In mathematics, a ''P''-multimagic cube is a

magic cube In mathematics, a magic cube is the dimension, 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, ...

that remains magic even if all its numbers are replaced by their ''k'' th

powers Powers may refer to: Arts and media * ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming ** ''Powers'' (American TV series), a 2015–2016 series based on the comics * ''Powers'' (British TV series), a 200 ...

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 Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...

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 bro ...

in 2000; it is a

cube of order 25 and

magic constant The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...

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 ...

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.

References

See also