Multimagic cube
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a P-multimagic cube is a magic cube
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...

 that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube
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...

 is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic. A P-multimagic cube is said to be semi-perfect if the k-th power cubes are perfect
Perfect magic cube
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant....

 for 1 ≤ k < P, and the P-th power cube is semiperfect
Semiperfect magic cube
In mathematics, a semiperfect magic cube is a magic cube that is not a perfect magic cube, i.e., a magic cube for which the cross section diagonals do not necessarily sum up to the cube's magic constant....

. If all P of the power cubes are perfect, the multimagic cube is said to be perfect.

The first known example of a bimagic cube was given by John Hendricks
John Hendricks
John Hendricks is the founder and chairman of Discovery Communications, a broadcasting and film production company which owns the Discovery Channel, TLC and Animal Planet cable tv networks, among other ventures.-Early life:...

 in 2000; it is a semiperfect
Semiperfect magic cube
In mathematics, a semiperfect magic cube is a magic cube that is not a perfect magic cube, i.e., a magic cube for which the cross section diagonals do not necessarily sum up to the cube's magic constant....

 cube of order 25 and magic constant
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....

 195325. In 2003, C. Bower discovered two semi-perfect bimagic cubes of order 16, and a perfect bimagic cube of order 32.

MathWorld
MathWorld
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...

reports that only two trimagic cubes are known, discovered by C. Bower in 2003; a semiperfect cube of order 64 and a perfect cube of order 256. It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.
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