Multimagic cube

In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic.[1] A P-multimagic 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. 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 in 2000; it is a semiperfect 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.[2]

MathWorld 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.[3] It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.[4]

References
  1. Weisstein, Eric W. "Multimagic cube". MathWorld.
  2. Weisstein, Eric W. "Bimagic Cube". MathWorld.
  3. Weisstein, Eric W. "Trimagic Cube". MathWorld.
  4. Weisstein, Eric W. "Tetramagic Cube". MathWorld.

See also


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