3 + 1)/2
A proper pantriagonal magic cube has 7m2 lines summing correctly. It contains NO magic squares.
Order 4 is the smallest Pantriagonal Magic Cube possible.
A pantriagonal magic cube is the 3-dimensional equivalent of the pandiagonal magic square. Only, 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.
Pickover, Clifford A., The Zen of Magic Squares, Circles and Stars, Princeton Univ. Press, 2002, 0-691-07041-5 page 178.
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.
J.R.Hendricks, Inlaid Magic Squares and Cubes 2nd Edition, 2000, 0-9684700-3-3..
Pickover, Clifford A., The Zen of Magic Squares, Circles and Stars, Princeton Univ. Press, 2002, 0-691-07041-5 Pages 97,121,175,268.
external Links
http://home.wanadoo.nl/aaledewinkel/Encyclopedia/ Aale de Winkel: Magic Encyclopedia
http://members.shaw.ca/hdhcubes/cube_perfect.htm Harvey Heinz: Perfect Magic Hypercubes