Pantriagonal magic cube
Encyclopedia
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 square
s 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.
Magic cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...
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 square
Simple magic square
A simple magic square is the lowest of two basic classes of magic square. It has the minimum requirements for a square to be considered magic. All lines parallel to the edges, plus the two main diagonals must sum to the magic constant...
s 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.
Further reading
- 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. PickoverClifford A. PickoverClifford A. Pickover is an American author, editor, and columnist in the fields of science, mathematics, and science fiction, and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.- Biography :He received his Ph.D...
(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