In mathematics, a ''P''-multimagic 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 ...

that remains magic even if all its numbers are replaced by their ''k'' th powers for 1 ≤ ''k'' ≤ ''P''. cubes are called bimagic, cubes are called trimagic, and cubes tetramagic. A cube is said to be semi-perfect if the ''k'' th power cubes are perfect for 1 ≤ ''k'' < ''P'', and the ''P'' th power cube is

semiperfect In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. ...

. If all ''P'' of the power cubes are perfect, the cube is said to be perfect. The first known example of a bimagic cube was given by

John Hendricks John Samuel Hendricks (born March 29, 1952)"John Hendricks: An Oral History," The Cable Center, September 2, 2003. is an American businessman and is the founder and former chairman of Discovery, Inc. (now a part of Warner Bros. Discovery) a broa ...

in 2000; it is a

cube of order 25 and magic constant 195325. In 2003, C. Bower discovered two semi-perfect bimagic cubes of order 16, and a perfect bimagic cube of order 32.

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 Di ...

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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See also