recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

, a pandiagonal magic cube is a

magic cube In mathematics, a magic cube is the 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, on each c ...

with the additional property that all

broken diagonal In recreational mathematics and the theory of magic squares, a broken diagonal is a set of ''n'' cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the squar ...

s (parallel to exactly two of the three coordinate axes) have the same sum as each other. Pandiagonal magic cubes are extensions of

diagonal magic cube The class of diagonal magic cubes is the second of the six magic cube classes (when ranked by the number of lines summing correctly), coming after the simple magic cubes. In a diagonal magic cube of order ''m'',Traditionally, ''n'' has been used ...

s (in which only the unbroken diagonals need to have the same sum as the rows of the cube) and generalize

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 squ ...

s to three dimensions. In a pandiagonal magic cube, all 3''m'' planar arrays must be panmagic squares. The 6 oblique squares are always

magic Magic or magick most commonly refers to: * Magic (supernatural), beliefs and actions employed to influence supernatural beings and forces ** ''Magick'' (with ''-ck'') can specifically refer to ceremonial magic * Magic (illusion), also known as sta ...

. Several of them may be panmagic squares. A proper pandiagonal magic cube has exactly 9''m''² lines plus the 4 main

space diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with '' face diagonals'', which connect vertices on the same face (b ...

s summing correctly (no broken space diagonals have the correct sum.) The smallest pandiagonal magic cube has order 7.

References

*Hendricks, J.R; ''Magic Squares to Tesseracts by Computer'', Self-published 1999. *Hendricks, J.R.; ''Perfect n-Dimensional Magic Hypercubes of Order 2n'', Self-published 1999.
Harvey Heinz: All about magic cubes
Magic squares {{combin-stub

See also

References