In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
Perfect commonly refers to:
* Perfection; completeness, and excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film and television
* ''Perfect'' (1985 film), a romantic drama
* ''Perfect'' (20 ...
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 the founder and former chairman of Discovery, Inc. (later Warner Bros. Discovery), a broadcasting ...
in 2000; it is a
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.
...
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
*
Magic square
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
*
Multimagic square
In mathematics, a ''P''-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their ''k''th powers for 1 ≤ ''k'' ≤ ''P''. squares are called bimagic, squares are call ...
Magic squares
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